{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0001","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79eef7298edbfb2c597bb1e8f469bac518b012f139adb43dacee660e435b8471","family_id":"qf-density17-order-three-rational-sort","seed":202610174001,"semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"0/1\",\"0/1\",\"0/1\"]}"},"fields":{"title":"Stable Rational Sort with Original Index Witnesses","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[0/1, 0/2, 0/3]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.","answer":{"kind":"expression","value":"([0/1, 0/1, 0/1], [0, 1, 2], [0/1, 0/1, 0/1], [=, =])","explanation":"Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0/1, 0/1, 0/1], [0, 1, 2], [0/1, 0/1, 0/1], [=, =])."},"skills":["Stable Rational Sort with Original Index Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0002","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2788ef51eb5812dafd72f1de79a497dde4a4415a1f91882b6a176123e226cdea","family_id":"qf-density17-order-three-rational-sort","seed":202610174002,"semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"-1/2\",\"1/3\",\"-1/2\"]}"},"fields":{"title":"Stable Rational Sort with Original Index Witnesses","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[-1/2, 1/3, -2/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.","answer":{"kind":"expression","value":"([-1/2, 1/3, -1/2], [0, 2, 1], [-1/2, -1/2, 1/3], [=, <])","explanation":"Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1/2, 1/3, -1/2], [0, 2, 1], [-1/2, -1/2, 1/3], [=, <])."},"skills":["Stable Rational Sort with Original Index Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0003","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6130d7f0ab07419d6609b46aa74f8d53cbe61e761c8ff136d807ee49e8656fa","family_id":"qf-density17-order-three-rational-sort","seed":202610174003,"semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"1/2\",\"-1/2\",\"0/1\"]}"},"fields":{"title":"Stable Rational Sort with Original Index Witnesses","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[1/2, -1/2, 0/1]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.","answer":{"kind":"expression","value":"([1/2, -1/2, 0/1], [1, 2, 0], [-1/2, 0/1, 1/2], [<, <])","explanation":"Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1/2, -1/2, 0/1], [1, 2, 0], [-1/2, 0/1, 1/2], [<, <])."},"skills":["Stable Rational Sort with Original Index Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0004","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03466932551f793819c19ef8626e19f8d44e97ed3bdcc1f5632f425b14c580cb","family_id":"qf-density17-order-three-rational-sort","seed":202610174004,"semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"-3/1\",\"1/2\",\"-1/4\"]}"},"fields":{"title":"Stable Rational Sort with Original Index Witnesses","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[-9/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.","answer":{"kind":"expression","value":"([-3/1, 1/2, -1/4], [0, 2, 1], [-3/1, -1/4, 1/2], [<, <])","explanation":"Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-3/1, 1/2, -1/4], [0, 2, 1], [-3/1, -1/4, 1/2], [<, <])."},"skills":["Stable Rational Sort with Original Index Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0005","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4fb14efccb068b8452998ee65d87a9119ad6f3830581a57e0f33c9fbc8aa6429","family_id":"qf-density17-order-three-rational-sort","seed":202610174005,"semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"-8/3\",\"1/2\",\"-1/4\"]}"},"fields":{"title":"Stable Rational Sort with Original Index Witnesses","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[-8/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.","answer":{"kind":"expression","value":"([-8/3, 1/2, -1/4], [0, 2, 1], [-8/3, -1/4, 1/2], [<, <])","explanation":"Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-8/3, 1/2, -1/4], [0, 2, 1], [-8/3, -1/4, 1/2], [<, <])."},"skills":["Stable Rational Sort with Original Index Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0006","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2bcad5b79800472fc729944322fda80ea8f910cfee2e4acd47e2672ed4ae58ce","family_id":"qf-density17-order-three-rational-sort","seed":202610174006,"semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"-7/3\",\"1/2\",\"-1/4\"]}"},"fields":{"title":"Stable Rational Sort with Original Index Witnesses","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[-7/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.","answer":{"kind":"expression","value":"([-7/3, 1/2, -1/4], [0, 2, 1], [-7/3, -1/4, 1/2], [<, <])","explanation":"Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-7/3, 1/2, -1/4], [0, 2, 1], [-7/3, -1/4, 1/2], [<, <])."},"skills":["Stable Rational Sort with Original Index Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0007","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a6e78f871d5b52ee7fddc5fb5d506bc8282df4130f941e5cbfa16e61ed5031d","family_id":"qf-density17-order-three-rational-sort","seed":202610174007,"semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"-2/1\",\"1/2\",\"-1/4\"]}"},"fields":{"title":"Stable Rational Sort with Original Index Witnesses","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[-6/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.","answer":{"kind":"expression","value":"([-2/1, 1/2, -1/4], [0, 2, 1], [-2/1, -1/4, 1/2], [<, <])","explanation":"Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-2/1, 1/2, -1/4], [0, 2, 1], [-2/1, -1/4, 1/2], [<, <])."},"skills":["Stable Rational Sort with Original Index Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0008","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99733d1c734fc911ad4ee45b420d51a7ef8b1b0e279f4f7de93c6f5254392b10","family_id":"qf-density17-order-three-rational-sort","seed":202610174008,"semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"-5/3\",\"1/2\",\"-1/4\"]}"},"fields":{"title":"Stable Rational Sort with Original Index Witnesses","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[-5/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.","answer":{"kind":"expression","value":"([-5/3, 1/2, -1/4], [0, 2, 1], [-5/3, -1/4, 1/2], [<, <])","explanation":"Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-5/3, 1/2, -1/4], [0, 2, 1], [-5/3, -1/4, 1/2], [<, <])."},"skills":["Stable Rational Sort with Original Index Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0009","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a0bd1ea1373e490f1b5f67cfda9a98297a39bfb8b5c162740b400be5857de0c9","family_id":"qf-density17-order-three-rational-sort","seed":202610174009,"semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"-4/3\",\"1/2\",\"-1/4\"]}"},"fields":{"title":"Stable Rational Sort with Original Index Witnesses","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[-4/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.","answer":{"kind":"expression","value":"([-4/3, 1/2, -1/4], [0, 2, 1], [-4/3, -1/4, 1/2], [<, <])","explanation":"Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-4/3, 1/2, -1/4], [0, 2, 1], [-4/3, -1/4, 1/2], [<, <])."},"skills":["Stable Rational Sort with Original Index Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0010","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"307d49a95599ea70aa4aa3dcecd8c1c4c7fd6da69b9305e136e2f234e36f3c59","family_id":"qf-density17-order-three-rational-sort","seed":202610174010,"semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"-1/1\",\"1/2\",\"-1/4\"]}"},"fields":{"title":"Stable Rational Sort with Original Index Witnesses","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[-3/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.","answer":{"kind":"expression","value":"([-1/1, 1/2, -1/4], [0, 2, 1], [-1/1, -1/4, 1/2], [<, <])","explanation":"Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1/1, 1/2, -1/4], [0, 2, 1], [-1/1, -1/4, 1/2], [<, <])."},"skills":["Stable Rational Sort with Original Index Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0011","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd7a2111c7036fa41f0cc0c1e484b982c0c4c07c3830c663d7e2b893d8a71606","family_id":"qf-density17-order-three-rational-sort","seed":202610174011,"semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"-2/3\",\"1/2\",\"-1/4\"]}"},"fields":{"title":"Stable Rational Sort with Original Index Witnesses","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[-2/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.","answer":{"kind":"expression","value":"([-2/3, 1/2, -1/4], [0, 2, 1], [-2/3, -1/4, 1/2], [<, <])","explanation":"Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-2/3, 1/2, -1/4], [0, 2, 1], [-2/3, -1/4, 1/2], [<, <])."},"skills":["Stable Rational Sort with Original Index Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0012","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48ccacc15535f73deec263d9d34c0e5cb76947980df634b6186de06cc3013ca8","family_id":"qf-density17-order-three-rational-sort","seed":202610174012,"semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"-1/3\",\"1/2\",\"-1/4\"]}"},"fields":{"title":"Stable Rational Sort with Original Index Witnesses","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[-1/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.","answer":{"kind":"expression","value":"([-1/3, 1/2, -1/4], [0, 2, 1], [-1/3, -1/4, 1/2], [<, <])","explanation":"Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1/3, 1/2, -1/4], [0, 2, 1], [-1/3, -1/4, 1/2], [<, <])."},"skills":["Stable Rational Sort with Original Index Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0013","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e4f882c1462a0b51eb1424afc3d6781b954092279fa9c4ab923b7e8737baaa4","family_id":"qf-density17-order-three-rational-sort","seed":202610174013,"semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"0/1\",\"1/2\",\"-1/4\"]}"},"fields":{"title":"Stable Rational Sort with Original Index Witnesses","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[0/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.","answer":{"kind":"expression","value":"([0/1, 1/2, -1/4], [2, 0, 1], [-1/4, 0/1, 1/2], [<, <])","explanation":"Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0/1, 1/2, -1/4], [2, 0, 1], [-1/4, 0/1, 1/2], [<, <])."},"skills":["Stable Rational Sort with Original Index Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0014","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e39ba2613d95f4a5737496f5ebd1529d4141cf7048d8c7300771b0a3bc9f5649","family_id":"qf-density17-order-three-rational-sort","seed":202610174014,"semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"1/3\",\"1/2\",\"-1/4\"]}"},"fields":{"title":"Stable Rational Sort with Original Index Witnesses","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[1/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.","answer":{"kind":"expression","value":"([1/3, 1/2, -1/4], [2, 0, 1], [-1/4, 1/3, 1/2], [<, <])","explanation":"Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1/3, 1/2, -1/4], [2, 0, 1], [-1/4, 1/3, 1/2], [<, <])."},"skills":["Stable Rational Sort with Original Index Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0015","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f22802050a0578d3b89a247686a16dbe3cc3af0313cc82d7b54565a688f06f0","family_id":"qf-density17-order-three-rational-sort","seed":202610174015,"semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"2/3\",\"1/2\",\"-1/4\"]}"},"fields":{"title":"Stable Rational Sort with Original Index Witnesses","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[2/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.","answer":{"kind":"expression","value":"([2/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 2/3], [<, <])","explanation":"Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 2/3], [<, <])."},"skills":["Stable Rational Sort with Original Index Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0016","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"858978f1ff3991550ef9aa5fe7134d83b1c77777d5b0df5184c3d5d75fb84098","family_id":"qf-density17-order-three-rational-sort","seed":202610174016,"semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"1/1\",\"1/2\",\"-1/4\"]}"},"fields":{"title":"Stable Rational Sort with Original Index Witnesses","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[3/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.","answer":{"kind":"expression","value":"([1/1, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 1/1], [<, <])","explanation":"Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1/1, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 1/1], [<, <])."},"skills":["Stable Rational Sort with Original Index Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0017","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"105e1943cd7c79ce6e16059bed75d8c3a25e34a80374cd79bd602fba3b07f78d","family_id":"qf-density17-order-three-rational-sort","seed":202610174017,"semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"4/3\",\"1/2\",\"-1/4\"]}"},"fields":{"title":"Stable Rational Sort with Original Index Witnesses","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[4/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.","answer":{"kind":"expression","value":"([4/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 4/3], [<, <])","explanation":"Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 4/3], [<, <])."},"skills":["Stable Rational Sort with Original Index Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0018","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c4533aec2c5b3dadf40635eb5547cfe9b64a6bb83f985d20fac273bdc5428b4","family_id":"qf-density17-order-three-rational-sort","seed":202610174018,"semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"5/3\",\"1/2\",\"-1/4\"]}"},"fields":{"title":"Stable Rational Sort with Original Index Witnesses","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[5/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.","answer":{"kind":"expression","value":"([5/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 5/3], [<, <])","explanation":"Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 5/3], [<, <])."},"skills":["Stable Rational Sort with Original Index Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0019","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19b0f58c80fe017583ac51d4c670a275def0b8453e2a464977e3de18f3bcf147","family_id":"qf-density17-order-three-rational-sort","seed":202610174019,"semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"2/1\",\"1/2\",\"-1/4\"]}"},"fields":{"title":"Stable Rational Sort with Original Index Witnesses","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[6/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.","answer":{"kind":"expression","value":"([2/1, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 2/1], [<, <])","explanation":"Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2/1, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 2/1], [<, <])."},"skills":["Stable Rational Sort with Original Index Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0020","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3322dcb719d309e27b1b8649d1db69114badc6491ca90bd79987bd215cedff01","family_id":"qf-density17-order-three-rational-sort","seed":202610174020,"semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"7/3\",\"1/2\",\"-1/4\"]}"},"fields":{"title":"Stable Rational Sort with Original Index Witnesses","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[7/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.","answer":{"kind":"expression","value":"([7/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 7/3], [<, <])","explanation":"Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([7/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 7/3], [<, <])."},"skills":["Stable Rational Sort with Original Index Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0021","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58050fc22a40ca032f1f4ddda21cf809a3ea775e768022f490335f5c7163de58","family_id":"qf-density17-order-three-rational-sort","seed":202610174021,"semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"8/3\",\"1/2\",\"-1/4\"]}"},"fields":{"title":"Stable Rational Sort with Original Index Witnesses","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[8/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.","answer":{"kind":"expression","value":"([8/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 8/3], [<, <])","explanation":"Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([8/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 8/3], [<, <])."},"skills":["Stable Rational Sort with Original Index Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0022","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea37988edc9c0d1784d91a1e0414cb25d95ecb1d2d0926369f93060b3a60659d","family_id":"qf-density17-order-three-rational-sort","seed":202610174022,"semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"3/1\",\"1/2\",\"-1/4\"]}"},"fields":{"title":"Stable Rational Sort with Original Index Witnesses","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[9/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.","answer":{"kind":"expression","value":"([3/1, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 3/1], [<, <])","explanation":"Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3/1, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 3/1], [<, <])."},"skills":["Stable Rational Sort with Original Index Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0023","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14b9c8f51a210a69a576fdb2bf9ecbaa1860e136dfdbe40d41d3305ad67b8cd9","family_id":"qf-density17-order-three-rational-sort","seed":202610174023,"semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"10/3\",\"1/2\",\"-1/4\"]}"},"fields":{"title":"Stable Rational Sort with Original Index Witnesses","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[10/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.","answer":{"kind":"expression","value":"([10/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 10/3], [<, <])","explanation":"Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([10/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 10/3], [<, <])."},"skills":["Stable Rational Sort with Original Index Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0024","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b087b6ff635c04037ce929a7a170bb6fab6ac2f549ead95bface54324889d896","family_id":"qf-density17-order-three-rational-sort","seed":202610174024,"semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"11/3\",\"1/2\",\"-1/4\"]}"},"fields":{"title":"Stable Rational Sort with Original Index Witnesses","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[11/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.","answer":{"kind":"expression","value":"([11/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 11/3], [<, <])","explanation":"Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([11/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 11/3], [<, <])."},"skills":["Stable Rational Sort with Original Index Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0025","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f04470d9b9f9ba967dd2814f48a711f46a553335bc21ca4cd679290dc055bbcf","family_id":"qf-density17-order-three-rational-sort","seed":202610174025,"semantic_template":"qf-density17-order-three-rational-sort/{\"values\":[\"4/1\",\"1/2\",\"-1/4\"]}"},"fields":{"title":"Stable Rational Sort with Original Index Witnesses","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Ordered inputs=[12/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.","answer":{"kind":"expression","value":"([4/1, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 4/1], [<, <])","explanation":"Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4/1, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 4/1], [<, <])."},"skills":["Stable Rational Sort with Original Index Witnesses"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0026","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7363a9385ccefe76f49004b41dcd285b19fa5246d5010056b43bf256e19834a9","family_id":"qf-density17-order-interval-intersection","seed":202610174026,"semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"0/1\",\"l2\":\"1/1\",\"u1\":\"1/1\",\"u2\":\"2/1\"}"},"fields":{"title":"Closed-Interval Intersection Including Touching Endpoints","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [0/1, 1/1] and [1/1, 2/1]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).","answer":{"kind":"expression","value":"([0/1, 1/1], [1/1, 2/1], 1/1, 1/1, Yes, [1/1, 1/1])","explanation":"Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0/1, 1/1], [1/1, 2/1], 1/1, 1/1, Yes, [1/1, 1/1])."},"skills":["Closed-Interval Intersection Including Touching Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0027","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e0b6405fcb85e803cf9eb791b9dca454ca40901a338703409720339a6204716","family_id":"qf-density17-order-interval-intersection","seed":202610174027,"semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"-2/1\",\"l2\":\"0/1\",\"u1\":\"-1/1\",\"u2\":\"1/1\"}"},"fields":{"title":"Closed-Interval Intersection Including Touching Endpoints","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [-2/1, -1/1] and [0/1, 1/1]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).","answer":{"kind":"expression","value":"([-2/1, -1/1], [0/1, 1/1], 0/1, -1/1, No, empty-set)","explanation":"Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-2/1, -1/1], [0/1, 1/1], 0/1, -1/1, No, empty-set)."},"skills":["Closed-Interval Intersection Including Touching Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0028","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4569171c2f2cff25c75be40621f75445d30a0cc51da3bc328324afd0c9cd38bf","family_id":"qf-density17-order-interval-intersection","seed":202610174028,"semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"0/1\",\"l2\":\"0/1\",\"u1\":\"0/1\",\"u2\":\"0/1\"}"},"fields":{"title":"Closed-Interval Intersection Including Touching Endpoints","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [0/1, 0/1] and [0/1, 0/1]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).","answer":{"kind":"expression","value":"([0/1, 0/1], [0/1, 0/1], 0/1, 0/1, Yes, [0/1, 0/1])","explanation":"Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0/1, 0/1], [0/1, 0/1], 0/1, 0/1, Yes, [0/1, 0/1])."},"skills":["Closed-Interval Intersection Including Touching Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0029","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ebbe92d8c577b457163e122b393789fb69795720e661ab2f868facf204d5944","family_id":"qf-density17-order-interval-intersection","seed":202610174029,"semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"-2/1\",\"l2\":\"-1/1\",\"u1\":\"2/1\",\"u2\":\"3/1\"}"},"fields":{"title":"Closed-Interval Intersection Including Touching Endpoints","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [-2/1, 2/1] and [-1/1, 3/1]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).","answer":{"kind":"expression","value":"([-2/1, 2/1], [-1/1, 3/1], -1/1, 2/1, Yes, [-1/1, 2/1])","explanation":"Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-2/1, 2/1], [-1/1, 3/1], -1/1, 2/1, Yes, [-1/1, 2/1])."},"skills":["Closed-Interval Intersection Including Touching Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0030","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d4c0a1a0e0cb52002d7cc1bd8a8c431bd5617e8248c28ec3e4beec0436ea4686","family_id":"qf-density17-order-interval-intersection","seed":202610174030,"semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"-4/1\",\"l2\":\"-5/2\",\"u1\":\"-1/1\",\"u2\":\"1/2\"}"},"fields":{"title":"Closed-Interval Intersection Including Touching Endpoints","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [-8/2, -2/2] and [-5/2, 1/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).","answer":{"kind":"expression","value":"([-4/1, -1/1], [-5/2, 1/2], -5/2, -1/1, Yes, [-5/2, -1/1])","explanation":"Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-4/1, -1/1], [-5/2, 1/2], -5/2, -1/1, Yes, [-5/2, -1/1])."},"skills":["Closed-Interval Intersection Including Touching Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0031","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e162ef30adc190bde83dda3b3f39adc16971d2f72b4035acc1dbafd36dc40b4","family_id":"qf-density17-order-interval-intersection","seed":202610174031,"semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"-7/2\",\"l2\":\"-2/1\",\"u1\":\"-1/2\",\"u2\":\"1/1\"}"},"fields":{"title":"Closed-Interval Intersection Including Touching Endpoints","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [-7/2, -1/2] and [-4/2, 2/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).","answer":{"kind":"expression","value":"([-7/2, -1/2], [-2/1, 1/1], -2/1, -1/2, Yes, [-2/1, -1/2])","explanation":"Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-7/2, -1/2], [-2/1, 1/1], -2/1, -1/2, Yes, [-2/1, -1/2])."},"skills":["Closed-Interval Intersection Including Touching Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0032","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02ce2625cc4933f046e15aac4480c3230bff40bde077eeafdc8e1cbca751686b","family_id":"qf-density17-order-interval-intersection","seed":202610174032,"semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"-3/1\",\"l2\":\"-3/2\",\"u1\":\"0/1\",\"u2\":\"3/2\"}"},"fields":{"title":"Closed-Interval Intersection Including Touching Endpoints","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [-6/2, 0/2] and [-3/2, 3/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).","answer":{"kind":"expression","value":"([-3/1, 0/1], [-3/2, 3/2], -3/2, 0/1, Yes, [-3/2, 0/1])","explanation":"Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-3/1, 0/1], [-3/2, 3/2], -3/2, 0/1, Yes, [-3/2, 0/1])."},"skills":["Closed-Interval Intersection Including Touching Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0033","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1cc0cba8b992a3ac80de50984bbb200291c071f05f49de3684e3627d1c520c3","family_id":"qf-density17-order-interval-intersection","seed":202610174033,"semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"-5/2\",\"l2\":\"-1/1\",\"u1\":\"1/2\",\"u2\":\"2/1\"}"},"fields":{"title":"Closed-Interval Intersection Including Touching Endpoints","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [-5/2, 1/2] and [-2/2, 4/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).","answer":{"kind":"expression","value":"([-5/2, 1/2], [-1/1, 2/1], -1/1, 1/2, Yes, [-1/1, 1/2])","explanation":"Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-5/2, 1/2], [-1/1, 2/1], -1/1, 1/2, Yes, [-1/1, 1/2])."},"skills":["Closed-Interval Intersection Including Touching Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0034","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f48e65d360c859d92cbb670e5132f5a6170e800764d3740f0b0476b09561a88e","family_id":"qf-density17-order-interval-intersection","seed":202610174034,"semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"-2/1\",\"l2\":\"-1/2\",\"u1\":\"1/1\",\"u2\":\"5/2\"}"},"fields":{"title":"Closed-Interval Intersection Including Touching Endpoints","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [-4/2, 2/2] and [-1/2, 5/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).","answer":{"kind":"expression","value":"([-2/1, 1/1], [-1/2, 5/2], -1/2, 1/1, Yes, [-1/2, 1/1])","explanation":"Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-2/1, 1/1], [-1/2, 5/2], -1/2, 1/1, Yes, [-1/2, 1/1])."},"skills":["Closed-Interval Intersection Including Touching Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0035","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43ecc6f7a05c364ce934e5c0dccb61ebf4fe39ce5f222c1412cec16bfb8bfbd8","family_id":"qf-density17-order-interval-intersection","seed":202610174035,"semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"-3/2\",\"l2\":\"0/1\",\"u1\":\"3/2\",\"u2\":\"3/1\"}"},"fields":{"title":"Closed-Interval Intersection Including Touching Endpoints","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [-3/2, 3/2] and [0/2, 6/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).","answer":{"kind":"expression","value":"([-3/2, 3/2], [0/1, 3/1], 0/1, 3/2, Yes, [0/1, 3/2])","explanation":"Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-3/2, 3/2], [0/1, 3/1], 0/1, 3/2, Yes, [0/1, 3/2])."},"skills":["Closed-Interval Intersection Including Touching Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0036","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00715031902030fac50a83e67b0f156ba1cc4ddd71978d0c3d33829c5b155a32","family_id":"qf-density17-order-interval-intersection","seed":202610174036,"semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"-1/1\",\"l2\":\"1/2\",\"u1\":\"2/1\",\"u2\":\"7/2\"}"},"fields":{"title":"Closed-Interval Intersection Including Touching Endpoints","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [-2/2, 4/2] and [1/2, 7/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).","answer":{"kind":"expression","value":"([-1/1, 2/1], [1/2, 7/2], 1/2, 2/1, Yes, [1/2, 2/1])","explanation":"Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1/1, 2/1], [1/2, 7/2], 1/2, 2/1, Yes, [1/2, 2/1])."},"skills":["Closed-Interval Intersection Including Touching Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0037","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b48b95a224a74d0d5464dc02d40ba0abd5fe0f74f41d9c97706957931c7105c0","family_id":"qf-density17-order-interval-intersection","seed":202610174037,"semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"-1/2\",\"l2\":\"1/1\",\"u1\":\"5/2\",\"u2\":\"4/1\"}"},"fields":{"title":"Closed-Interval Intersection Including Touching Endpoints","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [-1/2, 5/2] and [2/2, 8/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).","answer":{"kind":"expression","value":"([-1/2, 5/2], [1/1, 4/1], 1/1, 5/2, Yes, [1/1, 5/2])","explanation":"Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1/2, 5/2], [1/1, 4/1], 1/1, 5/2, Yes, [1/1, 5/2])."},"skills":["Closed-Interval Intersection Including Touching Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0038","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c3e9578e336340325170f5b01327eb3b166bf668a5c1d84c1cc23c91ececde95","family_id":"qf-density17-order-interval-intersection","seed":202610174038,"semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"0/1\",\"l2\":\"3/2\",\"u1\":\"3/1\",\"u2\":\"9/2\"}"},"fields":{"title":"Closed-Interval Intersection Including Touching Endpoints","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [0/2, 6/2] and [3/2, 9/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).","answer":{"kind":"expression","value":"([0/1, 3/1], [3/2, 9/2], 3/2, 3/1, Yes, [3/2, 3/1])","explanation":"Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0/1, 3/1], [3/2, 9/2], 3/2, 3/1, Yes, [3/2, 3/1])."},"skills":["Closed-Interval Intersection Including Touching Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0039","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46ad5a3c030bd005d66c3f23447bce34020d2c34871950c4b417c686b39c16f4","family_id":"qf-density17-order-interval-intersection","seed":202610174039,"semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"1/2\",\"l2\":\"2/1\",\"u1\":\"7/2\",\"u2\":\"5/1\"}"},"fields":{"title":"Closed-Interval Intersection Including Touching Endpoints","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [1/2, 7/2] and [4/2, 10/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).","answer":{"kind":"expression","value":"([1/2, 7/2], [2/1, 5/1], 2/1, 7/2, Yes, [2/1, 7/2])","explanation":"Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1/2, 7/2], [2/1, 5/1], 2/1, 7/2, Yes, [2/1, 7/2])."},"skills":["Closed-Interval Intersection Including Touching Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0040","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3391ed790ffb778dcda492cd8c624a8fdfac580914f6c619b311e9e19dca4b01","family_id":"qf-density17-order-interval-intersection","seed":202610174040,"semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"1/1\",\"l2\":\"5/2\",\"u1\":\"4/1\",\"u2\":\"11/2\"}"},"fields":{"title":"Closed-Interval Intersection Including Touching Endpoints","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [2/2, 8/2] and [5/2, 11/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).","answer":{"kind":"expression","value":"([1/1, 4/1], [5/2, 11/2], 5/2, 4/1, Yes, [5/2, 4/1])","explanation":"Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1/1, 4/1], [5/2, 11/2], 5/2, 4/1, Yes, [5/2, 4/1])."},"skills":["Closed-Interval Intersection Including Touching Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0041","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48eb2c36bba198604a7049017952935d858c2f40f02ff00a09da21cf5504d92f","family_id":"qf-density17-order-interval-intersection","seed":202610174041,"semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"3/2\",\"l2\":\"3/1\",\"u1\":\"9/2\",\"u2\":\"6/1\"}"},"fields":{"title":"Closed-Interval Intersection Including Touching Endpoints","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [3/2, 9/2] and [6/2, 12/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).","answer":{"kind":"expression","value":"([3/2, 9/2], [3/1, 6/1], 3/1, 9/2, Yes, [3/1, 9/2])","explanation":"Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3/2, 9/2], [3/1, 6/1], 3/1, 9/2, Yes, [3/1, 9/2])."},"skills":["Closed-Interval Intersection Including Touching Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0042","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e254cf0c89735a4772872b95db97c27cebeff00457e1da2de7c98da67ee51a23","family_id":"qf-density17-order-interval-intersection","seed":202610174042,"semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"2/1\",\"l2\":\"7/2\",\"u1\":\"5/1\",\"u2\":\"13/2\"}"},"fields":{"title":"Closed-Interval Intersection Including Touching Endpoints","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [4/2, 10/2] and [7/2, 13/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).","answer":{"kind":"expression","value":"([2/1, 5/1], [7/2, 13/2], 7/2, 5/1, Yes, [7/2, 5/1])","explanation":"Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2/1, 5/1], [7/2, 13/2], 7/2, 5/1, Yes, [7/2, 5/1])."},"skills":["Closed-Interval Intersection Including Touching Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0043","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5565d53892829a3a91aa4809196d7927b1c48a7fbeb28f5ff0486a3cb2e5f9b1","family_id":"qf-density17-order-interval-intersection","seed":202610174043,"semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"5/2\",\"l2\":\"4/1\",\"u1\":\"11/2\",\"u2\":\"7/1\"}"},"fields":{"title":"Closed-Interval Intersection Including Touching Endpoints","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [5/2, 11/2] and [8/2, 14/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).","answer":{"kind":"expression","value":"([5/2, 11/2], [4/1, 7/1], 4/1, 11/2, Yes, [4/1, 11/2])","explanation":"Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5/2, 11/2], [4/1, 7/1], 4/1, 11/2, Yes, [4/1, 11/2])."},"skills":["Closed-Interval Intersection Including Touching Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0044","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"866a7dfa2e396a4eb4a6f831b9bfb30bf397f61323ef970710afb360196ec4f1","family_id":"qf-density17-order-interval-intersection","seed":202610174044,"semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"3/1\",\"l2\":\"9/2\",\"u1\":\"6/1\",\"u2\":\"15/2\"}"},"fields":{"title":"Closed-Interval Intersection Including Touching Endpoints","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [6/2, 12/2] and [9/2, 15/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).","answer":{"kind":"expression","value":"([3/1, 6/1], [9/2, 15/2], 9/2, 6/1, Yes, [9/2, 6/1])","explanation":"Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3/1, 6/1], [9/2, 15/2], 9/2, 6/1, Yes, [9/2, 6/1])."},"skills":["Closed-Interval Intersection Including Touching Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0045","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3d5c8fafd894bcba2c5bfad0d1d53ca6e72f37a7d4e24518dc79c2fcb00b037","family_id":"qf-density17-order-interval-intersection","seed":202610174045,"semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"7/2\",\"l2\":\"5/1\",\"u1\":\"13/2\",\"u2\":\"8/1\"}"},"fields":{"title":"Closed-Interval Intersection Including Touching Endpoints","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [7/2, 13/2] and [10/2, 16/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).","answer":{"kind":"expression","value":"([7/2, 13/2], [5/1, 8/1], 5/1, 13/2, Yes, [5/1, 13/2])","explanation":"Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([7/2, 13/2], [5/1, 8/1], 5/1, 13/2, Yes, [5/1, 13/2])."},"skills":["Closed-Interval Intersection Including Touching Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0046","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"337f30cd2ba75f2c38c437c9e7d28ec186e6ff86aff17d7dd292ad8a00e1f02f","family_id":"qf-density17-order-interval-intersection","seed":202610174046,"semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"4/1\",\"l2\":\"11/2\",\"u1\":\"7/1\",\"u2\":\"17/2\"}"},"fields":{"title":"Closed-Interval Intersection Including Touching Endpoints","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [8/2, 14/2] and [11/2, 17/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).","answer":{"kind":"expression","value":"([4/1, 7/1], [11/2, 17/2], 11/2, 7/1, Yes, [11/2, 7/1])","explanation":"Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4/1, 7/1], [11/2, 17/2], 11/2, 7/1, Yes, [11/2, 7/1])."},"skills":["Closed-Interval Intersection Including Touching Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0047","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41dc89bde1da9a7424f3db3107ae07f6dd2874df0699b809e557d89bd36d15b0","family_id":"qf-density17-order-interval-intersection","seed":202610174047,"semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"9/2\",\"l2\":\"6/1\",\"u1\":\"15/2\",\"u2\":\"9/1\"}"},"fields":{"title":"Closed-Interval Intersection Including Touching Endpoints","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [9/2, 15/2] and [12/2, 18/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).","answer":{"kind":"expression","value":"([9/2, 15/2], [6/1, 9/1], 6/1, 15/2, Yes, [6/1, 15/2])","explanation":"Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([9/2, 15/2], [6/1, 9/1], 6/1, 15/2, Yes, [6/1, 15/2])."},"skills":["Closed-Interval Intersection Including Touching Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0048","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e3423a9763586a18fb64469cc494897f4637824cd38059b59ff619cb947cc33d","family_id":"qf-density17-order-interval-intersection","seed":202610174048,"semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"5/1\",\"l2\":\"13/2\",\"u1\":\"8/1\",\"u2\":\"19/2\"}"},"fields":{"title":"Closed-Interval Intersection Including Touching Endpoints","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [10/2, 16/2] and [13/2, 19/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).","answer":{"kind":"expression","value":"([5/1, 8/1], [13/2, 19/2], 13/2, 8/1, Yes, [13/2, 8/1])","explanation":"Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5/1, 8/1], [13/2, 19/2], 13/2, 8/1, Yes, [13/2, 8/1])."},"skills":["Closed-Interval Intersection Including Touching Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0049","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae094fa65377705cb753e12df45906abde2d02eaf64ddb15ea7c2171fc559d4d","family_id":"qf-density17-order-interval-intersection","seed":202610174049,"semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"11/2\",\"l2\":\"7/1\",\"u1\":\"17/2\",\"u2\":\"10/1\"}"},"fields":{"title":"Closed-Interval Intersection Including Touching Endpoints","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [11/2, 17/2] and [14/2, 20/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).","answer":{"kind":"expression","value":"([11/2, 17/2], [7/1, 10/1], 7/1, 17/2, Yes, [7/1, 17/2])","explanation":"Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([11/2, 17/2], [7/1, 10/1], 7/1, 17/2, Yes, [7/1, 17/2])."},"skills":["Closed-Interval Intersection Including Touching Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0050","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30db68967bdf35a3e581cb1d126c06beedc83f65440f2fd5644daaa5608ea48f","family_id":"qf-density17-order-interval-intersection","seed":202610174050,"semantic_template":"qf-density17-order-interval-intersection/{\"l1\":\"6/1\",\"l2\":\"15/2\",\"u1\":\"9/1\",\"u2\":\"21/2\"}"},"fields":{"title":"Closed-Interval Intersection Including Touching Endpoints","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Intervals are [12/2, 18/2] and [15/2, 21/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).","answer":{"kind":"expression","value":"([6/1, 9/1], [15/2, 21/2], 15/2, 9/1, Yes, [15/2, 9/1])","explanation":"Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([6/1, 9/1], [15/2, 21/2], 15/2, 9/1, Yes, [15/2, 9/1])."},"skills":["Closed-Interval Intersection Including Touching Endpoints"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0051","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"445e4597fe1109b397fd7b4844cc63c8a44faaacdf52d27fbba09d5640ef3fda","family_id":"qf-density17-order-three-membership-slacks","seed":202610174051,"semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"0/1\",\"queries\":[\"-1/1\",\"0/1\",\"1/1\"],\"u\":\"0/1\"}"},"fields":{"title":"Three Closed-Interval Membership Tests and Signed Slacks","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[0/1, 0/1], queries=[-1/1, 0/1, 1/1]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).","answer":{"kind":"expression","value":"([0/1, 0/1], [-1/1, 0/1, 1/1], [No, Yes, No], [[-1/1, 1/1], [0/1, 0/1], [1/1, -1/1]])","explanation":"Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0/1, 0/1], [-1/1, 0/1, 1/1], [No, Yes, No], [[-1/1, 1/1], [0/1, 0/1], [1/1, -1/1]])."},"skills":["Three Closed-Interval Membership Tests and Signed Slacks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0052","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5365ce8e8e4bc45bea0a6a32f1093720b5f4f63da5403cf896d798419c75e4c","family_id":"qf-density17-order-three-membership-slacks","seed":202610174052,"semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-1/2\",\"queries\":[\"-1/2\",\"1/2\",\"1/1\"],\"u\":\"1/2\"}"},"fields":{"title":"Three Closed-Interval Membership Tests and Signed Slacks","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-1/2, 1/2], queries=[-1/2, 1/2, 1/1]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).","answer":{"kind":"expression","value":"([-1/2, 1/2], [-1/2, 1/2, 1/1], [Yes, Yes, No], [[0/1, 1/1], [1/1, 0/1], [3/2, -1/2]])","explanation":"Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1/2, 1/2], [-1/2, 1/2, 1/1], [Yes, Yes, No], [[0/1, 1/1], [1/1, 0/1], [3/2, -1/2]])."},"skills":["Three Closed-Interval Membership Tests and Signed Slacks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0053","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8c28be66d36b09cf3845cb3539254580220aa0bdcb5bb46c6e2a29573abc977","family_id":"qf-density17-order-three-membership-slacks","seed":202610174053,"semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-2/3\",\"queries\":[\"0/1\",\"-2/3\",\"3/2\"],\"u\":\"1/1\"}"},"fields":{"title":"Three Closed-Interval Membership Tests and Signed Slacks","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-2/3, 2/2], queries=[0/1, -2/3, 3/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).","answer":{"kind":"expression","value":"([-2/3, 1/1], [0/1, -2/3, 3/2], [Yes, Yes, No], [[2/3, 1/1], [0/1, 5/3], [13/6, -1/2]])","explanation":"Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-2/3, 1/1], [0/1, -2/3, 3/2], [Yes, Yes, No], [[2/3, 1/1], [0/1, 5/3], [13/6, -1/2]])."},"skills":["Three Closed-Interval Membership Tests and Signed Slacks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0054","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"554942bc29865fa974970681f15e8adc9c08e1ee4b9647277e443e7e2f5840a0","family_id":"qf-density17-order-three-membership-slacks","seed":202610174054,"semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-1/1\",\"queries\":[\"0/1\",\"-1/1\",\"2/1\"],\"u\":\"3/2\"}"},"fields":{"title":"Three Closed-Interval Membership Tests and Signed Slacks","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-3/3, 3/2], queries=[0/1, -3/3, 4/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).","answer":{"kind":"expression","value":"([-1/1, 3/2], [0/1, -1/1, 2/1], [Yes, Yes, No], [[1/1, 3/2], [0/1, 5/2], [3/1, -1/2]])","explanation":"Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1/1, 3/2], [0/1, -1/1, 2/1], [Yes, Yes, No], [[1/1, 3/2], [0/1, 5/2], [3/1, -1/2]])."},"skills":["Three Closed-Interval Membership Tests and Signed Slacks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0055","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5ab352d68e747a2442b822bbbbb11a565cb9685764cb634a4df67cf9ac292b5","family_id":"qf-density17-order-three-membership-slacks","seed":202610174055,"semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-4/3\",\"queries\":[\"0/1\",\"-4/3\",\"5/2\"],\"u\":\"2/1\"}"},"fields":{"title":"Three Closed-Interval Membership Tests and Signed Slacks","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-4/3, 4/2], queries=[0/1, -4/3, 5/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).","answer":{"kind":"expression","value":"([-4/3, 2/1], [0/1, -4/3, 5/2], [Yes, Yes, No], [[4/3, 2/1], [0/1, 10/3], [23/6, -1/2]])","explanation":"Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-4/3, 2/1], [0/1, -4/3, 5/2], [Yes, Yes, No], [[4/3, 2/1], [0/1, 10/3], [23/6, -1/2]])."},"skills":["Three Closed-Interval Membership Tests and Signed Slacks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0056","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d089f01fe3651db27faf70c11dfcd7eadb5388a62ced1b8d8efdb841f3dc660d","family_id":"qf-density17-order-three-membership-slacks","seed":202610174056,"semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-5/3\",\"queries\":[\"0/1\",\"-5/3\",\"3/1\"],\"u\":\"5/2\"}"},"fields":{"title":"Three Closed-Interval Membership Tests and Signed Slacks","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-5/3, 5/2], queries=[0/1, -5/3, 6/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).","answer":{"kind":"expression","value":"([-5/3, 5/2], [0/1, -5/3, 3/1], [Yes, Yes, No], [[5/3, 5/2], [0/1, 25/6], [14/3, -1/2]])","explanation":"Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-5/3, 5/2], [0/1, -5/3, 3/1], [Yes, Yes, No], [[5/3, 5/2], [0/1, 25/6], [14/3, -1/2]])."},"skills":["Three Closed-Interval Membership Tests and Signed Slacks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0057","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4dc5b724b52fe0d9a9906f66de6603af5e85f106a676a113649fc733f1640f21","family_id":"qf-density17-order-three-membership-slacks","seed":202610174057,"semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-2/1\",\"queries\":[\"0/1\",\"-2/1\",\"7/2\"],\"u\":\"3/1\"}"},"fields":{"title":"Three Closed-Interval Membership Tests and Signed Slacks","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-6/3, 6/2], queries=[0/1, -6/3, 7/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).","answer":{"kind":"expression","value":"([-2/1, 3/1], [0/1, -2/1, 7/2], [Yes, Yes, No], [[2/1, 3/1], [0/1, 5/1], [11/2, -1/2]])","explanation":"Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-2/1, 3/1], [0/1, -2/1, 7/2], [Yes, Yes, No], [[2/1, 3/1], [0/1, 5/1], [11/2, -1/2]])."},"skills":["Three Closed-Interval Membership Tests and Signed Slacks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0058","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a5987c9d932912a71006bff1415e1e158b52dcb62a79b035852312667ac8c51","family_id":"qf-density17-order-three-membership-slacks","seed":202610174058,"semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-7/3\",\"queries\":[\"0/1\",\"-7/3\",\"4/1\"],\"u\":\"7/2\"}"},"fields":{"title":"Three Closed-Interval Membership Tests and Signed Slacks","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-7/3, 7/2], queries=[0/1, -7/3, 8/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).","answer":{"kind":"expression","value":"([-7/3, 7/2], [0/1, -7/3, 4/1], [Yes, Yes, No], [[7/3, 7/2], [0/1, 35/6], [19/3, -1/2]])","explanation":"Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-7/3, 7/2], [0/1, -7/3, 4/1], [Yes, Yes, No], [[7/3, 7/2], [0/1, 35/6], [19/3, -1/2]])."},"skills":["Three Closed-Interval Membership Tests and Signed Slacks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0059","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f5e18b96a077dc019871abe6458391446a12b651c45303950bf5dbce77cfab3","family_id":"qf-density17-order-three-membership-slacks","seed":202610174059,"semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-8/3\",\"queries\":[\"0/1\",\"-8/3\",\"9/2\"],\"u\":\"4/1\"}"},"fields":{"title":"Three Closed-Interval Membership Tests and Signed Slacks","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-8/3, 8/2], queries=[0/1, -8/3, 9/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).","answer":{"kind":"expression","value":"([-8/3, 4/1], [0/1, -8/3, 9/2], [Yes, Yes, No], [[8/3, 4/1], [0/1, 20/3], [43/6, -1/2]])","explanation":"Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-8/3, 4/1], [0/1, -8/3, 9/2], [Yes, Yes, No], [[8/3, 4/1], [0/1, 20/3], [43/6, -1/2]])."},"skills":["Three Closed-Interval Membership Tests and Signed Slacks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0060","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a857808974ff610623bd5d0a3c15bf58d2d72fe83a763d2951301c10c3981af","family_id":"qf-density17-order-three-membership-slacks","seed":202610174060,"semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-3/1\",\"queries\":[\"0/1\",\"-3/1\",\"5/1\"],\"u\":\"9/2\"}"},"fields":{"title":"Three Closed-Interval Membership Tests and Signed Slacks","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-9/3, 9/2], queries=[0/1, -9/3, 10/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).","answer":{"kind":"expression","value":"([-3/1, 9/2], [0/1, -3/1, 5/1], [Yes, Yes, No], [[3/1, 9/2], [0/1, 15/2], [8/1, -1/2]])","explanation":"Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-3/1, 9/2], [0/1, -3/1, 5/1], [Yes, Yes, No], [[3/1, 9/2], [0/1, 15/2], [8/1, -1/2]])."},"skills":["Three Closed-Interval Membership Tests and Signed Slacks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0061","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0afe8542d2210980cf0caede0287cb537a2fffdd17eea4985f8297685ee749c7","family_id":"qf-density17-order-three-membership-slacks","seed":202610174061,"semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-10/3\",\"queries\":[\"0/1\",\"-10/3\",\"11/2\"],\"u\":\"5/1\"}"},"fields":{"title":"Three Closed-Interval Membership Tests and Signed Slacks","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-10/3, 10/2], queries=[0/1, -10/3, 11/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).","answer":{"kind":"expression","value":"([-10/3, 5/1], [0/1, -10/3, 11/2], [Yes, Yes, No], [[10/3, 5/1], [0/1, 25/3], [53/6, -1/2]])","explanation":"Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-10/3, 5/1], [0/1, -10/3, 11/2], [Yes, Yes, No], [[10/3, 5/1], [0/1, 25/3], [53/6, -1/2]])."},"skills":["Three Closed-Interval Membership Tests and Signed Slacks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0062","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39fbc99a33de7a3694ad33a4bda2198a145c6c0d5779d82862b24224d55cc9c6","family_id":"qf-density17-order-three-membership-slacks","seed":202610174062,"semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-11/3\",\"queries\":[\"0/1\",\"-11/3\",\"6/1\"],\"u\":\"11/2\"}"},"fields":{"title":"Three Closed-Interval Membership Tests and Signed Slacks","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-11/3, 11/2], queries=[0/1, -11/3, 12/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).","answer":{"kind":"expression","value":"([-11/3, 11/2], [0/1, -11/3, 6/1], [Yes, Yes, No], [[11/3, 11/2], [0/1, 55/6], [29/3, -1/2]])","explanation":"Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-11/3, 11/2], [0/1, -11/3, 6/1], [Yes, Yes, No], [[11/3, 11/2], [0/1, 55/6], [29/3, -1/2]])."},"skills":["Three Closed-Interval Membership Tests and Signed Slacks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0063","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"426578daa406e38efbb00e19d6957e26761f674f844a693626637d5a97d9bd1a","family_id":"qf-density17-order-three-membership-slacks","seed":202610174063,"semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-4/1\",\"queries\":[\"0/1\",\"-4/1\",\"13/2\"],\"u\":\"6/1\"}"},"fields":{"title":"Three Closed-Interval Membership Tests and Signed Slacks","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-12/3, 12/2], queries=[0/1, -12/3, 13/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).","answer":{"kind":"expression","value":"([-4/1, 6/1], [0/1, -4/1, 13/2], [Yes, Yes, No], [[4/1, 6/1], [0/1, 10/1], [21/2, -1/2]])","explanation":"Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-4/1, 6/1], [0/1, -4/1, 13/2], [Yes, Yes, No], [[4/1, 6/1], [0/1, 10/1], [21/2, -1/2]])."},"skills":["Three Closed-Interval Membership Tests and Signed Slacks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0064","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5052def658129f12ee9bc5a9041283b271e1a465d935d50033df815e6feb7e3","family_id":"qf-density17-order-three-membership-slacks","seed":202610174064,"semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-13/3\",\"queries\":[\"0/1\",\"-13/3\",\"7/1\"],\"u\":\"13/2\"}"},"fields":{"title":"Three Closed-Interval Membership Tests and Signed Slacks","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-13/3, 13/2], queries=[0/1, -13/3, 14/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).","answer":{"kind":"expression","value":"([-13/3, 13/2], [0/1, -13/3, 7/1], [Yes, Yes, No], [[13/3, 13/2], [0/1, 65/6], [34/3, -1/2]])","explanation":"Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-13/3, 13/2], [0/1, -13/3, 7/1], [Yes, Yes, No], [[13/3, 13/2], [0/1, 65/6], [34/3, -1/2]])."},"skills":["Three Closed-Interval Membership Tests and Signed Slacks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0065","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6645565f5cf6a1a55be5969b16087babe74de31ff6481d324a6275bbc0938af1","family_id":"qf-density17-order-three-membership-slacks","seed":202610174065,"semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-14/3\",\"queries\":[\"0/1\",\"-14/3\",\"15/2\"],\"u\":\"7/1\"}"},"fields":{"title":"Three Closed-Interval Membership Tests and Signed Slacks","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-14/3, 14/2], queries=[0/1, -14/3, 15/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).","answer":{"kind":"expression","value":"([-14/3, 7/1], [0/1, -14/3, 15/2], [Yes, Yes, No], [[14/3, 7/1], [0/1, 35/3], [73/6, -1/2]])","explanation":"Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-14/3, 7/1], [0/1, -14/3, 15/2], [Yes, Yes, No], [[14/3, 7/1], [0/1, 35/3], [73/6, -1/2]])."},"skills":["Three Closed-Interval Membership Tests and Signed Slacks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0066","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd7b69a2c18f5ce7eb6e0486e8438b0146e7d05271404147a286524669be8c43","family_id":"qf-density17-order-three-membership-slacks","seed":202610174066,"semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-5/1\",\"queries\":[\"0/1\",\"-5/1\",\"8/1\"],\"u\":\"15/2\"}"},"fields":{"title":"Three Closed-Interval Membership Tests and Signed Slacks","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-15/3, 15/2], queries=[0/1, -15/3, 16/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).","answer":{"kind":"expression","value":"([-5/1, 15/2], [0/1, -5/1, 8/1], [Yes, Yes, No], [[5/1, 15/2], [0/1, 25/2], [13/1, -1/2]])","explanation":"Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-5/1, 15/2], [0/1, -5/1, 8/1], [Yes, Yes, No], [[5/1, 15/2], [0/1, 25/2], [13/1, -1/2]])."},"skills":["Three Closed-Interval Membership Tests and Signed Slacks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0067","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74956deff74d0ad21fae8d9d0a061c9d28e4dc10fbe21b78b5c752e5ffabce0d","family_id":"qf-density17-order-three-membership-slacks","seed":202610174067,"semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-16/3\",\"queries\":[\"0/1\",\"-16/3\",\"17/2\"],\"u\":\"8/1\"}"},"fields":{"title":"Three Closed-Interval Membership Tests and Signed Slacks","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-16/3, 16/2], queries=[0/1, -16/3, 17/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).","answer":{"kind":"expression","value":"([-16/3, 8/1], [0/1, -16/3, 17/2], [Yes, Yes, No], [[16/3, 8/1], [0/1, 40/3], [83/6, -1/2]])","explanation":"Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-16/3, 8/1], [0/1, -16/3, 17/2], [Yes, Yes, No], [[16/3, 8/1], [0/1, 40/3], [83/6, -1/2]])."},"skills":["Three Closed-Interval Membership Tests and Signed Slacks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0068","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b2a280d9721b01bb9b30e33acce81c8585ff16e90e87e38ec08a0fe8946d5aa","family_id":"qf-density17-order-three-membership-slacks","seed":202610174068,"semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-17/3\",\"queries\":[\"0/1\",\"-17/3\",\"9/1\"],\"u\":\"17/2\"}"},"fields":{"title":"Three Closed-Interval Membership Tests and Signed Slacks","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-17/3, 17/2], queries=[0/1, -17/3, 18/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).","answer":{"kind":"expression","value":"([-17/3, 17/2], [0/1, -17/3, 9/1], [Yes, Yes, No], [[17/3, 17/2], [0/1, 85/6], [44/3, -1/2]])","explanation":"Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-17/3, 17/2], [0/1, -17/3, 9/1], [Yes, Yes, No], [[17/3, 17/2], [0/1, 85/6], [44/3, -1/2]])."},"skills":["Three Closed-Interval Membership Tests and Signed Slacks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0069","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d340e319344c988e1f8e051c3b57688b330789ae33a563af41645d75c57d7ca8","family_id":"qf-density17-order-three-membership-slacks","seed":202610174069,"semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-6/1\",\"queries\":[\"0/1\",\"-6/1\",\"19/2\"],\"u\":\"9/1\"}"},"fields":{"title":"Three Closed-Interval Membership Tests and Signed Slacks","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-18/3, 18/2], queries=[0/1, -18/3, 19/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).","answer":{"kind":"expression","value":"([-6/1, 9/1], [0/1, -6/1, 19/2], [Yes, Yes, No], [[6/1, 9/1], [0/1, 15/1], [31/2, -1/2]])","explanation":"Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-6/1, 9/1], [0/1, -6/1, 19/2], [Yes, Yes, No], [[6/1, 9/1], [0/1, 15/1], [31/2, -1/2]])."},"skills":["Three Closed-Interval Membership Tests and Signed Slacks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0070","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a43e19322e789405a01ec13e3b5f13b8f15f583b52bb455bda8fee47ab0e82eb","family_id":"qf-density17-order-three-membership-slacks","seed":202610174070,"semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-19/3\",\"queries\":[\"0/1\",\"-19/3\",\"10/1\"],\"u\":\"19/2\"}"},"fields":{"title":"Three Closed-Interval Membership Tests and Signed Slacks","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-19/3, 19/2], queries=[0/1, -19/3, 20/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).","answer":{"kind":"expression","value":"([-19/3, 19/2], [0/1, -19/3, 10/1], [Yes, Yes, No], [[19/3, 19/2], [0/1, 95/6], [49/3, -1/2]])","explanation":"Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-19/3, 19/2], [0/1, -19/3, 10/1], [Yes, Yes, No], [[19/3, 19/2], [0/1, 95/6], [49/3, -1/2]])."},"skills":["Three Closed-Interval Membership Tests and Signed Slacks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0071","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aac236e2c9a85b1291e57372f17e1c1da1a1e79016b0613b45a3e4323bb176e9","family_id":"qf-density17-order-three-membership-slacks","seed":202610174071,"semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-20/3\",\"queries\":[\"0/1\",\"-20/3\",\"21/2\"],\"u\":\"10/1\"}"},"fields":{"title":"Three Closed-Interval Membership Tests and Signed Slacks","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-20/3, 20/2], queries=[0/1, -20/3, 21/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).","answer":{"kind":"expression","value":"([-20/3, 10/1], [0/1, -20/3, 21/2], [Yes, Yes, No], [[20/3, 10/1], [0/1, 50/3], [103/6, -1/2]])","explanation":"Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-20/3, 10/1], [0/1, -20/3, 21/2], [Yes, Yes, No], [[20/3, 10/1], [0/1, 50/3], [103/6, -1/2]])."},"skills":["Three Closed-Interval Membership Tests and Signed Slacks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0072","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2731f3709bd7b413ce23d205afada08976f5eb679e61d1ed4a731feeb0b3d602","family_id":"qf-density17-order-three-membership-slacks","seed":202610174072,"semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-7/1\",\"queries\":[\"0/1\",\"-7/1\",\"11/1\"],\"u\":\"21/2\"}"},"fields":{"title":"Three Closed-Interval Membership Tests and Signed Slacks","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-21/3, 21/2], queries=[0/1, -21/3, 22/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).","answer":{"kind":"expression","value":"([-7/1, 21/2], [0/1, -7/1, 11/1], [Yes, Yes, No], [[7/1, 21/2], [0/1, 35/2], [18/1, -1/2]])","explanation":"Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-7/1, 21/2], [0/1, -7/1, 11/1], [Yes, Yes, No], [[7/1, 21/2], [0/1, 35/2], [18/1, -1/2]])."},"skills":["Three Closed-Interval Membership Tests and Signed Slacks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0073","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4b3f88ef739353d7607a6faa7f95479d6694a741977f5a9dbbc9b78b76c8242","family_id":"qf-density17-order-three-membership-slacks","seed":202610174073,"semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-22/3\",\"queries\":[\"0/1\",\"-22/3\",\"23/2\"],\"u\":\"11/1\"}"},"fields":{"title":"Three Closed-Interval Membership Tests and Signed Slacks","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-22/3, 22/2], queries=[0/1, -22/3, 23/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).","answer":{"kind":"expression","value":"([-22/3, 11/1], [0/1, -22/3, 23/2], [Yes, Yes, No], [[22/3, 11/1], [0/1, 55/3], [113/6, -1/2]])","explanation":"Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-22/3, 11/1], [0/1, -22/3, 23/2], [Yes, Yes, No], [[22/3, 11/1], [0/1, 55/3], [113/6, -1/2]])."},"skills":["Three Closed-Interval Membership Tests and Signed Slacks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0074","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"56b84bbe3e2290f9481410e8378e576a9c86192daef589cc45c3ca18a37d595c","family_id":"qf-density17-order-three-membership-slacks","seed":202610174074,"semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-23/3\",\"queries\":[\"0/1\",\"-23/3\",\"12/1\"],\"u\":\"23/2\"}"},"fields":{"title":"Three Closed-Interval Membership Tests and Signed Slacks","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-23/3, 23/2], queries=[0/1, -23/3, 24/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).","answer":{"kind":"expression","value":"([-23/3, 23/2], [0/1, -23/3, 12/1], [Yes, Yes, No], [[23/3, 23/2], [0/1, 115/6], [59/3, -1/2]])","explanation":"Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-23/3, 23/2], [0/1, -23/3, 12/1], [Yes, Yes, No], [[23/3, 23/2], [0/1, 115/6], [59/3, -1/2]])."},"skills":["Three Closed-Interval Membership Tests and Signed Slacks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0075","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a077a5577e9bf63e2efab04a2a1a6de8cc3acffb38f710800e14fcbf6cc97054","family_id":"qf-density17-order-three-membership-slacks","seed":202610174075,"semantic_template":"qf-density17-order-three-membership-slacks/{\"l\":\"-8/1\",\"queries\":[\"0/1\",\"-8/1\",\"25/2\"],\"u\":\"12/1\"}"},"fields":{"title":"Three Closed-Interval Membership Tests and Signed Slacks","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-24/3, 24/2], queries=[0/1, -24/3, 25/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).","answer":{"kind":"expression","value":"([-8/1, 12/1], [0/1, -8/1, 25/2], [Yes, Yes, No], [[8/1, 12/1], [0/1, 20/1], [41/2, -1/2]])","explanation":"Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-8/1, 12/1], [0/1, -8/1, 25/2], [Yes, Yes, No], [[8/1, 12/1], [0/1, 20/1], [41/2, -1/2]])."},"skills":["Three Closed-Interval Membership Tests and Signed Slacks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0076","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1da86bfc26bc029409e0476efdbe446b7fcabedffaa5243465bd4761a6a9cad9","family_id":"qf-density17-order-scaling-direction","seed":202610174076,"semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"-1/2\",\"b\":\"1/3\",\"t\":\"0/1\"}"},"fields":{"title":"Scaled Rational Values and Order Direction","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=-1/2, b=1/3, t=0/1. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).","answer":{"kind":"expression","value":"(-1/2, 1/3, 0/1, 0/1, 0/1, <, =, collapsed)","explanation":"Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-1/2, 1/3, 0/1, 0/1, 0/1, <, =, collapsed)."},"skills":["Scaled Rational Values and Order Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0077","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72175edaec19c3bce3102634a6376a9dcb8f92a4b5e5cf2d6b8cc7d0a3451b3d","family_id":"qf-density17-order-scaling-direction","seed":202610174077,"semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"0/1\",\"b\":\"0/1\",\"t\":\"-2/1\"}"},"fields":{"title":"Scaled Rational Values and Order Direction","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=0/1, b=0/1, t=-2/1. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).","answer":{"kind":"expression","value":"(0/1, 0/1, -2/1, 0/1, 0/1, =, =, order-reversed)","explanation":"Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (0/1, 0/1, -2/1, 0/1, 0/1, =, =, order-reversed)."},"skills":["Scaled Rational Values and Order Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0078","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9eef85c546586b038d38f2f3eda9057c2b15b0e614f84027903555937ce3079","family_id":"qf-density17-order-scaling-direction","seed":202610174078,"semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"-10/3\",\"b\":\"-9/2\",\"t\":\"1/1\"}"},"fields":{"title":"Scaled Rational Values and Order Direction","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=-10/3, b=-9/2, t=3/3. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).","answer":{"kind":"expression","value":"(-10/3, -9/2, 1/1, -10/3, -9/2, >, >, order-preserved)","explanation":"Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-10/3, -9/2, 1/1, -10/3, -9/2, >, >, order-preserved)."},"skills":["Scaled Rational Values and Order Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0079","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84d44cde58b032884206247d5ad686b0b5c69ce0106664c2c6a421746c785a9f","family_id":"qf-density17-order-scaling-direction","seed":202610174079,"semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"-3/1\",\"b\":\"-4/1\",\"t\":\"0/1\"}"},"fields":{"title":"Scaled Rational Values and Order Direction","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=-9/3, b=-8/2, t=0/4. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).","answer":{"kind":"expression","value":"(-3/1, -4/1, 0/1, 0/1, 0/1, >, =, collapsed)","explanation":"Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-3/1, -4/1, 0/1, 0/1, 0/1, >, =, collapsed)."},"skills":["Scaled Rational Values and Order Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0080","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df2120b5bf19370ac319745a20facb33fc786eba8b8701d5ee0e11f3bdccedff","family_id":"qf-density17-order-scaling-direction","seed":202610174080,"semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"-8/3\",\"b\":\"-7/2\",\"t\":\"3/1\"}"},"fields":{"title":"Scaled Rational Values and Order Direction","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=-8/3, b=-7/2, t=3/1. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).","answer":{"kind":"expression","value":"(-8/3, -7/2, 3/1, -8/1, -21/2, >, >, order-preserved)","explanation":"Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-8/3, -7/2, 3/1, -8/1, -21/2, >, >, order-preserved)."},"skills":["Scaled Rational Values and Order Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0081","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"abec1318cdc47fd1fa7bcbf4db35a28b9537e3633f48eca8d65a81e87230a108","family_id":"qf-density17-order-scaling-direction","seed":202610174081,"semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"-7/3\",\"b\":\"-3/1\",\"t\":\"-1/1\"}"},"fields":{"title":"Scaled Rational Values and Order Direction","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=-7/3, b=-6/2, t=-2/2. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).","answer":{"kind":"expression","value":"(-7/3, -3/1, -1/1, 7/3, 3/1, >, <, order-reversed)","explanation":"Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-7/3, -3/1, -1/1, 7/3, 3/1, >, <, order-reversed)."},"skills":["Scaled Rational Values and Order Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0082","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aaff03bdc053e8c117fd9465560133b81dbbb229c0a5b8c08ef4960daff66900","family_id":"qf-density17-order-scaling-direction","seed":202610174082,"semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"-2/1\",\"b\":\"-5/2\",\"t\":\"0/1\"}"},"fields":{"title":"Scaled Rational Values and Order Direction","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=-6/3, b=-5/2, t=0/3. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).","answer":{"kind":"expression","value":"(-2/1, -5/2, 0/1, 0/1, 0/1, >, =, collapsed)","explanation":"Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-2/1, -5/2, 0/1, 0/1, 0/1, >, =, collapsed)."},"skills":["Scaled Rational Values and Order Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0083","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e539ce3d2611adec79cdd0303e749c133a55c40976e5f934aabfcf62f685024d","family_id":"qf-density17-order-scaling-direction","seed":202610174083,"semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"-5/3\",\"b\":\"-2/1\",\"t\":\"-1/2\"}"},"fields":{"title":"Scaled Rational Values and Order Direction","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=-5/3, b=-4/2, t=-2/4. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).","answer":{"kind":"expression","value":"(-5/3, -2/1, -1/2, 5/6, 1/1, >, <, order-reversed)","explanation":"Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-5/3, -2/1, -1/2, 5/6, 1/1, >, <, order-reversed)."},"skills":["Scaled Rational Values and Order Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0084","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f01894226c4b20e098f46e16dcd06608d3daade3ff03e2506bfa445eca026033","family_id":"qf-density17-order-scaling-direction","seed":202610174084,"semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"-4/3\",\"b\":\"-3/2\",\"t\":\"3/1\"}"},"fields":{"title":"Scaled Rational Values and Order Direction","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=-4/3, b=-3/2, t=3/1. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).","answer":{"kind":"expression","value":"(-4/3, -3/2, 3/1, -4/1, -9/2, >, >, order-preserved)","explanation":"Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-4/3, -3/2, 3/1, -4/1, -9/2, >, >, order-preserved)."},"skills":["Scaled Rational Values and Order Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0085","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04e62ddeeeff4b93d0fcfe1ca14c38503e9d0c153b0fa7fb2793fe294f25e0b2","family_id":"qf-density17-order-scaling-direction","seed":202610174085,"semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"-1/1\",\"b\":\"-1/1\",\"t\":\"0/1\"}"},"fields":{"title":"Scaled Rational Values and Order Direction","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=-3/3, b=-2/2, t=0/2. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).","answer":{"kind":"expression","value":"(-1/1, -1/1, 0/1, 0/1, 0/1, =, =, collapsed)","explanation":"Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-1/1, -1/1, 0/1, 0/1, 0/1, =, =, collapsed)."},"skills":["Scaled Rational Values and Order Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0086","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"877fd5467b3a5663e778380fc5e64478c31855464e5010e886674b3908f0df2b","family_id":"qf-density17-order-scaling-direction","seed":202610174086,"semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"-2/3\",\"b\":\"-1/2\",\"t\":\"1/1\"}"},"fields":{"title":"Scaled Rational Values and Order Direction","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=-2/3, b=-1/2, t=3/3. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).","answer":{"kind":"expression","value":"(-2/3, -1/2, 1/1, -2/3, -1/2, <, <, order-preserved)","explanation":"Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-2/3, -1/2, 1/1, -2/3, -1/2, <, <, order-preserved)."},"skills":["Scaled Rational Values and Order Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0087","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7afb76827ecae295e85303e7412654ad772d82809ecbc71112812591b6548faf","family_id":"qf-density17-order-scaling-direction","seed":202610174087,"semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"-1/3\",\"b\":\"0/1\",\"t\":\"-1/2\"}"},"fields":{"title":"Scaled Rational Values and Order Direction","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=-1/3, b=0/2, t=-2/4. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).","answer":{"kind":"expression","value":"(-1/3, 0/1, -1/2, 1/6, 0/1, <, >, order-reversed)","explanation":"Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-1/3, 0/1, -1/2, 1/6, 0/1, <, >, order-reversed)."},"skills":["Scaled Rational Values and Order Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0088","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"643365c434ffa3c660a6e94d16a2703246200a6b2062ea08bc374b9143266a6e","family_id":"qf-density17-order-scaling-direction","seed":202610174088,"semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"0/1\",\"b\":\"1/2\",\"t\":\"0/1\"}"},"fields":{"title":"Scaled Rational Values and Order Direction","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=0/3, b=1/2, t=0/1. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).","answer":{"kind":"expression","value":"(0/1, 1/2, 0/1, 0/1, 0/1, <, =, collapsed)","explanation":"Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (0/1, 1/2, 0/1, 0/1, 0/1, <, =, collapsed)."},"skills":["Scaled Rational Values and Order Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0089","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b78bf484075d8d56659ef9a28c0baf356bad06dbfba04cc956be7e37ae055bdd","family_id":"qf-density17-order-scaling-direction","seed":202610174089,"semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"1/3\",\"b\":\"1/1\",\"t\":\"-1/1\"}"},"fields":{"title":"Scaled Rational Values and Order Direction","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=1/3, b=2/2, t=-2/2. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).","answer":{"kind":"expression","value":"(1/3, 1/1, -1/1, -1/3, -1/1, <, >, order-reversed)","explanation":"Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1/3, 1/1, -1/1, -1/3, -1/1, <, >, order-reversed)."},"skills":["Scaled Rational Values and Order Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0090","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7b2eb43b3059ab9368ef7df77ae8741062bc3e027cb28d347a2bfc30cb0598e","family_id":"qf-density17-order-scaling-direction","seed":202610174090,"semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"2/3\",\"b\":\"3/2\",\"t\":\"1/1\"}"},"fields":{"title":"Scaled Rational Values and Order Direction","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=2/3, b=3/2, t=3/3. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).","answer":{"kind":"expression","value":"(2/3, 3/2, 1/1, 2/3, 3/2, <, <, order-preserved)","explanation":"Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (2/3, 3/2, 1/1, 2/3, 3/2, <, <, order-preserved)."},"skills":["Scaled Rational Values and Order Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0091","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d90ae014c1ded9128ca28ef1e4281902152bda76abb836135735d2b8a40c14a","family_id":"qf-density17-order-scaling-direction","seed":202610174091,"semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"1/1\",\"b\":\"2/1\",\"t\":\"0/1\"}"},"fields":{"title":"Scaled Rational Values and Order Direction","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=3/3, b=4/2, t=0/4. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).","answer":{"kind":"expression","value":"(1/1, 2/1, 0/1, 0/1, 0/1, <, =, collapsed)","explanation":"Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1/1, 2/1, 0/1, 0/1, 0/1, <, =, collapsed)."},"skills":["Scaled Rational Values and Order Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0092","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4cb1a9fcafe1ff97d22088141a18b86d08ff8d6033f1b141c9c7fd77956984cd","family_id":"qf-density17-order-scaling-direction","seed":202610174092,"semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"4/3\",\"b\":\"5/2\",\"t\":\"3/1\"}"},"fields":{"title":"Scaled Rational Values and Order Direction","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=4/3, b=5/2, t=3/1. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).","answer":{"kind":"expression","value":"(4/3, 5/2, 3/1, 4/1, 15/2, <, <, order-preserved)","explanation":"Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4/3, 5/2, 3/1, 4/1, 15/2, <, <, order-preserved)."},"skills":["Scaled Rational Values and Order Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0093","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2840cb7fe9e8b07db1930a7f586f5f77bbf04c9f0042015dad5d4f4d75f3f434","family_id":"qf-density17-order-scaling-direction","seed":202610174093,"semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"5/3\",\"b\":\"3/1\",\"t\":\"-1/1\"}"},"fields":{"title":"Scaled Rational Values and Order Direction","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=5/3, b=6/2, t=-2/2. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).","answer":{"kind":"expression","value":"(5/3, 3/1, -1/1, -5/3, -3/1, <, >, order-reversed)","explanation":"Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5/3, 3/1, -1/1, -5/3, -3/1, <, >, order-reversed)."},"skills":["Scaled Rational Values and Order Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0094","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"918171ed520a9a45c891ba190be4d875fd7645d4af19db320d3c3f27b4c61fe2","family_id":"qf-density17-order-scaling-direction","seed":202610174094,"semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"2/1\",\"b\":\"7/2\",\"t\":\"0/1\"}"},"fields":{"title":"Scaled Rational Values and Order Direction","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=6/3, b=7/2, t=0/3. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).","answer":{"kind":"expression","value":"(2/1, 7/2, 0/1, 0/1, 0/1, <, =, collapsed)","explanation":"Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (2/1, 7/2, 0/1, 0/1, 0/1, <, =, collapsed)."},"skills":["Scaled Rational Values and Order Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0095","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc46cfa50face39a1961cb42a57de198a4826d3582a138be092d13190ef80188","family_id":"qf-density17-order-scaling-direction","seed":202610174095,"semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"7/3\",\"b\":\"4/1\",\"t\":\"-1/2\"}"},"fields":{"title":"Scaled Rational Values and Order Direction","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=7/3, b=8/2, t=-2/4. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).","answer":{"kind":"expression","value":"(7/3, 4/1, -1/2, -7/6, -2/1, <, >, order-reversed)","explanation":"Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (7/3, 4/1, -1/2, -7/6, -2/1, <, >, order-reversed)."},"skills":["Scaled Rational Values and Order Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0096","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11751053f03529e833a035499cb0678783c4db7375bad26fb963faa12cd0d99f","family_id":"qf-density17-order-scaling-direction","seed":202610174096,"semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"8/3\",\"b\":\"9/2\",\"t\":\"3/1\"}"},"fields":{"title":"Scaled Rational Values and Order Direction","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=8/3, b=9/2, t=3/1. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).","answer":{"kind":"expression","value":"(8/3, 9/2, 3/1, 8/1, 27/2, <, <, order-preserved)","explanation":"Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (8/3, 9/2, 3/1, 8/1, 27/2, <, <, order-preserved)."},"skills":["Scaled Rational Values and Order Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0097","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2050bf98c4755e01b6c26ba4627393c6e6659f39a2d10e1b8b014d3536a412d6","family_id":"qf-density17-order-scaling-direction","seed":202610174097,"semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"3/1\",\"b\":\"5/1\",\"t\":\"0/1\"}"},"fields":{"title":"Scaled Rational Values and Order Direction","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=9/3, b=10/2, t=0/2. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).","answer":{"kind":"expression","value":"(3/1, 5/1, 0/1, 0/1, 0/1, <, =, collapsed)","explanation":"Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (3/1, 5/1, 0/1, 0/1, 0/1, <, =, collapsed)."},"skills":["Scaled Rational Values and Order Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0098","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a51095d807b53fd106db5438c0333d133a10715d81ba9ab8a1e2a6608e3c5b6","family_id":"qf-density17-order-scaling-direction","seed":202610174098,"semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"10/3\",\"b\":\"11/2\",\"t\":\"1/1\"}"},"fields":{"title":"Scaled Rational Values and Order Direction","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=10/3, b=11/2, t=3/3. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).","answer":{"kind":"expression","value":"(10/3, 11/2, 1/1, 10/3, 11/2, <, <, order-preserved)","explanation":"Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (10/3, 11/2, 1/1, 10/3, 11/2, <, <, order-preserved)."},"skills":["Scaled Rational Values and Order Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0099","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"abe626ecf76971ceee12dc07ad902bcf62c9d3efd0b3140e1482d78c59b55795","family_id":"qf-density17-order-scaling-direction","seed":202610174099,"semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"11/3\",\"b\":\"6/1\",\"t\":\"-1/2\"}"},"fields":{"title":"Scaled Rational Values and Order Direction","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=11/3, b=12/2, t=-2/4. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).","answer":{"kind":"expression","value":"(11/3, 6/1, -1/2, -11/6, -3/1, <, >, order-reversed)","explanation":"Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (11/3, 6/1, -1/2, -11/6, -3/1, <, >, order-reversed)."},"skills":["Scaled Rational Values and Order Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r04-order-relation-0100","batch_id":"batch-20261012-advance-20261017-r04-order-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc923ca8e31ff323fa21893390cfcba6d6ccb7263df1b6593e882d773a95d53e","family_id":"qf-density17-order-scaling-direction","seed":202610174100,"semantic_template":"qf-density17-order-scaling-direction/{\"a\":\"4/1\",\"b\":\"13/2\",\"t\":\"0/1\"}"},"fields":{"title":"Scaled Rational Values and Order Direction","stem":"Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. a=12/3, b=13/2, t=0/1. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).","answer":{"kind":"expression","value":"(4/1, 13/2, 0/1, 0/1, 0/1, <, =, collapsed)","explanation":"Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4/1, 13/2, 0/1, 0/1, 0/1, <, =, collapsed)."},"skills":["Scaled Rational Values and Order Direction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}