{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0001","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3bc58e6ee5919204af72c8c7b2d13d5db8f5988f53c894fb6442a7ddd2b7d8a","family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151001,"semantic_template":"qf-density15-fraction-sum-naive-audit/[\"0/1\",\"undefined\"]"},"fields":{"title":"True Fraction Sum versus Naive Representation Addition","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=1/2, b=1/-2. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).","answer":{"kind":"expression","value":"(0/1, No, undefined, undefined, No)","explanation":"Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (0/1, No, undefined, undefined, No)."},"skills":["True Fraction Sum versus Naive Representation Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0002","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12cbaa643dfc607296c3cdc09e9467c24f0178ebe517360b79bffb9ddf8d3caa","family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151002,"semantic_template":"qf-density15-fraction-sum-naive-audit/[\"1/1\",\"undefined\"]"},"fields":{"title":"True Fraction Sum versus Naive Representation Addition","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=1/2, b=-1/-2. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).","answer":{"kind":"expression","value":"(1/1, No, undefined, undefined, No)","explanation":"Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (1/1, No, undefined, undefined, No)."},"skills":["True Fraction Sum versus Naive Representation Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0003","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e16f4ab0e96b28d4651676291ae9aa5621f544721c581abe0d684b32ebe43efc","family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151003,"semantic_template":"qf-density15-fraction-sum-naive-audit/[\"0/1\",\"0/1\"]"},"fields":{"title":"True Fraction Sum versus Naive Representation Addition","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=0/2, b=0/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).","answer":{"kind":"expression","value":"(0/1, Yes, 0/1, 0/1, Yes)","explanation":"Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (0/1, Yes, 0/1, 0/1, Yes)."},"skills":["True Fraction Sum versus Naive Representation Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0004","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78b28aeabbc5183dfa4f2ab6030e42c0d2520e3442abdd939489f4ff5ae16de3","family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151004,"semantic_template":"qf-density15-fraction-sum-naive-audit/[\"3/4\",\"1/3\"]"},"fields":{"title":"True Fraction Sum versus Naive Representation Addition","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=1/2, b=1/4. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).","answer":{"kind":"expression","value":"(3/4, Yes, 1/3, -5/12, No)","explanation":"Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (3/4, Yes, 1/3, -5/12, No)."},"skills":["True Fraction Sum versus Naive Representation Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0005","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d461b7bf471b86854b387b5dc1afa9bc64ea330efbc44e4bafdad598e34f69ae","family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151005,"semantic_template":"qf-density15-fraction-sum-naive-audit/[\"-11/3\",\"-7/5\"]"},"fields":{"title":"True Fraction Sum versus Naive Representation Addition","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-8/2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).","answer":{"kind":"expression","value":"(-11/3, Yes, -7/5, 34/15, No)","explanation":"Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (-11/3, Yes, -7/5, 34/15, No)."},"skills":["True Fraction Sum versus Naive Representation Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0006","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c9e06fc0db7f0360c035a56b6e285187f885c4475bffad43db41739828067ff","family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151006,"semantic_template":"qf-density15-fraction-sum-naive-audit/[\"-19/6\",\"8/1\"]"},"fields":{"title":"True Fraction Sum versus Naive Representation Addition","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=7/-2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).","answer":{"kind":"expression","value":"(-19/6, Yes, 8/1, 67/6, No)","explanation":"Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (-19/6, Yes, 8/1, 67/6, No)."},"skills":["True Fraction Sum versus Naive Representation Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0007","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92dc2fa85d289fcbfe4f58e44869ffb00bcb9d42d66f982a7c049ee35a2aaa23","family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151007,"semantic_template":"qf-density15-fraction-sum-naive-audit/[\"-8/3\",\"-1/1\"]"},"fields":{"title":"True Fraction Sum versus Naive Representation Addition","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-6/2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).","answer":{"kind":"expression","value":"(-8/3, Yes, -1/1, 5/3, No)","explanation":"Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (-8/3, Yes, -1/1, 5/3, No)."},"skills":["True Fraction Sum versus Naive Representation Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0008","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e17da366f287af2d90a2be09f14429380711ede980da781ea5cd5e138a1acd4e","family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151008,"semantic_template":"qf-density15-fraction-sum-naive-audit/[\"-13/6\",\"6/1\"]"},"fields":{"title":"True Fraction Sum versus Naive Representation Addition","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=5/-2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).","answer":{"kind":"expression","value":"(-13/6, Yes, 6/1, 49/6, No)","explanation":"Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (-13/6, Yes, 6/1, 49/6, No)."},"skills":["True Fraction Sum versus Naive Representation Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0009","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c6fbf50fe3575445e8657181210a347b8cdb90a4925d4115e5fb295000201b84","family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151009,"semantic_template":"qf-density15-fraction-sum-naive-audit/[\"-5/3\",\"-3/5\"]"},"fields":{"title":"True Fraction Sum versus Naive Representation Addition","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-4/2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).","answer":{"kind":"expression","value":"(-5/3, Yes, -3/5, 16/15, No)","explanation":"Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (-5/3, Yes, -3/5, 16/15, No)."},"skills":["True Fraction Sum versus Naive Representation Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0010","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a44857248e7bd3a7d59b68dec4a207d072ce83019628f490f52c8967f2b69e1","family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151010,"semantic_template":"qf-density15-fraction-sum-naive-audit/[\"-7/6\",\"4/1\"]"},"fields":{"title":"True Fraction Sum versus Naive Representation Addition","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=3/-2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).","answer":{"kind":"expression","value":"(-7/6, Yes, 4/1, 31/6, No)","explanation":"Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (-7/6, Yes, 4/1, 31/6, No)."},"skills":["True Fraction Sum versus Naive Representation Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0011","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16319609231b876716c51bd34c73fc86bd99f4f728c312fadec5a5cb176e447c","family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151011,"semantic_template":"qf-density15-fraction-sum-naive-audit/[\"-2/3\",\"-1/5\"]"},"fields":{"title":"True Fraction Sum versus Naive Representation Addition","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-2/2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).","answer":{"kind":"expression","value":"(-2/3, Yes, -1/5, 7/15, No)","explanation":"Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (-2/3, Yes, -1/5, 7/15, No)."},"skills":["True Fraction Sum versus Naive Representation Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0012","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec220b0c472099adb6c3a7411b0d3594f330413a3ce7ee9f0c933a9ce7c12e4b","family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151012,"semantic_template":"qf-density15-fraction-sum-naive-audit/[\"-1/6\",\"2/1\"]"},"fields":{"title":"True Fraction Sum versus Naive Representation Addition","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=1/-2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).","answer":{"kind":"expression","value":"(-1/6, Yes, 2/1, 13/6, No)","explanation":"Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (-1/6, Yes, 2/1, 13/6, No)."},"skills":["True Fraction Sum versus Naive Representation Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0013","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32565fa386dd3c295a0be44c24940738350f226c50df8c4c22ae98a9bcd56470","family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151013,"semantic_template":"qf-density15-fraction-sum-naive-audit/[\"1/3\",\"1/5\"]"},"fields":{"title":"True Fraction Sum versus Naive Representation Addition","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=0/2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).","answer":{"kind":"expression","value":"(1/3, Yes, 1/5, -2/15, No)","explanation":"Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (1/3, Yes, 1/5, -2/15, No)."},"skills":["True Fraction Sum versus Naive Representation Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0014","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff852459273467c1ec7cdf9cbd00c7a309907ee2118ea479fa152c495a337a18","family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151014,"semantic_template":"qf-density15-fraction-sum-naive-audit/[\"5/6\",\"0/1\"]"},"fields":{"title":"True Fraction Sum versus Naive Representation Addition","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-1/-2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).","answer":{"kind":"expression","value":"(5/6, Yes, 0/1, -5/6, No)","explanation":"Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (5/6, Yes, 0/1, -5/6, No)."},"skills":["True Fraction Sum versus Naive Representation Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0015","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8438df8f84172961c755072b8a539f3bbf8c85501efcb5368dc8cfeba85ce50f","family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151015,"semantic_template":"qf-density15-fraction-sum-naive-audit/[\"4/3\",\"3/5\"]"},"fields":{"title":"True Fraction Sum versus Naive Representation Addition","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=2/2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).","answer":{"kind":"expression","value":"(4/3, Yes, 3/5, -11/15, No)","explanation":"Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (4/3, Yes, 3/5, -11/15, No)."},"skills":["True Fraction Sum versus Naive Representation Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0016","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef640643d0c463838729ae959824a40492d58475ce8a2b9d8eb3a7423d845b89","family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151016,"semantic_template":"qf-density15-fraction-sum-naive-audit/[\"11/6\",\"-2/1\"]"},"fields":{"title":"True Fraction Sum versus Naive Representation Addition","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-3/-2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).","answer":{"kind":"expression","value":"(11/6, Yes, -2/1, -23/6, No)","explanation":"Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (11/6, Yes, -2/1, -23/6, No)."},"skills":["True Fraction Sum versus Naive Representation Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0017","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d23ca8a1bea1ae708f0cb5692ff11de996a5b54a987e489453ecde9d772d24ce","family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151017,"semantic_template":"qf-density15-fraction-sum-naive-audit/[\"7/3\",\"1/1\"]"},"fields":{"title":"True Fraction Sum versus Naive Representation Addition","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=4/2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).","answer":{"kind":"expression","value":"(7/3, Yes, 1/1, -4/3, No)","explanation":"Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (7/3, Yes, 1/1, -4/3, No)."},"skills":["True Fraction Sum versus Naive Representation Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0018","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b6d84aeefca7ecba48e4dc6920f2fa18a6f70064b619c0ee0efa29480860e228","family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151018,"semantic_template":"qf-density15-fraction-sum-naive-audit/[\"17/6\",\"-4/1\"]"},"fields":{"title":"True Fraction Sum versus Naive Representation Addition","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-5/-2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).","answer":{"kind":"expression","value":"(17/6, Yes, -4/1, -41/6, No)","explanation":"Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (17/6, Yes, -4/1, -41/6, No)."},"skills":["True Fraction Sum versus Naive Representation Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0019","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c643f0e123068b34785bab6105dd04a208eb368584c51a3d12a67d8e4f50a190","family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151019,"semantic_template":"qf-density15-fraction-sum-naive-audit/[\"10/3\",\"7/5\"]"},"fields":{"title":"True Fraction Sum versus Naive Representation Addition","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=6/2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).","answer":{"kind":"expression","value":"(10/3, Yes, 7/5, -29/15, No)","explanation":"Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (10/3, Yes, 7/5, -29/15, No)."},"skills":["True Fraction Sum versus Naive Representation Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0020","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a25156e89e5316e1ee705d748545a144980dc17987710cfe6b886f85eea3e84","family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151020,"semantic_template":"qf-density15-fraction-sum-naive-audit/[\"23/6\",\"-6/1\"]"},"fields":{"title":"True Fraction Sum versus Naive Representation Addition","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-7/-2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).","answer":{"kind":"expression","value":"(23/6, Yes, -6/1, -59/6, No)","explanation":"Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (23/6, Yes, -6/1, -59/6, No)."},"skills":["True Fraction Sum versus Naive Representation Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0021","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5fe00c5d6764d47e4c5f973ac29f5834ff9711d66f8c249487d35cd1492b5915","family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151021,"semantic_template":"qf-density15-fraction-sum-naive-audit/[\"13/3\",\"9/5\"]"},"fields":{"title":"True Fraction Sum versus Naive Representation Addition","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=8/2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).","answer":{"kind":"expression","value":"(13/3, Yes, 9/5, -38/15, No)","explanation":"Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (13/3, Yes, 9/5, -38/15, No)."},"skills":["True Fraction Sum versus Naive Representation Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0022","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b5e76422b1c2dae80713c8555632f155894bf7cbbf2162749c44bc3db1f6e4c","family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151022,"semantic_template":"qf-density15-fraction-sum-naive-audit/[\"29/6\",\"-8/1\"]"},"fields":{"title":"True Fraction Sum versus Naive Representation Addition","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-9/-2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).","answer":{"kind":"expression","value":"(29/6, Yes, -8/1, -77/6, No)","explanation":"Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (29/6, Yes, -8/1, -77/6, No)."},"skills":["True Fraction Sum versus Naive Representation Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0023","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab3d35ab8dc0b550e6727025ff2f3930bc890e6eee4bff6b17260d7a4fce68ee","family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151023,"semantic_template":"qf-density15-fraction-sum-naive-audit/[\"16/3\",\"11/5\"]"},"fields":{"title":"True Fraction Sum versus Naive Representation Addition","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=10/2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).","answer":{"kind":"expression","value":"(16/3, Yes, 11/5, -47/15, No)","explanation":"Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (16/3, Yes, 11/5, -47/15, No)."},"skills":["True Fraction Sum versus Naive Representation Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0024","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"516de305e836b49abcbf60106021043ab45442396466cca01555f3684cb8ff48","family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151024,"semantic_template":"qf-density15-fraction-sum-naive-audit/[\"35/6\",\"-10/1\"]"},"fields":{"title":"True Fraction Sum versus Naive Representation Addition","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-11/-2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).","answer":{"kind":"expression","value":"(35/6, Yes, -10/1, -95/6, No)","explanation":"Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (35/6, Yes, -10/1, -95/6, No)."},"skills":["True Fraction Sum versus Naive Representation Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0025","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39734f43e47164da13f5aa8db142321d518c8768588ed349c218087a2bbc4b35","family_id":"qf-density15-fraction-sum-naive-audit","seed":202610151025,"semantic_template":"qf-density15-fraction-sum-naive-audit/[\"19/3\",\"13/5\"]"},"fields":{"title":"True Fraction Sum versus Naive Representation Addition","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=12/2, b=1/3. The naive candidate adds the original numerators and original denominators separately, before reduction. Give (true sum, is naive addition defined?, naive value or undefined, naive−true defect or undefined, is naive addition correct?).","answer":{"kind":"expression","value":"(19/3, Yes, 13/5, -56/15, No)","explanation":"Compute the true sum by common denominators; separately guard the original denominator sum before evaluating the naive candidate. Adding numerators and denominators separately is not fraction addition. The exact result, retaining the requested order, is (19/3, Yes, 13/5, -56/15, No)."},"skills":["True Fraction Sum versus Naive Representation Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0026","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f872c925683ad0485e4da6ae7707caaaeb407fab70ea6cd33c28d259716365f","family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151026,"semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"0/1\",\"0/1\",\"0/1\"]"},"fields":{"title":"Ordered Fraction Differences and Endpoint Check","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[0/1, 0/1, 0/1, 0/1]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.","answer":{"kind":"expression","value":"([0/1, 0/1, 0/1], 0/1, 0/1, Yes)","explanation":"Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([0/1, 0/1, 0/1], 0/1, 0/1, Yes)."},"skills":["Ordered Fraction Differences and Endpoint Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0027","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20372b44375a9cc8671dcee52766807d68216080ad4106a6d6d9f48227fcef60","family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151027,"semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"1/1\",\"-1/1\",\"1/1\"]"},"fields":{"title":"Ordered Fraction Differences and Endpoint Check","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[0/1, 1/1, 0/1, 1/1]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.","answer":{"kind":"expression","value":"([1/1, -1/1, 1/1], 1/1, 1/1, Yes)","explanation":"Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([1/1, -1/1, 1/1], 1/1, 1/1, Yes)."},"skills":["Ordered Fraction Differences and Endpoint Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0028","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4febbb2e484e7b85fbd889918811405714cfe87f99892b538a8c616d854aaa12","family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151028,"semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"10/3\",\"1/2\",\"1/1\"]"},"fields":{"title":"Ordered Fraction Differences and Endpoint Check","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[-10/3, 0/1, 1/2, 3/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.","answer":{"kind":"expression","value":"([10/3, 1/2, 1/1], 29/6, 29/6, Yes)","explanation":"Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([10/3, 1/2, 1/1], 29/6, 29/6, Yes)."},"skills":["Ordered Fraction Differences and Endpoint Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0029","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5c9aefd13990fd63cae702146f2240c185268577eb86a1dab18b3fbd3d8e409","family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151029,"semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"3/1\",\"1/2\",\"3/2\"]"},"fields":{"title":"Ordered Fraction Differences and Endpoint Check","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[9/-3, 0/1, 1/2, 4/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.","answer":{"kind":"expression","value":"([3/1, 1/2, 3/2], 5/1, 5/1, Yes)","explanation":"Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([3/1, 1/2, 3/2], 5/1, 5/1, Yes)."},"skills":["Ordered Fraction Differences and Endpoint Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0030","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d96536eda5dee62a5f94e76104320589e333be9e8c159844b21e03b3321869e8","family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151030,"semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"8/3\",\"1/2\",\"2/1\"]"},"fields":{"title":"Ordered Fraction Differences and Endpoint Check","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[-8/3, 0/1, 1/2, 5/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.","answer":{"kind":"expression","value":"([8/3, 1/2, 2/1], 31/6, 31/6, Yes)","explanation":"Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([8/3, 1/2, 2/1], 31/6, 31/6, Yes)."},"skills":["Ordered Fraction Differences and Endpoint Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0031","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"44d62266409c89ca0d775d57e0b78d7cb6c60d1ddf132d93d82669d8bae6b8db","family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151031,"semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"7/3\",\"1/2\",\"5/2\"]"},"fields":{"title":"Ordered Fraction Differences and Endpoint Check","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[7/-3, 0/1, 1/2, 6/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.","answer":{"kind":"expression","value":"([7/3, 1/2, 5/2], 16/3, 16/3, Yes)","explanation":"Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([7/3, 1/2, 5/2], 16/3, 16/3, Yes)."},"skills":["Ordered Fraction Differences and Endpoint Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0032","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f0ffe713973ab4a84c3182763a173383eb61a57619ecb0be16051263490d9d7","family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151032,"semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"2/1\",\"1/2\",\"3/1\"]"},"fields":{"title":"Ordered Fraction Differences and Endpoint Check","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[-6/3, 0/1, 1/2, 7/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.","answer":{"kind":"expression","value":"([2/1, 1/2, 3/1], 11/2, 11/2, Yes)","explanation":"Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([2/1, 1/2, 3/1], 11/2, 11/2, Yes)."},"skills":["Ordered Fraction Differences and Endpoint Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0033","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d64339b739c62af6cf1b8203f363a0c0e7805d895ac33cb1b977b27c3aa5dbe3","family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151033,"semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"5/3\",\"1/2\",\"7/2\"]"},"fields":{"title":"Ordered Fraction Differences and Endpoint Check","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[5/-3, 0/1, 1/2, 8/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.","answer":{"kind":"expression","value":"([5/3, 1/2, 7/2], 17/3, 17/3, Yes)","explanation":"Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([5/3, 1/2, 7/2], 17/3, 17/3, Yes)."},"skills":["Ordered Fraction Differences and Endpoint Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0034","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f4a225b63d041efce0a15927ec4f82bc2bf6b2a6ec6e01b6396c188705a67bc","family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151034,"semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"4/3\",\"1/2\",\"4/1\"]"},"fields":{"title":"Ordered Fraction Differences and Endpoint Check","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[-4/3, 0/1, 1/2, 9/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.","answer":{"kind":"expression","value":"([4/3, 1/2, 4/1], 35/6, 35/6, Yes)","explanation":"Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([4/3, 1/2, 4/1], 35/6, 35/6, Yes)."},"skills":["Ordered Fraction Differences and Endpoint Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0035","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19740737fb51aeecc79f42eb948f8736aba1e81144611d0b604f120bfcf63bc3","family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151035,"semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"1/1\",\"1/2\",\"9/2\"]"},"fields":{"title":"Ordered Fraction Differences and Endpoint Check","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[3/-3, 0/1, 1/2, 10/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.","answer":{"kind":"expression","value":"([1/1, 1/2, 9/2], 6/1, 6/1, Yes)","explanation":"Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([1/1, 1/2, 9/2], 6/1, 6/1, Yes)."},"skills":["Ordered Fraction Differences and Endpoint Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0036","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb224aa81f557ac8261d34790d17c97c2ea6a993a8000a127dadf22d7f95bea9","family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151036,"semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"2/3\",\"1/2\",\"5/1\"]"},"fields":{"title":"Ordered Fraction Differences and Endpoint Check","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[-2/3, 0/1, 1/2, 11/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.","answer":{"kind":"expression","value":"([2/3, 1/2, 5/1], 37/6, 37/6, Yes)","explanation":"Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([2/3, 1/2, 5/1], 37/6, 37/6, Yes)."},"skills":["Ordered Fraction Differences and Endpoint Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0037","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41ed3ecc3c19c3f9c4ff41d5357f5197dcaa2ffdaeb30c20a30fe5e03aaeaae3","family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151037,"semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"1/3\",\"1/2\",\"11/2\"]"},"fields":{"title":"Ordered Fraction Differences and Endpoint Check","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[1/-3, 0/1, 1/2, 12/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.","answer":{"kind":"expression","value":"([1/3, 1/2, 11/2], 19/3, 19/3, Yes)","explanation":"Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([1/3, 1/2, 11/2], 19/3, 19/3, Yes)."},"skills":["Ordered Fraction Differences and Endpoint Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0038","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57fcbba1c6a78e3d77b3ac0c6a8bb78417dae8496ff4734bdda820550d2b80cb","family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151038,"semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"0/1\",\"1/2\",\"6/1\"]"},"fields":{"title":"Ordered Fraction Differences and Endpoint Check","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[0/3, 0/1, 1/2, 13/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.","answer":{"kind":"expression","value":"([0/1, 1/2, 6/1], 13/2, 13/2, Yes)","explanation":"Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([0/1, 1/2, 6/1], 13/2, 13/2, Yes)."},"skills":["Ordered Fraction Differences and Endpoint Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0039","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91b0eb102e653bc4ade100b096c5354f055416afa7451363d4f504c83567ecdb","family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151039,"semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"-1/3\",\"1/2\",\"13/2\"]"},"fields":{"title":"Ordered Fraction Differences and Endpoint Check","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[-1/-3, 0/1, 1/2, 14/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.","answer":{"kind":"expression","value":"([-1/3, 1/2, 13/2], 20/3, 20/3, Yes)","explanation":"Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([-1/3, 1/2, 13/2], 20/3, 20/3, Yes)."},"skills":["Ordered Fraction Differences and Endpoint Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0040","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cebd009b51fbadd76757349006bb0302cf4c781e14d6a0a9c877d68cf1183d79","family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151040,"semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"-2/3\",\"1/2\",\"7/1\"]"},"fields":{"title":"Ordered Fraction Differences and Endpoint Check","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[2/3, 0/1, 1/2, 15/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.","answer":{"kind":"expression","value":"([-2/3, 1/2, 7/1], 41/6, 41/6, Yes)","explanation":"Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([-2/3, 1/2, 7/1], 41/6, 41/6, Yes)."},"skills":["Ordered Fraction Differences and Endpoint Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0041","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"911b21b7ccd7c2bdc918c98865b1347b7ea892c6b912d9fb0f89bf7e126c5207","family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151041,"semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"-1/1\",\"1/2\",\"15/2\"]"},"fields":{"title":"Ordered Fraction Differences and Endpoint Check","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[-3/-3, 0/1, 1/2, 16/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.","answer":{"kind":"expression","value":"([-1/1, 1/2, 15/2], 7/1, 7/1, Yes)","explanation":"Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([-1/1, 1/2, 15/2], 7/1, 7/1, Yes)."},"skills":["Ordered Fraction Differences and Endpoint Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0042","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e29d2ee643639b1b0288b286323e8652168863a7058e2343d5e42f4f8baa703","family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151042,"semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"-4/3\",\"1/2\",\"8/1\"]"},"fields":{"title":"Ordered Fraction Differences and Endpoint Check","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[4/3, 0/1, 1/2, 17/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.","answer":{"kind":"expression","value":"([-4/3, 1/2, 8/1], 43/6, 43/6, Yes)","explanation":"Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([-4/3, 1/2, 8/1], 43/6, 43/6, Yes)."},"skills":["Ordered Fraction Differences and Endpoint Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0043","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"10071c8497b005213192b10494e33bc56fc8f1856f2133ef1a3c26740f71afb3","family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151043,"semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"-5/3\",\"1/2\",\"17/2\"]"},"fields":{"title":"Ordered Fraction Differences and Endpoint Check","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[-5/-3, 0/1, 1/2, 18/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.","answer":{"kind":"expression","value":"([-5/3, 1/2, 17/2], 22/3, 22/3, Yes)","explanation":"Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([-5/3, 1/2, 17/2], 22/3, 22/3, Yes)."},"skills":["Ordered Fraction Differences and Endpoint Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0044","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"154d3fa0f787d9dbbba78daab42a5f6b64493a9e37cede7d4bed06116b5545cc","family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151044,"semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"-2/1\",\"1/2\",\"9/1\"]"},"fields":{"title":"Ordered Fraction Differences and Endpoint Check","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[6/3, 0/1, 1/2, 19/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.","answer":{"kind":"expression","value":"([-2/1, 1/2, 9/1], 15/2, 15/2, Yes)","explanation":"Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([-2/1, 1/2, 9/1], 15/2, 15/2, Yes)."},"skills":["Ordered Fraction Differences and Endpoint Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0045","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb912e5879036f0c9219b63b213fac305de96bef2799d6f318940b7f536c608e","family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151045,"semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"-7/3\",\"1/2\",\"19/2\"]"},"fields":{"title":"Ordered Fraction Differences and Endpoint Check","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[-7/-3, 0/1, 1/2, 20/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.","answer":{"kind":"expression","value":"([-7/3, 1/2, 19/2], 23/3, 23/3, Yes)","explanation":"Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([-7/3, 1/2, 19/2], 23/3, 23/3, Yes)."},"skills":["Ordered Fraction Differences and Endpoint Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0046","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"303ce981a653c37685e3f4b379379272d10a6095ac665f75d85558f77b5a9dc8","family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151046,"semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"-8/3\",\"1/2\",\"10/1\"]"},"fields":{"title":"Ordered Fraction Differences and Endpoint Check","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[8/3, 0/1, 1/2, 21/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.","answer":{"kind":"expression","value":"([-8/3, 1/2, 10/1], 47/6, 47/6, Yes)","explanation":"Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([-8/3, 1/2, 10/1], 47/6, 47/6, Yes)."},"skills":["Ordered Fraction Differences and Endpoint Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0047","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dfbddfe721c104878a234df843983c576dd442c464add5ec5715b0af3457f6b5","family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151047,"semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"-3/1\",\"1/2\",\"21/2\"]"},"fields":{"title":"Ordered Fraction Differences and Endpoint Check","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[-9/-3, 0/1, 1/2, 22/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.","answer":{"kind":"expression","value":"([-3/1, 1/2, 21/2], 8/1, 8/1, Yes)","explanation":"Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([-3/1, 1/2, 21/2], 8/1, 8/1, Yes)."},"skills":["Ordered Fraction Differences and Endpoint Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0048","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5cadefb1ba9842143b4296b44815554110c2224ded71c675abc333a92abeb1e","family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151048,"semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"-10/3\",\"1/2\",\"11/1\"]"},"fields":{"title":"Ordered Fraction Differences and Endpoint Check","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[10/3, 0/1, 1/2, 23/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.","answer":{"kind":"expression","value":"([-10/3, 1/2, 11/1], 49/6, 49/6, Yes)","explanation":"Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([-10/3, 1/2, 11/1], 49/6, 49/6, Yes)."},"skills":["Ordered Fraction Differences and Endpoint Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0049","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf901153054ae8b94f8174bc7049ee0eefa47d7ae3bb7329f075a67b1d359889","family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151049,"semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"-11/3\",\"1/2\",\"23/2\"]"},"fields":{"title":"Ordered Fraction Differences and Endpoint Check","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[-11/-3, 0/1, 1/2, 24/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.","answer":{"kind":"expression","value":"([-11/3, 1/2, 23/2], 25/3, 25/3, Yes)","explanation":"Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([-11/3, 1/2, 23/2], 25/3, 25/3, Yes)."},"skills":["Ordered Fraction Differences and Endpoint Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0050","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd2618ed20c6faeb2107d955fecee42fd694987a484471cabe8ff4406e169696","family_id":"qf-density15-fraction-adjacent-difference-chain","seed":202610151050,"semantic_template":"qf-density15-fraction-adjacent-difference-chain/[\"-4/1\",\"1/2\",\"12/1\"]"},"fields":{"title":"Ordered Fraction Differences and Endpoint Check","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Four ordered values=[12/3, 0/1, 1/2, 25/2]. Give (three following-minus-previous differences, their sum, last−first, are those last two results equal?); do not sort.","answer":{"kind":"expression","value":"([-4/1, 1/2, 12/1], 17/2, 17/2, Yes)","explanation":"Subtract adjacent values in original order, add the differences, and independently compute the endpoint difference. Finite arithmetic is not a quotient well-definedness proof. The exact result, retaining the requested order, is ([-4/1, 1/2, 12/1], 17/2, 17/2, Yes)."},"skills":["Ordered Fraction Differences and Endpoint Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0051","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2104d56cb03afd2998fbe31e31c357836de2849e6edafd65bf720df39b20006f","family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151051,"semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"0/1\",\"0/1\"]"},"fields":{"title":"Fraction Product and Two Reciprocal Guards","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=0/-2, b=0/3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).","answer":{"kind":"expression","value":"(0/1, [No, No], [undefined, undefined], undefined)","explanation":"Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (0/1, [No, No], [undefined, undefined], undefined)."},"skills":["Fraction Product and Two Reciprocal Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0052","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77448389192ffcaa8bb5e154f71990758d02ed58b800a8cfe61d09692b274cc3","family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151052,"semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"0/1\",\"2/3\"]"},"fields":{"title":"Fraction Product and Two Reciprocal Guards","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=0/1, b=2/3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).","answer":{"kind":"expression","value":"(0/1, [No, Yes], [undefined, 3/2], undefined)","explanation":"Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (0/1, [No, Yes], [undefined, 3/2], undefined)."},"skills":["Fraction Product and Two Reciprocal Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0053","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c38f0d0c1c0a5875f4a9405090d4ea6040bd52cef616783345551e2a4e029d99","family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151053,"semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"-1/2\",\"0/1\"]"},"fields":{"title":"Fraction Product and Two Reciprocal Guards","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-1/2, b=0/-5. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).","answer":{"kind":"expression","value":"(0/1, [Yes, No], [-2/1, undefined], undefined)","explanation":"Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (0/1, [Yes, No], [-2/1, undefined], undefined)."},"skills":["Fraction Product and Two Reciprocal Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0054","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b62f0163dda20276e7cf37f1623105b009615fd323c7a831734aa75cccb93b40","family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151054,"semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"-3/1\",\"-2/3\"]"},"fields":{"title":"Fraction Product and Two Reciprocal Guards","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-9/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).","answer":{"kind":"expression","value":"(2/1, [Yes, Yes], [-1/3, -3/2], 1/2)","explanation":"Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (2/1, [Yes, Yes], [-1/3, -3/2], 1/2)."},"skills":["Fraction Product and Two Reciprocal Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0055","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6aa4326f0eb732741a7974fb300e290c6722ee0b5d037fdf14c69cc6d9f5468e","family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151055,"semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"-8/3\",\"-2/3\"]"},"fields":{"title":"Fraction Product and Two Reciprocal Guards","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-8/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).","answer":{"kind":"expression","value":"(16/9, [Yes, Yes], [-3/8, -3/2], 9/16)","explanation":"Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (16/9, [Yes, Yes], [-3/8, -3/2], 9/16)."},"skills":["Fraction Product and Two Reciprocal Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0056","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb8976bd82910d51415434f07b9b33901bdd79cec75c811976a20dc35d09d97c","family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151056,"semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"-7/3\",\"-2/3\"]"},"fields":{"title":"Fraction Product and Two Reciprocal Guards","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-7/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).","answer":{"kind":"expression","value":"(14/9, [Yes, Yes], [-3/7, -3/2], 9/14)","explanation":"Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (14/9, [Yes, Yes], [-3/7, -3/2], 9/14)."},"skills":["Fraction Product and Two Reciprocal Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0057","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ddb82a9338b69e0cdfc5b4760a822cf00282681090c8e34645b4bf56a1c1eb4c","family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151057,"semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"-2/1\",\"-2/3\"]"},"fields":{"title":"Fraction Product and Two Reciprocal Guards","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-6/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).","answer":{"kind":"expression","value":"(4/3, [Yes, Yes], [-1/2, -3/2], 3/4)","explanation":"Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (4/3, [Yes, Yes], [-1/2, -3/2], 3/4)."},"skills":["Fraction Product and Two Reciprocal Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0058","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9506dc8ba0af7ca0e1071ec67e902e12142b6ed87cfe90d5867962733854fe0d","family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151058,"semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"-5/3\",\"-2/3\"]"},"fields":{"title":"Fraction Product and Two Reciprocal Guards","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-5/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).","answer":{"kind":"expression","value":"(10/9, [Yes, Yes], [-3/5, -3/2], 9/10)","explanation":"Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (10/9, [Yes, Yes], [-3/5, -3/2], 9/10)."},"skills":["Fraction Product and Two Reciprocal Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0059","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"825a4cefb6c2723a1efd2d85a410930bbd0ac873d71555ff9b3494f1a9604834","family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151059,"semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"-4/3\",\"-2/3\"]"},"fields":{"title":"Fraction Product and Two Reciprocal Guards","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-4/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).","answer":{"kind":"expression","value":"(8/9, [Yes, Yes], [-3/4, -3/2], 9/8)","explanation":"Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (8/9, [Yes, Yes], [-3/4, -3/2], 9/8)."},"skills":["Fraction Product and Two Reciprocal Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0060","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8027584c382b94f843a5f83a3e24c98afccc7e746a7244d43c092fa97781876d","family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151060,"semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"-1/1\",\"-2/3\"]"},"fields":{"title":"Fraction Product and Two Reciprocal Guards","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-3/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).","answer":{"kind":"expression","value":"(2/3, [Yes, Yes], [-1/1, -3/2], 3/2)","explanation":"Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (2/3, [Yes, Yes], [-1/1, -3/2], 3/2)."},"skills":["Fraction Product and Two Reciprocal Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0061","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e187a4d7b4618635c40ec2740d1d67db62115915f9b69f188c5394f24acaa75","family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151061,"semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"-2/3\",\"-2/3\"]"},"fields":{"title":"Fraction Product and Two Reciprocal Guards","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-2/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).","answer":{"kind":"expression","value":"(4/9, [Yes, Yes], [-3/2, -3/2], 9/4)","explanation":"Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (4/9, [Yes, Yes], [-3/2, -3/2], 9/4)."},"skills":["Fraction Product and Two Reciprocal Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0062","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"de6914277ac3aaea7adfccc76fe6a8580f30f935553d1d3a247261e6a61cf289","family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151062,"semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"-1/3\",\"-2/3\"]"},"fields":{"title":"Fraction Product and Two Reciprocal Guards","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=-1/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).","answer":{"kind":"expression","value":"(2/9, [Yes, Yes], [-3/1, -3/2], 9/2)","explanation":"Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (2/9, [Yes, Yes], [-3/1, -3/2], 9/2)."},"skills":["Fraction Product and Two Reciprocal Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0063","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae8defd19be7dca30d453c20bd93ba737c971c20c54f689d2491deb032b876e8","family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151063,"semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"0/1\",\"-2/3\"]"},"fields":{"title":"Fraction Product and Two Reciprocal Guards","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=0/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).","answer":{"kind":"expression","value":"(0/1, [No, Yes], [undefined, -3/2], undefined)","explanation":"Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (0/1, [No, Yes], [undefined, -3/2], undefined)."},"skills":["Fraction Product and Two Reciprocal Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0064","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c2fd6142878644828d9334b76e439b1d05c394af7fabec21fcd8b636541869f","family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151064,"semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"1/3\",\"-2/3\"]"},"fields":{"title":"Fraction Product and Two Reciprocal Guards","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=1/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).","answer":{"kind":"expression","value":"(-2/9, [Yes, Yes], [3/1, -3/2], -9/2)","explanation":"Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (-2/9, [Yes, Yes], [3/1, -3/2], -9/2)."},"skills":["Fraction Product and Two Reciprocal Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0065","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8419856b468f53dbeaf61c737959a7f0295fe6fde5b39488595579205133e00d","family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151065,"semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"2/3\",\"-2/3\"]"},"fields":{"title":"Fraction Product and Two Reciprocal Guards","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=2/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).","answer":{"kind":"expression","value":"(-4/9, [Yes, Yes], [3/2, -3/2], -9/4)","explanation":"Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (-4/9, [Yes, Yes], [3/2, -3/2], -9/4)."},"skills":["Fraction Product and Two Reciprocal Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0066","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4ec71b34184d40c69a1d82331877ba66e2e1a77c1cbc44093a9b77b3b5f86a3","family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151066,"semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"1/1\",\"-2/3\"]"},"fields":{"title":"Fraction Product and Two Reciprocal Guards","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=3/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).","answer":{"kind":"expression","value":"(-2/3, [Yes, Yes], [1/1, -3/2], -3/2)","explanation":"Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (-2/3, [Yes, Yes], [1/1, -3/2], -3/2)."},"skills":["Fraction Product and Two Reciprocal Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0067","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6fd3ac9ef6e92ae67b0f182c8277a09b46d95f7dada4416c61b207d05952cd69","family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151067,"semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"4/3\",\"-2/3\"]"},"fields":{"title":"Fraction Product and Two Reciprocal Guards","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=4/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).","answer":{"kind":"expression","value":"(-8/9, [Yes, Yes], [3/4, -3/2], -9/8)","explanation":"Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (-8/9, [Yes, Yes], [3/4, -3/2], -9/8)."},"skills":["Fraction Product and Two Reciprocal Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0068","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79a8303a725b7088d0c129b165790fe6b26d16ea34844e923c5716ef3b1daecb","family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151068,"semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"5/3\",\"-2/3\"]"},"fields":{"title":"Fraction Product and Two Reciprocal Guards","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=5/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).","answer":{"kind":"expression","value":"(-10/9, [Yes, Yes], [3/5, -3/2], -9/10)","explanation":"Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (-10/9, [Yes, Yes], [3/5, -3/2], -9/10)."},"skills":["Fraction Product and Two Reciprocal Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0069","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b354f60f7b95f15735026b3fc465972caf3a54c2abaa02efc4ee9170b0e9c0a","family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151069,"semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"2/1\",\"-2/3\"]"},"fields":{"title":"Fraction Product and Two Reciprocal Guards","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=6/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).","answer":{"kind":"expression","value":"(-4/3, [Yes, Yes], [1/2, -3/2], -3/4)","explanation":"Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (-4/3, [Yes, Yes], [1/2, -3/2], -3/4)."},"skills":["Fraction Product and Two Reciprocal Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0070","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2eb96f26394cba2013596c7a8d2ad0665e508bc13382d28b92d9e1e2e7fb9f0","family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151070,"semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"7/3\",\"-2/3\"]"},"fields":{"title":"Fraction Product and Two Reciprocal Guards","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=7/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).","answer":{"kind":"expression","value":"(-14/9, [Yes, Yes], [3/7, -3/2], -9/14)","explanation":"Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (-14/9, [Yes, Yes], [3/7, -3/2], -9/14)."},"skills":["Fraction Product and Two Reciprocal Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0071","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8d1aecf5dba8a0a6a309a55a9ab443f1780bf087f4855ca298539a11a84483d","family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151071,"semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"8/3\",\"-2/3\"]"},"fields":{"title":"Fraction Product and Two Reciprocal Guards","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=8/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).","answer":{"kind":"expression","value":"(-16/9, [Yes, Yes], [3/8, -3/2], -9/16)","explanation":"Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (-16/9, [Yes, Yes], [3/8, -3/2], -9/16)."},"skills":["Fraction Product and Two Reciprocal Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0072","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e48255591eb08d0585e74849a3dcefefc985c98d6187580c1e9ed5d74c6345d8","family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151072,"semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"3/1\",\"-2/3\"]"},"fields":{"title":"Fraction Product and Two Reciprocal Guards","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=9/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).","answer":{"kind":"expression","value":"(-2/1, [Yes, Yes], [1/3, -3/2], -1/2)","explanation":"Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (-2/1, [Yes, Yes], [1/3, -3/2], -1/2)."},"skills":["Fraction Product and Two Reciprocal Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0073","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c93445f0d999d6a15a92c8a2425e3004e20d753100c116503c8637fedbbe7a0b","family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151073,"semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"10/3\",\"-2/3\"]"},"fields":{"title":"Fraction Product and Two Reciprocal Guards","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=10/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).","answer":{"kind":"expression","value":"(-20/9, [Yes, Yes], [3/10, -3/2], -9/20)","explanation":"Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (-20/9, [Yes, Yes], [3/10, -3/2], -9/20)."},"skills":["Fraction Product and Two Reciprocal Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0074","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8261310e1505acc7cd49a16f6c1768549bc5f83730add3950e52e8113b846113","family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151074,"semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"11/3\",\"-2/3\"]"},"fields":{"title":"Fraction Product and Two Reciprocal Guards","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=11/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).","answer":{"kind":"expression","value":"(-22/9, [Yes, Yes], [3/11, -3/2], -9/22)","explanation":"Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (-22/9, [Yes, Yes], [3/11, -3/2], -9/22)."},"skills":["Fraction Product and Two Reciprocal Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0075","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e48acc6da8dcb22ebb0b1c592dfaea697c4afe5900de0bfd964636b2b30fb34c","family_id":"qf-density15-fraction-inverse-pair-guard","seed":202610151075,"semantic_template":"qf-density15-fraction-inverse-pair-guard/[\"4/1\",\"-2/3\"]"},"fields":{"title":"Fraction Product and Two Reciprocal Guards","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. a=12/3, b=2/-3. Give (product, nonzero eligibility list for a and b separately, their reciprocals or undefined, reciprocal product or undefined).","answer":{"kind":"expression","value":"(-8/3, [Yes, Yes], [1/4, -3/2], -3/8)","explanation":"Check each operand’s value for zero before taking its reciprocal; exchange normalized numerator/denominator only for a nonzero value. The exact result, retaining the requested order, is (-8/3, [Yes, Yes], [1/4, -3/2], -3/8)."},"skills":["Fraction Product and Two Reciprocal Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0076","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64e22b5de05f2660f5aa65cffda5c7a7281af22687d7a302316be1f6710b4ccc","family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151076,"semantic_template":"qf-density15-fraction-two-scaled-representations/[\"0/1\",[[\"0\",\"0\"],[\"0\",\"2\"]]]"},"fields":{"title":"Two Raw Scaled Fraction Representations","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=0, d=1; ordered scale factors [s,t]=[0,2]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.","answer":{"kind":"expression","value":"(0/1, [[0, 0], [0, 2]], [No, Yes], [undefined, 0/1])","explanation":"A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (0/1, [[0, 0], [0, 2]], [No, Yes], [undefined, 0/1])."},"skills":["Two Raw Scaled Fraction Representations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0077","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee73bb9d2b186616db083be3610b0b0927e95e8f29c69edda162ee15f9a9e568","family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151077,"semantic_template":"qf-density15-fraction-two-scaled-representations/[\"-2/3\",[[\"0\",\"0\"],[\"-2\",\"3\"]]]"},"fields":{"title":"Two Raw Scaled Fraction Representations","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=2, d=-3; ordered scale factors [s,t]=[0,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.","answer":{"kind":"expression","value":"(-2/3, [[0, 0], [-2, 3]], [No, Yes], [undefined, -2/3])","explanation":"A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (-2/3, [[0, 0], [-2, 3]], [No, Yes], [undefined, -2/3])."},"skills":["Two Raw Scaled Fraction Representations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0078","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2dd9393fe8053372ae4517397b31f014bb0f99a89616c7cfc4b5de45343d9388","family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151078,"semantic_template":"qf-density15-fraction-two-scaled-representations/[\"1/2\",[[\"1\",\"2\"],[\"0\",\"0\"]]]"},"fields":{"title":"Two Raw Scaled Fraction Representations","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=1, d=2; ordered scale factors [s,t]=[1,0]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.","answer":{"kind":"expression","value":"(1/2, [[1, 2], [0, 0]], [Yes, No], [1/2, undefined])","explanation":"A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (1/2, [[1, 2], [0, 0]], [Yes, No], [1/2, undefined])."},"skills":["Two Raw Scaled Fraction Representations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0079","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b6d1249e329299f1ee2f780c243b9e2268ead7f9abf8ededc55586218a5014a8","family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151079,"semantic_template":"qf-density15-fraction-two-scaled-representations/[\"-9007199254740991/7\",[[\"81129638414606663681390495662081\",\"-63050394783186937\"],[\"-18014398509481982\",\"14\"]]]"},"fields":{"title":"Two Raw Scaled Fraction Representations","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=9007199254740991, d=-7; ordered scale factors [s,t]=[9007199254740991,-2]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.","answer":{"kind":"expression","value":"(-9007199254740991/7, [[81129638414606663681390495662081, -63050394783186937], [-18014398509481982, 14]], [Yes, Yes], [-9007199254740991/7, -9007199254740991/7])","explanation":"A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (-9007199254740991/7, [[81129638414606663681390495662081, -63050394783186937], [-18014398509481982, 14]], [Yes, Yes], [-9007199254740991/7, -9007199254740991/7])."},"skills":["Two Raw Scaled Fraction Representations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0080","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1682ea7c369cd4a29bea8d897bdce4ea57ebfd099e54110b0bde8241138953bc","family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151080,"semantic_template":"qf-density15-fraction-two-scaled-representations/[\"-4/1\",[[\"-16\",\"4\"],[\"8\",\"-2\"]]]"},"fields":{"title":"Two Raw Scaled Fraction Representations","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=-8, d=2; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.","answer":{"kind":"expression","value":"(-4/1, [[-16, 4], [8, -2]], [Yes, Yes], [-4/1, -4/1])","explanation":"A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (-4/1, [[-16, 4], [8, -2]], [Yes, Yes], [-4/1, -4/1])."},"skills":["Two Raw Scaled Fraction Representations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0081","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41faab37e21ed3e2858dd9e1c1125d6a0a429e83fd713698fc2b45769bb61d04","family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151081,"semantic_template":"qf-density15-fraction-two-scaled-representations/[\"7/3\",[[\"-14\",\"-6\"],[\"7\",\"3\"]]]"},"fields":{"title":"Two Raw Scaled Fraction Representations","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=-7, d=-3; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.","answer":{"kind":"expression","value":"(7/3, [[-14, -6], [7, 3]], [Yes, Yes], [7/3, 7/3])","explanation":"A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (7/3, [[-14, -6], [7, 3]], [Yes, Yes], [7/3, 7/3])."},"skills":["Two Raw Scaled Fraction Representations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0082","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"445be8a6d672c41ccffe6f03a32ee95662fc17b1b9ca2d9babb1c9d7f5e1a30e","family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151082,"semantic_template":"qf-density15-fraction-two-scaled-representations/[\"-3/1\",[[\"-12\",\"4\"],[\"6\",\"-2\"]]]"},"fields":{"title":"Two Raw Scaled Fraction Representations","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=-6, d=2; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.","answer":{"kind":"expression","value":"(-3/1, [[-12, 4], [6, -2]], [Yes, Yes], [-3/1, -3/1])","explanation":"A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (-3/1, [[-12, 4], [6, -2]], [Yes, Yes], [-3/1, -3/1])."},"skills":["Two Raw Scaled Fraction Representations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0083","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24fbac59a8d7f4cb87bf1731987db931b72a4579c51dae2b4e0db6b681f99c46","family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151083,"semantic_template":"qf-density15-fraction-two-scaled-representations/[\"5/3\",[[\"-10\",\"-6\"],[\"5\",\"3\"]]]"},"fields":{"title":"Two Raw Scaled Fraction Representations","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=-5, d=-3; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.","answer":{"kind":"expression","value":"(5/3, [[-10, -6], [5, 3]], [Yes, Yes], [5/3, 5/3])","explanation":"A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (5/3, [[-10, -6], [5, 3]], [Yes, Yes], [5/3, 5/3])."},"skills":["Two Raw Scaled Fraction Representations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0084","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f40daa08f9abc5bd37b3cf95d4dcbaf162e0cd5d55be0e3771ece6ed220bfeb","family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151084,"semantic_template":"qf-density15-fraction-two-scaled-representations/[\"-2/1\",[[\"-8\",\"4\"],[\"4\",\"-2\"]]]"},"fields":{"title":"Two Raw Scaled Fraction Representations","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=-4, d=2; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.","answer":{"kind":"expression","value":"(-2/1, [[-8, 4], [4, -2]], [Yes, Yes], [-2/1, -2/1])","explanation":"A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (-2/1, [[-8, 4], [4, -2]], [Yes, Yes], [-2/1, -2/1])."},"skills":["Two Raw Scaled Fraction Representations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0085","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"82fb3ae501fa5046dbea9ec3f944139ff8d63984b9d8d7f61bc95bf4687436ce","family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151085,"semantic_template":"qf-density15-fraction-two-scaled-representations/[\"1/1\",[[\"-6\",\"-6\"],[\"3\",\"3\"]]]"},"fields":{"title":"Two Raw Scaled Fraction Representations","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=-3, d=-3; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.","answer":{"kind":"expression","value":"(1/1, [[-6, -6], [3, 3]], [Yes, Yes], [1/1, 1/1])","explanation":"A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (1/1, [[-6, -6], [3, 3]], [Yes, Yes], [1/1, 1/1])."},"skills":["Two Raw Scaled Fraction Representations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0086","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c8b052eadc5cae21b47bea7661af4d6bc2244c4a8df5bbef84bbf3900374d52","family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151086,"semantic_template":"qf-density15-fraction-two-scaled-representations/[\"-1/1\",[[\"-4\",\"4\"],[\"2\",\"-2\"]]]"},"fields":{"title":"Two Raw Scaled Fraction Representations","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=-2, d=2; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.","answer":{"kind":"expression","value":"(-1/1, [[-4, 4], [2, -2]], [Yes, Yes], [-1/1, -1/1])","explanation":"A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (-1/1, [[-4, 4], [2, -2]], [Yes, Yes], [-1/1, -1/1])."},"skills":["Two Raw Scaled Fraction Representations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0087","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28c752c09141a5dbbf242918aab15ff8d6e39945fcfd3faad09518e0a5a53b7e","family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151087,"semantic_template":"qf-density15-fraction-two-scaled-representations/[\"1/3\",[[\"-2\",\"-6\"],[\"1\",\"3\"]]]"},"fields":{"title":"Two Raw Scaled Fraction Representations","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=-1, d=-3; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.","answer":{"kind":"expression","value":"(1/3, [[-2, -6], [1, 3]], [Yes, Yes], [1/3, 1/3])","explanation":"A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (1/3, [[-2, -6], [1, 3]], [Yes, Yes], [1/3, 1/3])."},"skills":["Two Raw Scaled Fraction Representations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0088","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00bbfa964c9137c86ab56af6d58d235823edca7090ad518c6bae808c984d8879","family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151088,"semantic_template":"qf-density15-fraction-two-scaled-representations/[\"0/1\",[[\"0\",\"4\"],[\"0\",\"-2\"]]]"},"fields":{"title":"Two Raw Scaled Fraction Representations","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=0, d=2; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.","answer":{"kind":"expression","value":"(0/1, [[0, 4], [0, -2]], [Yes, Yes], [0/1, 0/1])","explanation":"A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (0/1, [[0, 4], [0, -2]], [Yes, Yes], [0/1, 0/1])."},"skills":["Two Raw Scaled Fraction Representations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0089","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3bf4478e8144d753607622815f1baeceffd3a972a525452deb47ddf1148985a","family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151089,"semantic_template":"qf-density15-fraction-two-scaled-representations/[\"-1/3\",[[\"2\",\"-6\"],[\"-1\",\"3\"]]]"},"fields":{"title":"Two Raw Scaled Fraction Representations","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=1, d=-3; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.","answer":{"kind":"expression","value":"(-1/3, [[2, -6], [-1, 3]], [Yes, Yes], [-1/3, -1/3])","explanation":"A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (-1/3, [[2, -6], [-1, 3]], [Yes, Yes], [-1/3, -1/3])."},"skills":["Two Raw Scaled Fraction Representations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0090","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88496a365dab31b558fb0a136036a62450abf81430a8a0ea7e5b35bf2f4d656f","family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151090,"semantic_template":"qf-density15-fraction-two-scaled-representations/[\"1/1\",[[\"4\",\"4\"],[\"-2\",\"-2\"]]]"},"fields":{"title":"Two Raw Scaled Fraction Representations","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=2, d=2; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.","answer":{"kind":"expression","value":"(1/1, [[4, 4], [-2, -2]], [Yes, Yes], [1/1, 1/1])","explanation":"A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (1/1, [[4, 4], [-2, -2]], [Yes, Yes], [1/1, 1/1])."},"skills":["Two Raw Scaled Fraction Representations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0091","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b40e0caea23fbfb9cd40c83f5bfb374c7c16586fc111e63e22e58ae53ffddac9","family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151091,"semantic_template":"qf-density15-fraction-two-scaled-representations/[\"-1/1\",[[\"6\",\"-6\"],[\"-3\",\"3\"]]]"},"fields":{"title":"Two Raw Scaled Fraction Representations","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=3, d=-3; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.","answer":{"kind":"expression","value":"(-1/1, [[6, -6], [-3, 3]], [Yes, Yes], [-1/1, -1/1])","explanation":"A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (-1/1, [[6, -6], [-3, 3]], [Yes, Yes], [-1/1, -1/1])."},"skills":["Two Raw Scaled Fraction Representations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0092","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f925e1b513977dcbf4a1d00cecc1009d6fd58bf5f570198fa5840e3aac59fce7","family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151092,"semantic_template":"qf-density15-fraction-two-scaled-representations/[\"2/1\",[[\"8\",\"4\"],[\"-4\",\"-2\"]]]"},"fields":{"title":"Two Raw Scaled Fraction Representations","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=4, d=2; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.","answer":{"kind":"expression","value":"(2/1, [[8, 4], [-4, -2]], [Yes, Yes], [2/1, 2/1])","explanation":"A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (2/1, [[8, 4], [-4, -2]], [Yes, Yes], [2/1, 2/1])."},"skills":["Two Raw Scaled Fraction Representations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0093","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"897f1cf0342bd660b7df6f6f4be4f771df31d6e2d8d0eb685d72b8661e9d7d7f","family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151093,"semantic_template":"qf-density15-fraction-two-scaled-representations/[\"-5/3\",[[\"10\",\"-6\"],[\"-5\",\"3\"]]]"},"fields":{"title":"Two Raw Scaled Fraction Representations","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=5, d=-3; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.","answer":{"kind":"expression","value":"(-5/3, [[10, -6], [-5, 3]], [Yes, Yes], [-5/3, -5/3])","explanation":"A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (-5/3, [[10, -6], [-5, 3]], [Yes, Yes], [-5/3, -5/3])."},"skills":["Two Raw Scaled Fraction Representations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0094","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c05be753b5818a436eebcd1570da4755b88f37a92121a393d88da6a2eb8e5fb7","family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151094,"semantic_template":"qf-density15-fraction-two-scaled-representations/[\"3/1\",[[\"12\",\"4\"],[\"-6\",\"-2\"]]]"},"fields":{"title":"Two Raw Scaled Fraction Representations","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=6, d=2; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.","answer":{"kind":"expression","value":"(3/1, [[12, 4], [-6, -2]], [Yes, Yes], [3/1, 3/1])","explanation":"A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (3/1, [[12, 4], [-6, -2]], [Yes, Yes], [3/1, 3/1])."},"skills":["Two Raw Scaled Fraction Representations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0095","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73cd5d611c42f80aee7dc3d8eb378b231ac19f88e7782083166bd3978e1c670c","family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151095,"semantic_template":"qf-density15-fraction-two-scaled-representations/[\"-7/3\",[[\"14\",\"-6\"],[\"-7\",\"3\"]]]"},"fields":{"title":"Two Raw Scaled Fraction Representations","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=7, d=-3; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.","answer":{"kind":"expression","value":"(-7/3, [[14, -6], [-7, 3]], [Yes, Yes], [-7/3, -7/3])","explanation":"A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (-7/3, [[14, -6], [-7, 3]], [Yes, Yes], [-7/3, -7/3])."},"skills":["Two Raw Scaled Fraction Representations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0096","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b5e02c7ed8b4b3e585c6e50fcaf6ff1cc5efdb905f9bae213047ea36a3888d7","family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151096,"semantic_template":"qf-density15-fraction-two-scaled-representations/[\"4/1\",[[\"16\",\"4\"],[\"-8\",\"-2\"]]]"},"fields":{"title":"Two Raw Scaled Fraction Representations","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=8, d=2; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.","answer":{"kind":"expression","value":"(4/1, [[16, 4], [-8, -2]], [Yes, Yes], [4/1, 4/1])","explanation":"A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (4/1, [[16, 4], [-8, -2]], [Yes, Yes], [4/1, 4/1])."},"skills":["Two Raw Scaled Fraction Representations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0097","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53c5649952c924d6792d3000aaff5d188dc3174f313f0d89763c0aea5d53d009","family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151097,"semantic_template":"qf-density15-fraction-two-scaled-representations/[\"-3/1\",[[\"18\",\"-6\"],[\"-9\",\"3\"]]]"},"fields":{"title":"Two Raw Scaled Fraction Representations","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=9, d=-3; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.","answer":{"kind":"expression","value":"(-3/1, [[18, -6], [-9, 3]], [Yes, Yes], [-3/1, -3/1])","explanation":"A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (-3/1, [[18, -6], [-9, 3]], [Yes, Yes], [-3/1, -3/1])."},"skills":["Two Raw Scaled Fraction Representations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0098","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da69e6fc87d39d70004e6bcfa3b990b77a50887993e0e92b413ecd331a6b64e6","family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151098,"semantic_template":"qf-density15-fraction-two-scaled-representations/[\"5/1\",[[\"20\",\"4\"],[\"-10\",\"-2\"]]]"},"fields":{"title":"Two Raw Scaled Fraction Representations","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=10, d=2; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.","answer":{"kind":"expression","value":"(5/1, [[20, 4], [-10, -2]], [Yes, Yes], [5/1, 5/1])","explanation":"A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (5/1, [[20, 4], [-10, -2]], [Yes, Yes], [5/1, 5/1])."},"skills":["Two Raw Scaled Fraction Representations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0099","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da7fbdd92a9dee8468092de937575816a74bbf049d6ea0d0c96c27fb2234ffc4","family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151099,"semantic_template":"qf-density15-fraction-two-scaled-representations/[\"-11/3\",[[\"22\",\"-6\"],[\"-11\",\"3\"]]]"},"fields":{"title":"Two Raw Scaled Fraction Representations","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=11, d=-3; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.","answer":{"kind":"expression","value":"(-11/3, [[22, -6], [-11, 3]], [Yes, Yes], [-11/3, -11/3])","explanation":"A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (-11/3, [[22, -6], [-11, 3]], [Yes, Yes], [-11/3, -11/3])."},"skills":["Two Raw Scaled Fraction Representations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261011-advance-20261015-r01-fraction-0100","batch_id":"batch-20261011-advance-20261015-r01-fraction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85064131af8c65825eb90597e316a7f351556ea8037a694627c2191119c203e5","family_id":"qf-density15-fraction-two-scaled-representations","seed":202610151100,"semantic_template":"qf-density15-fraction-two-scaled-representations/[\"6/1\",[[\"24\",\"4\"],[\"-12\",\"-2\"]]]"},"fields":{"title":"Two Raw Scaled Fraction Representations","stem":"Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=12, d=2; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.","answer":{"kind":"expression","value":"(6/1, [[24, 4], [-12, -2]], [Yes, Yes], [6/1, 6/1])","explanation":"A nonzero scale preserves value; a zero scale creates 0/0, which is not a legal representation. Compute raw products before reducing legal representations. The exact result, retaining the requested order, is (6/1, [[24, 4], [-12, -2]], [Yes, Yes], [6/1, 6/1])."},"skills":["Two Raw Scaled Fraction Representations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}