{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0001","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b2366d313c24d92b973ff52999e54f808b817ac6dda9aa9369dfe22d5dfd6ec","family_id":"qf-density-primitive-coefficient","seed":202610095001,"semantic_template":"qf-density-primitive-coefficient/[0,1]"},"fields":{"title":"Specified Coefficient of the Zero-Constant Primitive","stem":"On all R, f has ascending coefficients [0], constant term first. For its zero-constant primitive F0, find the coefficient of x^1. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"0/1","explanation":"Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain 0/1. The order is ascending and the representative constant is zero."},"skills":["Specified Coefficient of the Zero-Constant Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0002","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05bb9a7830428d8e83b321a277841ecb42c5c7d6910ca09f28008037cb7decf8","family_id":"qf-density-primitive-coefficient","seed":202610095002,"semantic_template":"qf-density-primitive-coefficient/[0,4]"},"fields":{"title":"Specified Coefficient of the Zero-Constant Primitive","stem":"On all R, f has ascending coefficients [2], constant term first. For its zero-constant primitive F0, find the coefficient of x^4. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"0/1","explanation":"Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain 0/1. The order is ascending and the representative constant is zero."},"skills":["Specified Coefficient of the Zero-Constant Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0003","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b8b49e7242b12139e3f110fd30140c15f94a22e835d39bd64e2c98bcfd3494e","family_id":"qf-density-primitive-coefficient","seed":202610095003,"semantic_template":"qf-density-primitive-coefficient/[-4,3]"},"fields":{"title":"Specified Coefficient of the Zero-Constant Primitive","stem":"On all R, f has ascending coefficients [-10, 3, -4, 5], constant term first. For its zero-constant primitive F0, find the coefficient of x^3. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-4/3","explanation":"Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain -4/3. The order is ascending and the representative constant is zero."},"skills":["Specified Coefficient of the Zero-Constant Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0004","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3747626e0bfc1b95845b80b5f422f33b8d6033959d133751b73955312dfb764","family_id":"qf-density-primitive-coefficient","seed":202610095004,"semantic_template":"qf-density-primitive-coefficient/[-6,4]"},"fields":{"title":"Specified Coefficient of the Zero-Constant Primitive","stem":"On all R, f has ascending coefficients [-9, 4, -5, -6], constant term first. For its zero-constant primitive F0, find the coefficient of x^4. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-3/2","explanation":"Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain -3/2. The order is ascending and the representative constant is zero."},"skills":["Specified Coefficient of the Zero-Constant Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0005","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc3cd9ac3061d50a5f086cf2b70ace1b8d8490567e6c38664d03e3c8b1337e8a","family_id":"qf-density-primitive-coefficient","seed":202610095005,"semantic_template":"qf-density-primitive-coefficient/[-8,1]"},"fields":{"title":"Specified Coefficient of the Zero-Constant Primitive","stem":"On all R, f has ascending coefficients [-8, 5, -6, 7], constant term first. For its zero-constant primitive F0, find the coefficient of x^1. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-8/1","explanation":"Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain -8/1. The order is ascending and the representative constant is zero."},"skills":["Specified Coefficient of the Zero-Constant Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0006","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a1298ca7e12b96769cff1d75fb3a9f5f8df94770efd072958594a0e981421da","family_id":"qf-density-primitive-coefficient","seed":202610095006,"semantic_template":"qf-density-primitive-coefficient/[6,2]"},"fields":{"title":"Specified Coefficient of the Zero-Constant Primitive","stem":"On all R, f has ascending coefficients [-7, 6, -7, -8], constant term first. For its zero-constant primitive F0, find the coefficient of x^2. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"3/1","explanation":"Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain 3/1. The order is ascending and the representative constant is zero."},"skills":["Specified Coefficient of the Zero-Constant Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0007","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83234ef4368e369f7f6b0d93528da6c0c4d62d815e47fc0968df260c687216d1","family_id":"qf-density-primitive-coefficient","seed":202610095007,"semantic_template":"qf-density-primitive-coefficient/[-8,3]"},"fields":{"title":"Specified Coefficient of the Zero-Constant Primitive","stem":"On all R, f has ascending coefficients [-6, 7, -8, 9], constant term first. For its zero-constant primitive F0, find the coefficient of x^3. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-8/3","explanation":"Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain -8/3. The order is ascending and the representative constant is zero."},"skills":["Specified Coefficient of the Zero-Constant Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0008","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5419d97e207b5b0ba9908a130761a35b8f0a26f53c0ed8d23bbbb7d9117d2a98","family_id":"qf-density-primitive-coefficient","seed":202610095008,"semantic_template":"qf-density-primitive-coefficient/[-10,4]"},"fields":{"title":"Specified Coefficient of the Zero-Constant Primitive","stem":"On all R, f has ascending coefficients [-5, 8, -9, -10], constant term first. For its zero-constant primitive F0, find the coefficient of x^4. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-5/2","explanation":"Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain -5/2. The order is ascending and the representative constant is zero."},"skills":["Specified Coefficient of the Zero-Constant Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0009","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8113110e16331c748024eecbaf9f42faa94d6dc8cb227252b8e253cb72b647e0","family_id":"qf-density-primitive-coefficient","seed":202610095009,"semantic_template":"qf-density-primitive-coefficient/[-4,1]"},"fields":{"title":"Specified Coefficient of the Zero-Constant Primitive","stem":"On all R, f has ascending coefficients [-4, 9, -10, 11], constant term first. For its zero-constant primitive F0, find the coefficient of x^1. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-4/1","explanation":"Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain -4/1. The order is ascending and the representative constant is zero."},"skills":["Specified Coefficient of the Zero-Constant Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0010","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53f9fcc280c30b7e7779e36869fd47c6c388e224b30034e8cebf1168360dfdea","family_id":"qf-density-primitive-coefficient","seed":202610095010,"semantic_template":"qf-density-primitive-coefficient/[10,2]"},"fields":{"title":"Specified Coefficient of the Zero-Constant Primitive","stem":"On all R, f has ascending coefficients [-3, 10, -11, -12], constant term first. For its zero-constant primitive F0, find the coefficient of x^2. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"5/1","explanation":"Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain 5/1. The order is ascending and the representative constant is zero."},"skills":["Specified Coefficient of the Zero-Constant Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0011","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"331d4be9968054cc928044ebfee503af7d0593c9709006cc1e515fff737dced3","family_id":"qf-density-primitive-coefficient","seed":202610095011,"semantic_template":"qf-density-primitive-coefficient/[-12,3]"},"fields":{"title":"Specified Coefficient of the Zero-Constant Primitive","stem":"On all R, f has ascending coefficients [-2, 11, -12, 13], constant term first. For its zero-constant primitive F0, find the coefficient of x^3. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-4/1","explanation":"Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain -4/1. The order is ascending and the representative constant is zero."},"skills":["Specified Coefficient of the Zero-Constant Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0012","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef286d98e5072b790c315f588ef0372030ad53f55c0fb18a7b46c6e14420f7c3","family_id":"qf-density-primitive-coefficient","seed":202610095012,"semantic_template":"qf-density-primitive-coefficient/[-14,4]"},"fields":{"title":"Specified Coefficient of the Zero-Constant Primitive","stem":"On all R, f has ascending coefficients [-1, 12, -13, -14], constant term first. For its zero-constant primitive F0, find the coefficient of x^4. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-7/2","explanation":"Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain -7/2. The order is ascending and the representative constant is zero."},"skills":["Specified Coefficient of the Zero-Constant Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0013","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5cea0f578d47e27b22fcd633f35cd81dd94bbfa37c6931b6e76270f43f2def0","family_id":"qf-density-primitive-coefficient","seed":202610095013,"semantic_template":"qf-density-primitive-coefficient/[14,2]"},"fields":{"title":"Specified Coefficient of the Zero-Constant Primitive","stem":"On all R, f has ascending coefficients [1, 14, -15, -16], constant term first. For its zero-constant primitive F0, find the coefficient of x^2. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"7/1","explanation":"Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain 7/1. The order is ascending and the representative constant is zero."},"skills":["Specified Coefficient of the Zero-Constant Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0014","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b31b548288290969a0c6f16bc5cb05a83b2308eeca3157a4a958486b4027a47b","family_id":"qf-density-primitive-coefficient","seed":202610095014,"semantic_template":"qf-density-primitive-coefficient/[-16,3]"},"fields":{"title":"Specified Coefficient of the Zero-Constant Primitive","stem":"On all R, f has ascending coefficients [2, 15, -16, 17], constant term first. For its zero-constant primitive F0, find the coefficient of x^3. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-16/3","explanation":"Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain -16/3. The order is ascending and the representative constant is zero."},"skills":["Specified Coefficient of the Zero-Constant Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0015","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4499c9cc6a65b7060a2ab2ac65a156db9ae92c2c4bdeedb67a6ff370a522f42","family_id":"qf-density-primitive-coefficient","seed":202610095015,"semantic_template":"qf-density-primitive-coefficient/[-18,4]"},"fields":{"title":"Specified Coefficient of the Zero-Constant Primitive","stem":"On all R, f has ascending coefficients [3, 16, -17, -18], constant term first. For its zero-constant primitive F0, find the coefficient of x^4. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-9/2","explanation":"Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain -9/2. The order is ascending and the representative constant is zero."},"skills":["Specified Coefficient of the Zero-Constant Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0016","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2c0097eabcbd11ae2c917aa993a7a75e0436b5ba34c00c6a80743abb3f3ec8d","family_id":"qf-density-primitive-coefficient","seed":202610095016,"semantic_template":"qf-density-primitive-coefficient/[4,1]"},"fields":{"title":"Specified Coefficient of the Zero-Constant Primitive","stem":"On all R, f has ascending coefficients [4, 17, -18, 19], constant term first. For its zero-constant primitive F0, find the coefficient of x^1. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"4/1","explanation":"Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain 4/1. The order is ascending and the representative constant is zero."},"skills":["Specified Coefficient of the Zero-Constant Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0017","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b7db98569e0a8e0206d09f589b69bd908477a6fe8e8d7c3d30e9a941a3777c9","family_id":"qf-density-primitive-coefficient","seed":202610095017,"semantic_template":"qf-density-primitive-coefficient/[18,2]"},"fields":{"title":"Specified Coefficient of the Zero-Constant Primitive","stem":"On all R, f has ascending coefficients [5, 18, -19, -20], constant term first. For its zero-constant primitive F0, find the coefficient of x^2. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"9/1","explanation":"Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain 9/1. The order is ascending and the representative constant is zero."},"skills":["Specified Coefficient of the Zero-Constant Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0018","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f705b1e495dadd776295d7bb12e34c4cd00a74321bf1146ff51de86e442f7c4","family_id":"qf-density-primitive-coefficient","seed":202610095018,"semantic_template":"qf-density-primitive-coefficient/[-20,3]"},"fields":{"title":"Specified Coefficient of the Zero-Constant Primitive","stem":"On all R, f has ascending coefficients [6, 19, -20, 21], constant term first. For its zero-constant primitive F0, find the coefficient of x^3. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-20/3","explanation":"Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain -20/3. The order is ascending and the representative constant is zero."},"skills":["Specified Coefficient of the Zero-Constant Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0019","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8675167f8b31dafb449d85c25c53b10e124422c89809d26aec51003648e6631","family_id":"qf-density-primitive-coefficient","seed":202610095019,"semantic_template":"qf-density-primitive-coefficient/[-22,4]"},"fields":{"title":"Specified Coefficient of the Zero-Constant Primitive","stem":"On all R, f has ascending coefficients [7, 20, -21, -22], constant term first. For its zero-constant primitive F0, find the coefficient of x^4. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-11/2","explanation":"Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain -11/2. The order is ascending and the representative constant is zero."},"skills":["Specified Coefficient of the Zero-Constant Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0020","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c9a824fdfd3cce058ac21474af0e49a5f2642b05908c0f4f03f41cec4c99762","family_id":"qf-density-primitive-coefficient","seed":202610095020,"semantic_template":"qf-density-primitive-coefficient/[8,1]"},"fields":{"title":"Specified Coefficient of the Zero-Constant Primitive","stem":"On all R, f has ascending coefficients [8, 21, -22, 23], constant term first. For its zero-constant primitive F0, find the coefficient of x^1. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"8/1","explanation":"Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain 8/1. The order is ascending and the representative constant is zero."},"skills":["Specified Coefficient of the Zero-Constant Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0021","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a46692588e152b6600c2cffa7b6e0c8311c16bf5db149023dd5e3e1e4357db9","family_id":"qf-density-primitive-coefficient","seed":202610095021,"semantic_template":"qf-density-primitive-coefficient/[22,2]"},"fields":{"title":"Specified Coefficient of the Zero-Constant Primitive","stem":"On all R, f has ascending coefficients [9, 22, -23, -24], constant term first. For its zero-constant primitive F0, find the coefficient of x^2. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"11/1","explanation":"Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain 11/1. The order is ascending and the representative constant is zero."},"skills":["Specified Coefficient of the Zero-Constant Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0022","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23ec2f8c56d575ecfe349f9fbe2aa3b60825267b66be3b0be0a8ae503ff6d0cb","family_id":"qf-density-primitive-coefficient","seed":202610095022,"semantic_template":"qf-density-primitive-coefficient/[-24,3]"},"fields":{"title":"Specified Coefficient of the Zero-Constant Primitive","stem":"On all R, f has ascending coefficients [10, 23, -24, 25], constant term first. For its zero-constant primitive F0, find the coefficient of x^3. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-8/1","explanation":"Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain -8/1. The order is ascending and the representative constant is zero."},"skills":["Specified Coefficient of the Zero-Constant Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0023","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55bfbab0f37e01dac4621d02a2424edbdbb0f42c75ad47a52acb7485f459dfb6","family_id":"qf-density-primitive-coefficient","seed":202610095023,"semantic_template":"qf-density-primitive-coefficient/[-26,4]"},"fields":{"title":"Specified Coefficient of the Zero-Constant Primitive","stem":"On all R, f has ascending coefficients [11, 24, -25, -26], constant term first. For its zero-constant primitive F0, find the coefficient of x^4. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-13/2","explanation":"Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain -13/2. The order is ascending and the representative constant is zero."},"skills":["Specified Coefficient of the Zero-Constant Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0024","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"483d3990ae12ac0d3e2fdcf8d430a0b89a45cf76137f9afee2a6644fd699b745","family_id":"qf-density-primitive-coefficient","seed":202610095024,"semantic_template":"qf-density-primitive-coefficient/[12,1]"},"fields":{"title":"Specified Coefficient of the Zero-Constant Primitive","stem":"On all R, f has ascending coefficients [12, 25, -26, 27], constant term first. For its zero-constant primitive F0, find the coefficient of x^1. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"12/1","explanation":"Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain 12/1. The order is ascending and the representative constant is zero."},"skills":["Specified Coefficient of the Zero-Constant Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0025","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b8a4d29c6e0e4b31e4dba9b251ac1c38a219b066d3b00e7a8aa860eb2d8e191","family_id":"qf-density-primitive-coefficient","seed":202610095025,"semantic_template":"qf-density-primitive-coefficient/[26,2]"},"fields":{"title":"Specified Coefficient of the Zero-Constant Primitive","stem":"On all R, f has ascending coefficients [13, 26, -27, -28], constant term first. For its zero-constant primitive F0, find the coefficient of x^2. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"13/1","explanation":"Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain 13/1. The order is ascending and the representative constant is zero."},"skills":["Specified Coefficient of the Zero-Constant Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0026","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf4c4b030414e5d38cf8b80636429b915fe25feb0272e04023bbc315429710e3","family_id":"qf-density-primitive-constant-difference","seed":202610095026,"semantic_template":"qf-density-primitive-constant-difference/[\"-12\"]"},"fields":{"title":"Difference of Constants for a Shared Primitive","stem":"Two full-real polynomial primitives share F0: F1=F0+(-12) and F2=F0+(0). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-12/1","explanation":"The shared primitive cancels, leaving C1−C2=-12/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains."},"skills":["Difference of Constants for a Shared Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0027","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1828a50698f5933e76d1149e65f7a5d4f055aa8b4d4e041530100ef7b70714ba","family_id":"qf-density-primitive-constant-difference","seed":202610095027,"semantic_template":"qf-density-primitive-constant-difference/[\"-10\"]"},"fields":{"title":"Difference of Constants for a Shared Primitive","stem":"Two full-real polynomial primitives share F0: F1=F0+(-11) and F2=F0+(-1). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-10/1","explanation":"The shared primitive cancels, leaving C1−C2=-10/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains."},"skills":["Difference of Constants for a Shared Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0028","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0f1409c7e2ccd6f2edb32e6b6e77884cece9fa79c02d368f06e0f4c8edefb8f","family_id":"qf-density-primitive-constant-difference","seed":202610095028,"semantic_template":"qf-density-primitive-constant-difference/[\"-8\"]"},"fields":{"title":"Difference of Constants for a Shared Primitive","stem":"Two full-real polynomial primitives share F0: F1=F0+(-10) and F2=F0+(-2). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-8/1","explanation":"The shared primitive cancels, leaving C1−C2=-8/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains."},"skills":["Difference of Constants for a Shared Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0029","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2292d933e6a6c2dd0e613cc86dd6e2811ac3056edd9edc0b26e43413858f326","family_id":"qf-density-primitive-constant-difference","seed":202610095029,"semantic_template":"qf-density-primitive-constant-difference/[\"-6\"]"},"fields":{"title":"Difference of Constants for a Shared Primitive","stem":"Two full-real polynomial primitives share F0: F1=F0+(-9) and F2=F0+(-3). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-6/1","explanation":"The shared primitive cancels, leaving C1−C2=-6/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains."},"skills":["Difference of Constants for a Shared Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0030","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"182912e41856f63dca1e3e6e8c35561c5196cac95b8b00d48b5cd82052a06c9e","family_id":"qf-density-primitive-constant-difference","seed":202610095030,"semantic_template":"qf-density-primitive-constant-difference/[\"-4\"]"},"fields":{"title":"Difference of Constants for a Shared Primitive","stem":"Two full-real polynomial primitives share F0: F1=F0+(-8) and F2=F0+(-4). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-4/1","explanation":"The shared primitive cancels, leaving C1−C2=-4/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains."},"skills":["Difference of Constants for a Shared Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0031","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f432b9ed54c6d76bb42547f62b541284ca34d696f2d2ba53cba63c437e26e76c","family_id":"qf-density-primitive-constant-difference","seed":202610095031,"semantic_template":"qf-density-primitive-constant-difference/[\"-2\"]"},"fields":{"title":"Difference of Constants for a Shared Primitive","stem":"Two full-real polynomial primitives share F0: F1=F0+(-7) and F2=F0+(-5). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-2/1","explanation":"The shared primitive cancels, leaving C1−C2=-2/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains."},"skills":["Difference of Constants for a Shared Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0032","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c19c5721c27dc2ca4b22885e708092a3c10d2d65e7e8204cd340beef91fe2d9c","family_id":"qf-density-primitive-constant-difference","seed":202610095032,"semantic_template":"qf-density-primitive-constant-difference/[\"0\"]"},"fields":{"title":"Difference of Constants for a Shared Primitive","stem":"Two full-real polynomial primitives share F0: F1=F0+(-6) and F2=F0+(-6). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"0/1","explanation":"The shared primitive cancels, leaving C1−C2=0/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains."},"skills":["Difference of Constants for a Shared Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0033","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c55a97d38e4478d72d3571a8424e97a74f98efdcdcbbf2edb2f5d0d111fee80","family_id":"qf-density-primitive-constant-difference","seed":202610095033,"semantic_template":"qf-density-primitive-constant-difference/[\"2\"]"},"fields":{"title":"Difference of Constants for a Shared Primitive","stem":"Two full-real polynomial primitives share F0: F1=F0+(-5) and F2=F0+(-7). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"2/1","explanation":"The shared primitive cancels, leaving C1−C2=2/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains."},"skills":["Difference of Constants for a Shared Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0034","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ebac117c0e53a8c384a2bd92c57c204cb8065d8895321f269d0ce42e77a933e","family_id":"qf-density-primitive-constant-difference","seed":202610095034,"semantic_template":"qf-density-primitive-constant-difference/[\"4\"]"},"fields":{"title":"Difference of Constants for a Shared Primitive","stem":"Two full-real polynomial primitives share F0: F1=F0+(-4) and F2=F0+(-8). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"4/1","explanation":"The shared primitive cancels, leaving C1−C2=4/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains."},"skills":["Difference of Constants for a Shared Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0035","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41293e363ee695d1753f48169e4fefb4e8f00df4fef4e397c9e2d7e8a11e6c4f","family_id":"qf-density-primitive-constant-difference","seed":202610095035,"semantic_template":"qf-density-primitive-constant-difference/[\"6\"]"},"fields":{"title":"Difference of Constants for a Shared Primitive","stem":"Two full-real polynomial primitives share F0: F1=F0+(-3) and F2=F0+(-9). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"6/1","explanation":"The shared primitive cancels, leaving C1−C2=6/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains."},"skills":["Difference of Constants for a Shared Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0036","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd36a83776396ce110c0c15cee2c3f80d9c8c58b4777208c8145940b2af5d8d9","family_id":"qf-density-primitive-constant-difference","seed":202610095036,"semantic_template":"qf-density-primitive-constant-difference/[\"8\"]"},"fields":{"title":"Difference of Constants for a Shared Primitive","stem":"Two full-real polynomial primitives share F0: F1=F0+(-2) and F2=F0+(-10). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"8/1","explanation":"The shared primitive cancels, leaving C1−C2=8/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains."},"skills":["Difference of Constants for a Shared Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0037","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68fe4efd251f50297424e1e69958ea76c5d2d98fd8506edf3465e921d71b3e8c","family_id":"qf-density-primitive-constant-difference","seed":202610095037,"semantic_template":"qf-density-primitive-constant-difference/[\"10\"]"},"fields":{"title":"Difference of Constants for a Shared Primitive","stem":"Two full-real polynomial primitives share F0: F1=F0+(-1) and F2=F0+(-11). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"10/1","explanation":"The shared primitive cancels, leaving C1−C2=10/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains."},"skills":["Difference of Constants for a Shared Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0038","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2278cf660411990304a22180965948c8b7b098aa73a9a853ec2bbd23c646702b","family_id":"qf-density-primitive-constant-difference","seed":202610095038,"semantic_template":"qf-density-primitive-constant-difference/[\"12\"]"},"fields":{"title":"Difference of Constants for a Shared Primitive","stem":"Two full-real polynomial primitives share F0: F1=F0+(0) and F2=F0+(-12). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"12/1","explanation":"The shared primitive cancels, leaving C1−C2=12/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains."},"skills":["Difference of Constants for a Shared Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0039","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d83c28e8c53444da0dec33574e68266d67fa1bdc880c99ee2f70e6130b7fbe41","family_id":"qf-density-primitive-constant-difference","seed":202610095039,"semantic_template":"qf-density-primitive-constant-difference/[\"14\"]"},"fields":{"title":"Difference of Constants for a Shared Primitive","stem":"Two full-real polynomial primitives share F0: F1=F0+(1) and F2=F0+(-13). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"14/1","explanation":"The shared primitive cancels, leaving C1−C2=14/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains."},"skills":["Difference of Constants for a Shared Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0040","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2babee28f642c767fe01fea6ca4735b3a04cbce02987ff3b146f87e609c5c7c6","family_id":"qf-density-primitive-constant-difference","seed":202610095040,"semantic_template":"qf-density-primitive-constant-difference/[\"16\"]"},"fields":{"title":"Difference of Constants for a Shared Primitive","stem":"Two full-real polynomial primitives share F0: F1=F0+(2) and F2=F0+(-14). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"16/1","explanation":"The shared primitive cancels, leaving C1−C2=16/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains."},"skills":["Difference of Constants for a Shared Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0041","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8934b0b7805919c40e0dd3870c88ac792001f0a26b8d220f2f89db50dbc482b6","family_id":"qf-density-primitive-constant-difference","seed":202610095041,"semantic_template":"qf-density-primitive-constant-difference/[\"18\"]"},"fields":{"title":"Difference of Constants for a Shared Primitive","stem":"Two full-real polynomial primitives share F0: F1=F0+(3) and F2=F0+(-15). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"18/1","explanation":"The shared primitive cancels, leaving C1−C2=18/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains."},"skills":["Difference of Constants for a Shared Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0042","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e45b28a5af4b26d21b7fb62f0e8f634b8574e7d344475c8d8eb7cec0da9c7d48","family_id":"qf-density-primitive-constant-difference","seed":202610095042,"semantic_template":"qf-density-primitive-constant-difference/[\"20\"]"},"fields":{"title":"Difference of Constants for a Shared Primitive","stem":"Two full-real polynomial primitives share F0: F1=F0+(4) and F2=F0+(-16). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"20/1","explanation":"The shared primitive cancels, leaving C1−C2=20/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains."},"skills":["Difference of Constants for a Shared Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0043","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97b45ae2643c1760d6a7b7ec49661fb38de04f3f5e8c0a4192975ad51a096036","family_id":"qf-density-primitive-constant-difference","seed":202610095043,"semantic_template":"qf-density-primitive-constant-difference/[\"22\"]"},"fields":{"title":"Difference of Constants for a Shared Primitive","stem":"Two full-real polynomial primitives share F0: F1=F0+(5) and F2=F0+(-17). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"22/1","explanation":"The shared primitive cancels, leaving C1−C2=22/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains."},"skills":["Difference of Constants for a Shared Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0044","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ca81fca8cd1283f8597cb1be519e5a31c639ea16b186528811422f33ac93423","family_id":"qf-density-primitive-constant-difference","seed":202610095044,"semantic_template":"qf-density-primitive-constant-difference/[\"24\"]"},"fields":{"title":"Difference of Constants for a Shared Primitive","stem":"Two full-real polynomial primitives share F0: F1=F0+(6) and F2=F0+(-18). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"24/1","explanation":"The shared primitive cancels, leaving C1−C2=24/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains."},"skills":["Difference of Constants for a Shared Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0045","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95e0b482d9d8b45be687169234e1693b1947b0a9c0461f0bce4c3c7287b1e978","family_id":"qf-density-primitive-constant-difference","seed":202610095045,"semantic_template":"qf-density-primitive-constant-difference/[\"26\"]"},"fields":{"title":"Difference of Constants for a Shared Primitive","stem":"Two full-real polynomial primitives share F0: F1=F0+(7) and F2=F0+(-19). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"26/1","explanation":"The shared primitive cancels, leaving C1−C2=26/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains."},"skills":["Difference of Constants for a Shared Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0046","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b7aabbf0b25558d586c17e63cd4eaa28051d5792ec6cefeec2e958527030a9d","family_id":"qf-density-primitive-constant-difference","seed":202610095046,"semantic_template":"qf-density-primitive-constant-difference/[\"28\"]"},"fields":{"title":"Difference of Constants for a Shared Primitive","stem":"Two full-real polynomial primitives share F0: F1=F0+(8) and F2=F0+(-20). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"28/1","explanation":"The shared primitive cancels, leaving C1−C2=28/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains."},"skills":["Difference of Constants for a Shared Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0047","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f65c0e02383af0443c57f8032c7be03dd6991c9bacc28191c298125b221a3408","family_id":"qf-density-primitive-constant-difference","seed":202610095047,"semantic_template":"qf-density-primitive-constant-difference/[\"30\"]"},"fields":{"title":"Difference of Constants for a Shared Primitive","stem":"Two full-real polynomial primitives share F0: F1=F0+(9) and F2=F0+(-21). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"30/1","explanation":"The shared primitive cancels, leaving C1−C2=30/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains."},"skills":["Difference of Constants for a Shared Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0048","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"038fb9fcb5bf5c3c7bd798894eec93306bc29c7487d3fa081bea4f18eabe65db","family_id":"qf-density-primitive-constant-difference","seed":202610095048,"semantic_template":"qf-density-primitive-constant-difference/[\"32\"]"},"fields":{"title":"Difference of Constants for a Shared Primitive","stem":"Two full-real polynomial primitives share F0: F1=F0+(10) and F2=F0+(-22). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"32/1","explanation":"The shared primitive cancels, leaving C1−C2=32/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains."},"skills":["Difference of Constants for a Shared Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0049","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7965d7cd22a471d1a5632b7c1a50672ef3aa8a639af517199cf217f0535ed5d","family_id":"qf-density-primitive-constant-difference","seed":202610095049,"semantic_template":"qf-density-primitive-constant-difference/[\"34\"]"},"fields":{"title":"Difference of Constants for a Shared Primitive","stem":"Two full-real polynomial primitives share F0: F1=F0+(11) and F2=F0+(-23). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"34/1","explanation":"The shared primitive cancels, leaving C1−C2=34/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains."},"skills":["Difference of Constants for a Shared Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0050","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eac8bb17a5fad9825fb71a1e831e10a71a6f72007773a11e6f099d2367a8b276","family_id":"qf-density-primitive-constant-difference","seed":202610095050,"semantic_template":"qf-density-primitive-constant-difference/[\"36\"]"},"fields":{"title":"Difference of Constants for a Shared Primitive","stem":"Two full-real polynomial primitives share F0: F1=F0+(12) and F2=F0+(-24). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"36/1","explanation":"The shared primitive cancels, leaving C1−C2=36/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains."},"skills":["Difference of Constants for a Shared Primitive"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0051","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad3c2e709600c8bc12f276c9ccbfb1a7481a3bbbafdac57094a1ee1416d0b900","family_id":"qf-density-primitive-anchor-constant","seed":202610095051,"semantic_template":"qf-density-primitive-anchor-constant/[\"zero_primitive_at_anchor\",\"1/2\"]"},"fields":{"title":"Primitive Constant Selected by an Explicit Anchor","stem":"On all R, f has ascending coefficients [1, 1], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(0)=1/2. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"1/2","explanation":"Compute C=V−F0(a) exactly to obtain 1/2. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation."},"skills":["Primitive Constant Selected by an Explicit Anchor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0052","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7cf8eec87d7d434c2e6ba760a53a976ec99d70d4580f305900241188f103cd06","family_id":"qf-density-primitive-anchor-constant","seed":202610095052,"semantic_template":"qf-density-primitive-anchor-constant/[\"zero_primitive_at_anchor\",\"-2/1\"]"},"fields":{"title":"Primitive Constant Selected by an Explicit Anchor","stem":"On all R, f has ascending coefficients [0], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(3)=-2/1. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-2/1","explanation":"Compute C=V−F0(a) exactly to obtain -2/1. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation."},"skills":["Primitive Constant Selected by an Explicit Anchor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0053","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"073572f305cfaba7c75fdb6a0d5d36f1251646646f2428a4b3bde79186572237","family_id":"qf-density-primitive-anchor-constant","seed":202610095053,"semantic_template":"qf-density-primitive-anchor-constant/[[-10,3,1,3],-3,\"-10/3\"]"},"fields":{"title":"Primitive Constant Selected by an Explicit Anchor","stem":"On all R, f has ascending coefficients [-10, 3, 1, 3], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-3)=-10/3. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-1183/12","explanation":"Compute C=V−F0(a) exactly to obtain -1183/12. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation."},"skills":["Primitive Constant Selected by an Explicit Anchor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0054","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01ee21bb9a816388b1c593d7863683f7bbe1c183d1b4d442ef57f4aac84126d7","family_id":"qf-density-primitive-anchor-constant","seed":202610095054,"semantic_template":"qf-density-primitive-anchor-constant/[[-9,4,-1,4],-4,\"-9/1\"]"},"fields":{"title":"Primitive Constant Selected by an Explicit Anchor","stem":"On all R, f has ascending coefficients [-9, 4, -1, 4], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-4)=-9/1. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-1063/3","explanation":"Compute C=V−F0(a) exactly to obtain -1063/3. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation."},"skills":["Primitive Constant Selected by an Explicit Anchor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0055","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03125eea129e3d61b377d449ba835c859da3042390972758a31de3c3f06d0268","family_id":"qf-density-primitive-anchor-constant","seed":202610095055,"semantic_template":"qf-density-primitive-anchor-constant/[[-8,5,0,1],-1,\"-4/1\"]"},"fields":{"title":"Primitive Constant Selected by an Explicit Anchor","stem":"On all R, f has ascending coefficients [-8, 5, 0, 1], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-1)=-8/2. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-59/4","explanation":"Compute C=V−F0(a) exactly to obtain -59/4. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation."},"skills":["Primitive Constant Selected by an Explicit Anchor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0056","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae0046f7411d237defe9f523686427b7e1a5e66559c3cd87d71b455dce7351c2","family_id":"qf-density-primitive-anchor-constant","seed":202610095056,"semantic_template":"qf-density-primitive-anchor-constant/[[-7,6,1,2],-2,\"-7/3\"]"},"fields":{"title":"Primitive Constant Selected by an Explicit Anchor","stem":"On all R, f has ascending coefficients [-7, 6, 1, 2], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-2)=-7/3. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-101/3","explanation":"Compute C=V−F0(a) exactly to obtain -101/3. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation."},"skills":["Primitive Constant Selected by an Explicit Anchor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0057","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1c2d558999ad252b444d3ba7f2a69032fd2c5932f1f94f86bb2a4fa066cfafc","family_id":"qf-density-primitive-anchor-constant","seed":202610095057,"semantic_template":"qf-density-primitive-anchor-constant/[[-6,7,-1,3],-3,\"-6/1\"]"},"fields":{"title":"Primitive Constant Selected by an Explicit Anchor","stem":"On all R, f has ascending coefficients [-6, 7, -1, 3], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-3)=-6/1. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-501/4","explanation":"Compute C=V−F0(a) exactly to obtain -501/4. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation."},"skills":["Primitive Constant Selected by an Explicit Anchor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0058","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3f661648cc1bb4c81ee8c0677ed68076fa4f1e90208b4a92940d08e7e33abe3","family_id":"qf-density-primitive-anchor-constant","seed":202610095058,"semantic_template":"qf-density-primitive-anchor-constant/[[-5,8,0,4],-4,\"-5/2\"]"},"fields":{"title":"Primitive Constant Selected by an Explicit Anchor","stem":"On all R, f has ascending coefficients [-5, 8, 0, 4], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-4)=-5/2. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-685/2","explanation":"Compute C=V−F0(a) exactly to obtain -685/2. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation."},"skills":["Primitive Constant Selected by an Explicit Anchor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0059","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ff30684765d0dce3ab48c72c9127c1cfef7e63717481ef3a92d3b3c1a334ca7","family_id":"qf-density-primitive-anchor-constant","seed":202610095059,"semantic_template":"qf-density-primitive-anchor-constant/[[-4,9,1,1],-1,\"-4/3\"]"},"fields":{"title":"Primitive Constant Selected by an Explicit Anchor","stem":"On all R, f has ascending coefficients [-4, 9, 1, 1], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-1)=-4/3. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-39/4","explanation":"Compute C=V−F0(a) exactly to obtain -39/4. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation."},"skills":["Primitive Constant Selected by an Explicit Anchor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0060","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c12958dbcbbf877bcf0c56f13c83699c1c9118e1cdcc8e7bd58f6989e62e135","family_id":"qf-density-primitive-anchor-constant","seed":202610095060,"semantic_template":"qf-density-primitive-anchor-constant/[[-3,10,-1,2],-2,\"-3/1\"]"},"fields":{"title":"Primitive Constant Selected by an Explicit Anchor","stem":"On all R, f has ascending coefficients [-3, 10, -1, 2], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-2)=-3/1. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-119/3","explanation":"Compute C=V−F0(a) exactly to obtain -119/3. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation."},"skills":["Primitive Constant Selected by an Explicit Anchor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0061","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ffb072e65d9404e1311455551287213a883353fc0e04e8fd8792abaad4ee73d","family_id":"qf-density-primitive-anchor-constant","seed":202610095061,"semantic_template":"qf-density-primitive-anchor-constant/[[-2,11,0,3],-3,\"-1/1\"]"},"fields":{"title":"Primitive Constant Selected by an Explicit Anchor","stem":"On all R, f has ascending coefficients [-2, 11, 0, 3], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-3)=-2/2. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-469/4","explanation":"Compute C=V−F0(a) exactly to obtain -469/4. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation."},"skills":["Primitive Constant Selected by an Explicit Anchor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0062","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"948c54223806204d041df3bb86cce3d61935bebe8b6f89b50ed91b63f891bfc7","family_id":"qf-density-primitive-anchor-constant","seed":202610095062,"semantic_template":"qf-density-primitive-anchor-constant/[[-1,12,1,4],-4,\"-1/3\"]"},"fields":{"title":"Primitive Constant Selected by an Explicit Anchor","stem":"On all R, f has ascending coefficients [-1, 12, 1, 4], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-4)=-1/3. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-335/1","explanation":"Compute C=V−F0(a) exactly to obtain -335/1. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation."},"skills":["Primitive Constant Selected by an Explicit Anchor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0063","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f9753b77235b71a45ce5be005a2515f96df300c8d8bb70ab8dc4896c74e1779","family_id":"qf-density-primitive-anchor-constant","seed":202610095063,"semantic_template":"qf-density-primitive-anchor-constant/[[0,13,-1,1],-1,\"0/1\"]"},"fields":{"title":"Primitive Constant Selected by an Explicit Anchor","stem":"On all R, f has ascending coefficients [0, 13, -1, 1], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-1)=0/1. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-85/12","explanation":"Compute C=V−F0(a) exactly to obtain -85/12. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation."},"skills":["Primitive Constant Selected by an Explicit Anchor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0064","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7cf8099fec93546c6a5424378ece0d086a9a39beedd24da78ab8cf4a03b6a879","family_id":"qf-density-primitive-anchor-constant","seed":202610095064,"semantic_template":"qf-density-primitive-anchor-constant/[[1,14,0,2],-2,\"1/2\"]"},"fields":{"title":"Primitive Constant Selected by an Explicit Anchor","stem":"On all R, f has ascending coefficients [1, 14, 0, 2], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-2)=1/2. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-67/2","explanation":"Compute C=V−F0(a) exactly to obtain -67/2. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation."},"skills":["Primitive Constant Selected by an Explicit Anchor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0065","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93c7d47d3258dc8d4d5335f6456c6243d51ab19e5085135e31418297788e726c","family_id":"qf-density-primitive-anchor-constant","seed":202610095065,"semantic_template":"qf-density-primitive-anchor-constant/[[2,15,1,3],-3,\"2/3\"]"},"fields":{"title":"Primitive Constant Selected by an Explicit Anchor","stem":"On all R, f has ascending coefficients [2, 15, 1, 3], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-3)=2/3. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-1351/12","explanation":"Compute C=V−F0(a) exactly to obtain -1351/12. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation."},"skills":["Primitive Constant Selected by an Explicit Anchor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0066","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e37b40683d412c68d44e718d9ae071e74fb7258575bc871e6e1f544e765608b","family_id":"qf-density-primitive-anchor-constant","seed":202610095066,"semantic_template":"qf-density-primitive-anchor-constant/[[3,16,-1,4],-4,\"3/1\"]"},"fields":{"title":"Primitive Constant Selected by an Explicit Anchor","stem":"On all R, f has ascending coefficients [3, 16, -1, 4], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-4)=3/1. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-1171/3","explanation":"Compute C=V−F0(a) exactly to obtain -1171/3. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation."},"skills":["Primitive Constant Selected by an Explicit Anchor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0067","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"541a15335cc1f79f647420cb15e5ebce927bc196727fde027546ab9b3f008ab2","family_id":"qf-density-primitive-anchor-constant","seed":202610095067,"semantic_template":"qf-density-primitive-anchor-constant/[[4,17,0,1],-1,\"2/1\"]"},"fields":{"title":"Primitive Constant Selected by an Explicit Anchor","stem":"On all R, f has ascending coefficients [4, 17, 0, 1], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-1)=4/2. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-11/4","explanation":"Compute C=V−F0(a) exactly to obtain -11/4. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation."},"skills":["Primitive Constant Selected by an Explicit Anchor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0068","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5a43e9d77d0f4011030497e1f37aeadbd7717114d97c34f1c52d92b5a7c27d0","family_id":"qf-density-primitive-anchor-constant","seed":202610095068,"semantic_template":"qf-density-primitive-anchor-constant/[[5,18,1,2],-2,\"5/3\"]"},"fields":{"title":"Primitive Constant Selected by an Explicit Anchor","stem":"On all R, f has ascending coefficients [5, 18, 1, 2], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-2)=5/3. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-89/3","explanation":"Compute C=V−F0(a) exactly to obtain -89/3. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation."},"skills":["Primitive Constant Selected by an Explicit Anchor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0069","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"509b7e5dc87b22d9f94e55f4e953d345ab4207ac45a5f64a8a2d3c289d72c250","family_id":"qf-density-primitive-anchor-constant","seed":202610095069,"semantic_template":"qf-density-primitive-anchor-constant/[[6,19,-1,3],-3,\"6/1\"]"},"fields":{"title":"Primitive Constant Selected by an Explicit Anchor","stem":"On all R, f has ascending coefficients [6, 19, -1, 3], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-3)=6/1. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-525/4","explanation":"Compute C=V−F0(a) exactly to obtain -525/4. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation."},"skills":["Primitive Constant Selected by an Explicit Anchor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0070","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be615320f56afd1f221e6ef6b19df825ab53a8bb1d4f12f595566d937d9fcbf6","family_id":"qf-density-primitive-anchor-constant","seed":202610095070,"semantic_template":"qf-density-primitive-anchor-constant/[[7,20,0,4],-4,\"7/2\"]"},"fields":{"title":"Primitive Constant Selected by an Explicit Anchor","stem":"On all R, f has ascending coefficients [7, 20, 0, 4], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-4)=7/2. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-769/2","explanation":"Compute C=V−F0(a) exactly to obtain -769/2. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation."},"skills":["Primitive Constant Selected by an Explicit Anchor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0071","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48f2253be176fee0c8bb4c95d602d5013fa6cec8cad4bf11bbcc1756d6922ecd","family_id":"qf-density-primitive-anchor-constant","seed":202610095071,"semantic_template":"qf-density-primitive-anchor-constant/[[8,21,1,1],-1,\"8/3\"]"},"fields":{"title":"Primitive Constant Selected by an Explicit Anchor","stem":"On all R, f has ascending coefficients [8, 21, 1, 1], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-1)=8/3. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"1/4","explanation":"Compute C=V−F0(a) exactly to obtain 1/4. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation."},"skills":["Primitive Constant Selected by an Explicit Anchor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0072","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"548240c777afe557961ec650313be80693286b8344513ab3d10f3e255bb7329c","family_id":"qf-density-primitive-anchor-constant","seed":202610095072,"semantic_template":"qf-density-primitive-anchor-constant/[[9,22,-1,2],-2,\"9/1\"]"},"fields":{"title":"Primitive Constant Selected by an Explicit Anchor","stem":"On all R, f has ascending coefficients [9, 22, -1, 2], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-2)=9/1. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-83/3","explanation":"Compute C=V−F0(a) exactly to obtain -83/3. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation."},"skills":["Primitive Constant Selected by an Explicit Anchor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0073","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ccc7d5027f0776c27a33e871e81e9450ae13fad4b6711930ec41b670e6bc041","family_id":"qf-density-primitive-anchor-constant","seed":202610095073,"semantic_template":"qf-density-primitive-anchor-constant/[[10,23,0,3],-3,\"5/1\"]"},"fields":{"title":"Primitive Constant Selected by an Explicit Anchor","stem":"On all R, f has ascending coefficients [10, 23, 0, 3], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-3)=10/2. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-517/4","explanation":"Compute C=V−F0(a) exactly to obtain -517/4. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation."},"skills":["Primitive Constant Selected by an Explicit Anchor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0074","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8c1076624e5f292b06433e116e5131f2e3301e3eba555c85c80a38cd41f9aab","family_id":"qf-density-primitive-anchor-constant","seed":202610095074,"semantic_template":"qf-density-primitive-anchor-constant/[[11,24,1,4],-4,\"11/3\"]"},"fields":{"title":"Primitive Constant Selected by an Explicit Anchor","stem":"On all R, f has ascending coefficients [11, 24, 1, 4], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-4)=11/3. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"-379/1","explanation":"Compute C=V−F0(a) exactly to obtain -379/1. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation."},"skills":["Primitive Constant Selected by an Explicit Anchor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0075","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93ead07c1d1f2564b1d429e16c37a98729aa01b2c5c58eb103fcc4f360819505","family_id":"qf-density-primitive-anchor-constant","seed":202610095075,"semantic_template":"qf-density-primitive-anchor-constant/[[12,25,-1,1],-1,\"12/1\"]"},"fields":{"title":"Primitive Constant Selected by an Explicit Anchor","stem":"On all R, f has ascending coefficients [12, 25, -1, 1], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-1)=12/1. Use reduced positive-denominator fractions, retaining /1 for integers.","answer":{"kind":"expression","value":"131/12","explanation":"Compute C=V−F0(a) exactly to obtain 131/12. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation."},"skills":["Primitive Constant Selected by an Explicit Anchor"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0076","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36142e6335a1ba8331e9581836aa200313a2b353440e34d934c1d2f4d96ea657","family_id":"qf-density-primitive-candidate-derivative","seed":202610095076,"semantic_template":"qf-density-primitive-candidate-derivative/[]"},"fields":{"title":"Derivative Coefficients of a Rational Candidate","stem":"On all R, F has ascending rational coefficients [7/1]. Give its complete normalized ascending derivative coefficients as reduced fractions.","answer":{"kind":"expression","value":"[0/1]","explanation":"Differentiate coefficientwise to obtain [0/1]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check."},"skills":["Derivative Coefficients of a Rational Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0077","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"544e978db9d8d39f6148b05930042fe8a11b91965c46e1b2bfcfc1a2441d2ee7","family_id":"qf-density-primitive-candidate-derivative","seed":202610095077,"semantic_template":"qf-density-primitive-candidate-derivative/[\"2/3\"]"},"fields":{"title":"Derivative Coefficients of a Rational Candidate","stem":"On all R, F has ascending rational coefficients [3/1, 2/3]. Give its complete normalized ascending derivative coefficients as reduced fractions.","answer":{"kind":"expression","value":"[2/3]","explanation":"Differentiate coefficientwise to obtain [2/3]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check."},"skills":["Derivative Coefficients of a Rational Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0078","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc4a5e8783d08e6c35c0a45bfb7bf691fba9e34e77d0645cd3ed47cf91e67ff4","family_id":"qf-density-primitive-candidate-derivative","seed":202610095078,"semantic_template":"qf-density-primitive-candidate-derivative/[\"3/2\",\"-1/3\",\"1/1\",\"2/5\"]"},"fields":{"title":"Derivative Coefficients of a Rational Candidate","stem":"On all R, F has ascending rational coefficients [-10/1, 3/2, -1/3, 1/1, 2/5]. Give its complete normalized ascending derivative coefficients as reduced fractions.","answer":{"kind":"expression","value":"[3/2, -2/3, 3/1, 8/5]","explanation":"Differentiate coefficientwise to obtain [3/2, -2/3, 3/1, 8/5]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check."},"skills":["Derivative Coefficients of a Rational Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0079","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"257aa3621c48d7f588f977f742d72d07513ed11dc4fd8d7658934514ac345401","family_id":"qf-density-primitive-candidate-derivative","seed":202610095079,"semantic_template":"qf-density-primitive-candidate-derivative/[\"2/1\",\"-2/3\",\"5/4\",\"-3/5\"]"},"fields":{"title":"Derivative Coefficients of a Rational Candidate","stem":"On all R, F has ascending rational coefficients [-9/1, 2/1, -2/3, 5/4, -3/5]. Give its complete normalized ascending derivative coefficients as reduced fractions.","answer":{"kind":"expression","value":"[2/1, -4/3, 15/4, -12/5]","explanation":"Differentiate coefficientwise to obtain [2/1, -4/3, 15/4, -12/5]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check."},"skills":["Derivative Coefficients of a Rational Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0080","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a5a5a8ac9e0d91c0a496fd067b780d27e5236ab9bcf9111373c658299c0be71","family_id":"qf-density-primitive-candidate-derivative","seed":202610095080,"semantic_template":"qf-density-primitive-candidate-derivative/[\"5/2\",\"-1/1\",\"3/2\",\"4/5\"]"},"fields":{"title":"Derivative Coefficients of a Rational Candidate","stem":"On all R, F has ascending rational coefficients [-8/1, 5/2, -1/1, 3/2, 4/5]. Give its complete normalized ascending derivative coefficients as reduced fractions.","answer":{"kind":"expression","value":"[5/2, -2/1, 9/2, 16/5]","explanation":"Differentiate coefficientwise to obtain [5/2, -2/1, 9/2, 16/5]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check."},"skills":["Derivative Coefficients of a Rational Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0081","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28fcd11c30ac9f5e7b159c833b16392912d482923606f2f0327e93139ef72c6e","family_id":"qf-density-primitive-candidate-derivative","seed":202610095081,"semantic_template":"qf-density-primitive-candidate-derivative/[\"3/1\",\"-4/3\",\"7/4\",\"-1/1\"]"},"fields":{"title":"Derivative Coefficients of a Rational Candidate","stem":"On all R, F has ascending rational coefficients [-7/1, 3/1, -4/3, 7/4, -1/1]. Give its complete normalized ascending derivative coefficients as reduced fractions.","answer":{"kind":"expression","value":"[3/1, -8/3, 21/4, -4/1]","explanation":"Differentiate coefficientwise to obtain [3/1, -8/3, 21/4, -4/1]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check."},"skills":["Derivative Coefficients of a Rational Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0082","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4289a180e2dfef9b76da744dec4b70c826d1b20785fcd319fd5db593472bf42f","family_id":"qf-density-primitive-candidate-derivative","seed":202610095082,"semantic_template":"qf-density-primitive-candidate-derivative/[\"7/2\",\"-5/3\",\"2/1\",\"6/5\"]"},"fields":{"title":"Derivative Coefficients of a Rational Candidate","stem":"On all R, F has ascending rational coefficients [-6/1, 7/2, -5/3, 2/1, 6/5]. Give its complete normalized ascending derivative coefficients as reduced fractions.","answer":{"kind":"expression","value":"[7/2, -10/3, 6/1, 24/5]","explanation":"Differentiate coefficientwise to obtain [7/2, -10/3, 6/1, 24/5]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check."},"skills":["Derivative Coefficients of a Rational Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0083","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd791ab4265fde7999567697d723bfbb3736f9e04088b9921ddf104f73e8f7f6","family_id":"qf-density-primitive-candidate-derivative","seed":202610095083,"semantic_template":"qf-density-primitive-candidate-derivative/[\"4/1\",\"-2/1\",\"9/4\",\"-7/5\"]"},"fields":{"title":"Derivative Coefficients of a Rational Candidate","stem":"On all R, F has ascending rational coefficients [-5/1, 4/1, -2/1, 9/4, -7/5]. Give its complete normalized ascending derivative coefficients as reduced fractions.","answer":{"kind":"expression","value":"[4/1, -4/1, 27/4, -28/5]","explanation":"Differentiate coefficientwise to obtain [4/1, -4/1, 27/4, -28/5]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check."},"skills":["Derivative Coefficients of a Rational Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0084","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e13235746f08ef930eeb52911de5c132c3583716adc2d4a62cc3397326e48f1d","family_id":"qf-density-primitive-candidate-derivative","seed":202610095084,"semantic_template":"qf-density-primitive-candidate-derivative/[\"9/2\",\"-7/3\",\"5/2\",\"8/5\"]"},"fields":{"title":"Derivative Coefficients of a Rational Candidate","stem":"On all R, F has ascending rational coefficients [-4/1, 9/2, -7/3, 5/2, 8/5]. Give its complete normalized ascending derivative coefficients as reduced fractions.","answer":{"kind":"expression","value":"[9/2, -14/3, 15/2, 32/5]","explanation":"Differentiate coefficientwise to obtain [9/2, -14/3, 15/2, 32/5]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check."},"skills":["Derivative Coefficients of a Rational Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0085","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"886e53c1bcb94e5f8fa9f903a4ab81b32d23f265a2284d071dc9d56aa5b9b641","family_id":"qf-density-primitive-candidate-derivative","seed":202610095085,"semantic_template":"qf-density-primitive-candidate-derivative/[\"5/1\",\"-8/3\",\"11/4\",\"-9/5\"]"},"fields":{"title":"Derivative Coefficients of a Rational Candidate","stem":"On all R, F has ascending rational coefficients [-3/1, 5/1, -8/3, 11/4, -9/5]. Give its complete normalized ascending derivative coefficients as reduced fractions.","answer":{"kind":"expression","value":"[5/1, -16/3, 33/4, -36/5]","explanation":"Differentiate coefficientwise to obtain [5/1, -16/3, 33/4, -36/5]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check."},"skills":["Derivative Coefficients of a Rational Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0086","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"027f343bfaf899bbb303b7669d4aaffade7cb7be9e7520360fa49e6f3f25fc39","family_id":"qf-density-primitive-candidate-derivative","seed":202610095086,"semantic_template":"qf-density-primitive-candidate-derivative/[\"11/2\",\"-3/1\",\"3/1\",\"2/1\"]"},"fields":{"title":"Derivative Coefficients of a Rational Candidate","stem":"On all R, F has ascending rational coefficients [-2/1, 11/2, -3/1, 3/1, 2/1]. Give its complete normalized ascending derivative coefficients as reduced fractions.","answer":{"kind":"expression","value":"[11/2, -6/1, 9/1, 8/1]","explanation":"Differentiate coefficientwise to obtain [11/2, -6/1, 9/1, 8/1]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check."},"skills":["Derivative Coefficients of a Rational Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0087","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1999b582456749681ec12766609f7443e9270cc6bae756b273fea95ed1b29f20","family_id":"qf-density-primitive-candidate-derivative","seed":202610095087,"semantic_template":"qf-density-primitive-candidate-derivative/[\"6/1\",\"-10/3\",\"13/4\",\"-11/5\"]"},"fields":{"title":"Derivative Coefficients of a Rational Candidate","stem":"On all R, F has ascending rational coefficients [-1/1, 6/1, -10/3, 13/4, -11/5]. Give its complete normalized ascending derivative coefficients as reduced fractions.","answer":{"kind":"expression","value":"[6/1, -20/3, 39/4, -44/5]","explanation":"Differentiate coefficientwise to obtain [6/1, -20/3, 39/4, -44/5]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check."},"skills":["Derivative Coefficients of a Rational Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0088","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46f0c55e75281b25fef56c9ae48b7d8d526e3286a1e92af5f032c5a34b1befbc","family_id":"qf-density-primitive-candidate-derivative","seed":202610095088,"semantic_template":"qf-density-primitive-candidate-derivative/[\"13/2\",\"-11/3\",\"7/2\",\"12/5\"]"},"fields":{"title":"Derivative Coefficients of a Rational Candidate","stem":"On all R, F has ascending rational coefficients [0/1, 13/2, -11/3, 7/2, 12/5]. Give its complete normalized ascending derivative coefficients as reduced fractions.","answer":{"kind":"expression","value":"[13/2, -22/3, 21/2, 48/5]","explanation":"Differentiate coefficientwise to obtain [13/2, -22/3, 21/2, 48/5]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check."},"skills":["Derivative Coefficients of a Rational Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0089","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cce3a354dfcbd8f20dcc42028fa339beb05cc70c9a9be9569b30837e56a23384","family_id":"qf-density-primitive-candidate-derivative","seed":202610095089,"semantic_template":"qf-density-primitive-candidate-derivative/[\"7/1\",\"-4/1\",\"15/4\",\"-13/5\"]"},"fields":{"title":"Derivative Coefficients of a Rational Candidate","stem":"On all R, F has ascending rational coefficients [1/1, 7/1, -4/1, 15/4, -13/5]. Give its complete normalized ascending derivative coefficients as reduced fractions.","answer":{"kind":"expression","value":"[7/1, -8/1, 45/4, -52/5]","explanation":"Differentiate coefficientwise to obtain [7/1, -8/1, 45/4, -52/5]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check."},"skills":["Derivative Coefficients of a Rational Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0090","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c547d46971fbf23a0b3e1df6e2f0f9cd077433897bc55beb05f92044e7a02941","family_id":"qf-density-primitive-candidate-derivative","seed":202610095090,"semantic_template":"qf-density-primitive-candidate-derivative/[\"15/2\",\"-13/3\",\"4/1\",\"14/5\"]"},"fields":{"title":"Derivative Coefficients of a Rational Candidate","stem":"On all R, F has ascending rational coefficients [2/1, 15/2, -13/3, 4/1, 14/5]. Give its complete normalized ascending derivative coefficients as reduced fractions.","answer":{"kind":"expression","value":"[15/2, -26/3, 12/1, 56/5]","explanation":"Differentiate coefficientwise to obtain [15/2, -26/3, 12/1, 56/5]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check."},"skills":["Derivative Coefficients of a Rational Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0091","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"191466434efdb2144b7e136226b07d0e53cd5de3420f1e4484f1b700c8e01a03","family_id":"qf-density-primitive-candidate-derivative","seed":202610095091,"semantic_template":"qf-density-primitive-candidate-derivative/[\"8/1\",\"-14/3\",\"17/4\",\"-3/1\"]"},"fields":{"title":"Derivative Coefficients of a Rational Candidate","stem":"On all R, F has ascending rational coefficients [3/1, 8/1, -14/3, 17/4, -3/1]. Give its complete normalized ascending derivative coefficients as reduced fractions.","answer":{"kind":"expression","value":"[8/1, -28/3, 51/4, -12/1]","explanation":"Differentiate coefficientwise to obtain [8/1, -28/3, 51/4, -12/1]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check."},"skills":["Derivative Coefficients of a Rational Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0092","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"938cb8076995fded2e6f472bf3db48e741d17736dc602b8498464fea102b525c","family_id":"qf-density-primitive-candidate-derivative","seed":202610095092,"semantic_template":"qf-density-primitive-candidate-derivative/[\"17/2\",\"-5/1\",\"9/2\",\"16/5\"]"},"fields":{"title":"Derivative Coefficients of a Rational Candidate","stem":"On all R, F has ascending rational coefficients [4/1, 17/2, -5/1, 9/2, 16/5]. Give its complete normalized ascending derivative coefficients as reduced fractions.","answer":{"kind":"expression","value":"[17/2, -10/1, 27/2, 64/5]","explanation":"Differentiate coefficientwise to obtain [17/2, -10/1, 27/2, 64/5]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check."},"skills":["Derivative Coefficients of a Rational Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0093","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4540a665fdfe4de8e4c4c02b12909b48161a5ee9c463d39286ee1854c0973d2f","family_id":"qf-density-primitive-candidate-derivative","seed":202610095093,"semantic_template":"qf-density-primitive-candidate-derivative/[\"9/1\",\"-16/3\",\"19/4\",\"-17/5\"]"},"fields":{"title":"Derivative Coefficients of a Rational Candidate","stem":"On all R, F has ascending rational coefficients [5/1, 9/1, -16/3, 19/4, -17/5]. Give its complete normalized ascending derivative coefficients as reduced fractions.","answer":{"kind":"expression","value":"[9/1, -32/3, 57/4, -68/5]","explanation":"Differentiate coefficientwise to obtain [9/1, -32/3, 57/4, -68/5]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check."},"skills":["Derivative Coefficients of a Rational Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0094","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae2f6c47f2c1d3b3d32afdd07fdf7c0773dd3b6999439ca601e4d67eab637aec","family_id":"qf-density-primitive-candidate-derivative","seed":202610095094,"semantic_template":"qf-density-primitive-candidate-derivative/[\"19/2\",\"-17/3\",\"5/1\",\"18/5\"]"},"fields":{"title":"Derivative Coefficients of a Rational Candidate","stem":"On all R, F has ascending rational coefficients [6/1, 19/2, -17/3, 5/1, 18/5]. Give its complete normalized ascending derivative coefficients as reduced fractions.","answer":{"kind":"expression","value":"[19/2, -34/3, 15/1, 72/5]","explanation":"Differentiate coefficientwise to obtain [19/2, -34/3, 15/1, 72/5]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check."},"skills":["Derivative Coefficients of a Rational Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0095","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55f3d0d3dc642eb8d99ff251bb933fd8ebd17671f544812932857ad4d03ffe61","family_id":"qf-density-primitive-candidate-derivative","seed":202610095095,"semantic_template":"qf-density-primitive-candidate-derivative/[\"10/1\",\"-6/1\",\"21/4\",\"-19/5\"]"},"fields":{"title":"Derivative Coefficients of a Rational Candidate","stem":"On all R, F has ascending rational coefficients [7/1, 10/1, -6/1, 21/4, -19/5]. Give its complete normalized ascending derivative coefficients as reduced fractions.","answer":{"kind":"expression","value":"[10/1, -12/1, 63/4, -76/5]","explanation":"Differentiate coefficientwise to obtain [10/1, -12/1, 63/4, -76/5]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check."},"skills":["Derivative Coefficients of a Rational Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0096","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d617cbb7895aa51dd52dafdeccd0058fefcf3c5d6d62a032e7aa5de76e91547","family_id":"qf-density-primitive-candidate-derivative","seed":202610095096,"semantic_template":"qf-density-primitive-candidate-derivative/[\"21/2\",\"-19/3\",\"11/2\",\"4/1\"]"},"fields":{"title":"Derivative Coefficients of a Rational Candidate","stem":"On all R, F has ascending rational coefficients [8/1, 21/2, -19/3, 11/2, 4/1]. Give its complete normalized ascending derivative coefficients as reduced fractions.","answer":{"kind":"expression","value":"[21/2, -38/3, 33/2, 16/1]","explanation":"Differentiate coefficientwise to obtain [21/2, -38/3, 33/2, 16/1]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check."},"skills":["Derivative Coefficients of a Rational Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0097","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7515dfbd11eff67fd03e4fd4c5ffd53761d14b173e3e14a9c250491740ecfd6f","family_id":"qf-density-primitive-candidate-derivative","seed":202610095097,"semantic_template":"qf-density-primitive-candidate-derivative/[\"11/1\",\"-20/3\",\"23/4\",\"-21/5\"]"},"fields":{"title":"Derivative Coefficients of a Rational Candidate","stem":"On all R, F has ascending rational coefficients [9/1, 11/1, -20/3, 23/4, -21/5]. Give its complete normalized ascending derivative coefficients as reduced fractions.","answer":{"kind":"expression","value":"[11/1, -40/3, 69/4, -84/5]","explanation":"Differentiate coefficientwise to obtain [11/1, -40/3, 69/4, -84/5]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check."},"skills":["Derivative Coefficients of a Rational Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0098","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9cfe0961727f6bc0983daef2133b2ac13696a692e47e5ffa79c1816e4cb662f7","family_id":"qf-density-primitive-candidate-derivative","seed":202610095098,"semantic_template":"qf-density-primitive-candidate-derivative/[\"23/2\",\"-7/1\",\"6/1\",\"22/5\"]"},"fields":{"title":"Derivative Coefficients of a Rational Candidate","stem":"On all R, F has ascending rational coefficients [10/1, 23/2, -7/1, 6/1, 22/5]. Give its complete normalized ascending derivative coefficients as reduced fractions.","answer":{"kind":"expression","value":"[23/2, -14/1, 18/1, 88/5]","explanation":"Differentiate coefficientwise to obtain [23/2, -14/1, 18/1, 88/5]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check."},"skills":["Derivative Coefficients of a Rational Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0099","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d58ae3ef6eb11cd35e0c6f9ed99e8e74123cef3ca0355d986cb0e304c178e0ed","family_id":"qf-density-primitive-candidate-derivative","seed":202610095099,"semantic_template":"qf-density-primitive-candidate-derivative/[\"12/1\",\"-22/3\",\"25/4\",\"-23/5\"]"},"fields":{"title":"Derivative Coefficients of a Rational Candidate","stem":"On all R, F has ascending rational coefficients [11/1, 12/1, -22/3, 25/4, -23/5]. Give its complete normalized ascending derivative coefficients as reduced fractions.","answer":{"kind":"expression","value":"[12/1, -44/3, 75/4, -92/5]","explanation":"Differentiate coefficientwise to obtain [12/1, -44/3, 75/4, -92/5]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check."},"skills":["Derivative Coefficients of a Rational Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r05-antiderivative-0100","batch_id":"batch-20261007-advance-20261009-r05-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93863b9a0c98561645b939ffb00191ba2deeb69ab93b76055a229dbbe29f2220","family_id":"qf-density-primitive-candidate-derivative","seed":202610095100,"semantic_template":"qf-density-primitive-candidate-derivative/[\"25/2\",\"-23/3\",\"13/2\",\"24/5\"]"},"fields":{"title":"Derivative Coefficients of a Rational Candidate","stem":"On all R, F has ascending rational coefficients [12/1, 25/2, -23/3, 13/2, 24/5]. Give its complete normalized ascending derivative coefficients as reduced fractions.","answer":{"kind":"expression","value":"[25/2, -46/3, 39/2, 96/5]","explanation":"Differentiate coefficientwise to obtain [25/2, -46/3, 39/2, 96/5]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check."},"skills":["Derivative Coefficients of a Rational Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}