{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261008-r06-antiderivative-0001","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e1065ddc634f7d006fe53ea6cb72077e55794d61504b8d91d2f433511452e059","family_id":"qf-antiderivative-monomial","seed":202610086001,"semantic_template":"qf-antiderivative-monomial/{\"inputs\":[\"zero_primitive\"],\"operation\":\"monomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Monomial","stem":"On all R, f(x)=(0)x^0, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.","answer":{"kind":"expression","value":"[0/1]","explanation":"Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family."},"skills":["Zero-Constant Antiderivative of a Monomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0002","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a79b72a507564dd352331f189136bcf5eaae28a40e37af9d01a81eed8c3d30e","family_id":"qf-antiderivative-monomial","seed":202610086002,"semantic_template":"qf-antiderivative-monomial/{\"inputs\":[\"-2\",1],\"operation\":\"monomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Monomial","stem":"On all R, f(x)=(-2)x^1, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.","answer":{"kind":"expression","value":"[0/1, 0/1, -1/1]","explanation":"Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 0/1, -1/1]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family."},"skills":["Zero-Constant Antiderivative of a Monomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0003","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"907362ddbd1f9eff437b6d03219f351ea82debf27e7e150f40bcf635a5b604dc","family_id":"qf-antiderivative-monomial","seed":202610086003,"semantic_template":"qf-antiderivative-monomial/{\"inputs\":[\"3\",2],\"operation\":\"monomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Monomial","stem":"On all R, f(x)=(3)x^2, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.","answer":{"kind":"expression","value":"[0/1, 0/1, 0/1, 1/1]","explanation":"Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 0/1, 0/1, 1/1]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family."},"skills":["Zero-Constant Antiderivative of a Monomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0004","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27ba745edba54efe777430218148a274852b5d2dfd94dac4d914776e8f663d2b","family_id":"qf-antiderivative-monomial","seed":202610086004,"semantic_template":"qf-antiderivative-monomial/{\"inputs\":[\"-4\",3],\"operation\":\"monomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Monomial","stem":"On all R, f(x)=(-4)x^3, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.","answer":{"kind":"expression","value":"[0/1, 0/1, 0/1, 0/1, -1/1]","explanation":"Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 0/1, 0/1, 0/1, -1/1]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family."},"skills":["Zero-Constant Antiderivative of a Monomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0005","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b46553a6e82f804f0abffa30a3e914556fc46bd7d51c88148be6904a4634b4a9","family_id":"qf-antiderivative-monomial","seed":202610086005,"semantic_template":"qf-antiderivative-monomial/{\"inputs\":[\"5\",4],\"operation\":\"monomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Monomial","stem":"On all R, f(x)=(5)x^4, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.","answer":{"kind":"expression","value":"[0/1, 0/1, 0/1, 0/1, 0/1, 1/1]","explanation":"Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 0/1, 0/1, 0/1, 0/1, 1/1]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family."},"skills":["Zero-Constant Antiderivative of a Monomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0006","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"034756fccea522e1ec88aedc4f63e4c7e14a74bec8148ec779be930eefb5316d","family_id":"qf-antiderivative-monomial","seed":202610086006,"semantic_template":"qf-antiderivative-monomial/{\"inputs\":[\"-6\",5],\"operation\":\"monomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Monomial","stem":"On all R, f(x)=(-6)x^5, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.","answer":{"kind":"expression","value":"[0/1, 0/1, 0/1, 0/1, 0/1, 0/1, -1/1]","explanation":"Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 0/1, 0/1, 0/1, 0/1, 0/1, -1/1]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family."},"skills":["Zero-Constant Antiderivative of a Monomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0007","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07c52f9005be47111ec0d8772a5decb01677b119be3d218ff7ce0086eaf7fd04","family_id":"qf-antiderivative-monomial","seed":202610086007,"semantic_template":"qf-antiderivative-monomial/{\"inputs\":[\"7\",0],\"operation\":\"monomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Monomial","stem":"On all R, f(x)=(7)x^0, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.","answer":{"kind":"expression","value":"[0/1, 7/1]","explanation":"Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 7/1]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family."},"skills":["Zero-Constant Antiderivative of a Monomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0008","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5902c9b2fc0f4072b2c943c409aeec440628fd9f0f03e7bc4c607f525aa62aa","family_id":"qf-antiderivative-monomial","seed":202610086008,"semantic_template":"qf-antiderivative-monomial/{\"inputs\":[\"-8\",1],\"operation\":\"monomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Monomial","stem":"On all R, f(x)=(-8)x^1, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.","answer":{"kind":"expression","value":"[0/1, 0/1, -4/1]","explanation":"Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 0/1, -4/1]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family."},"skills":["Zero-Constant Antiderivative of a Monomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0009","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e432d8b2de7438aec14b29921e926022fe9fbff96592189b3980d222357faaf","family_id":"qf-antiderivative-monomial","seed":202610086009,"semantic_template":"qf-antiderivative-monomial/{\"inputs\":[\"9\",2],\"operation\":\"monomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Monomial","stem":"On all R, f(x)=(9)x^2, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.","answer":{"kind":"expression","value":"[0/1, 0/1, 0/1, 3/1]","explanation":"Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 0/1, 0/1, 3/1]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family."},"skills":["Zero-Constant Antiderivative of a Monomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0010","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e4175589ab7301459dbf597d77d79ba018c66ae5ef399f56ecbc020bf89f284","family_id":"qf-antiderivative-monomial","seed":202610086010,"semantic_template":"qf-antiderivative-monomial/{\"inputs\":[\"-10\",3],\"operation\":\"monomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Monomial","stem":"On all R, f(x)=(-10)x^3, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.","answer":{"kind":"expression","value":"[0/1, 0/1, 0/1, 0/1, -5/2]","explanation":"Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 0/1, 0/1, 0/1, -5/2]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family."},"skills":["Zero-Constant Antiderivative of a Monomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0011","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08187ffab4e08f594ebc4a746fb1ad5b99ae37b0dfdaa15fe7607c9068363353","family_id":"qf-antiderivative-monomial","seed":202610086011,"semantic_template":"qf-antiderivative-monomial/{\"inputs\":[\"11\",4],\"operation\":\"monomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Monomial","stem":"On all R, f(x)=(11)x^4, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.","answer":{"kind":"expression","value":"[0/1, 0/1, 0/1, 0/1, 0/1, 11/5]","explanation":"Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 0/1, 0/1, 0/1, 0/1, 11/5]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family."},"skills":["Zero-Constant Antiderivative of a Monomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0012","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc3cee62be6845f92ea4e4b6b2edfb25eb8a60b335036e56a387030772cc0568","family_id":"qf-antiderivative-monomial","seed":202610086012,"semantic_template":"qf-antiderivative-monomial/{\"inputs\":[\"-12\",5],\"operation\":\"monomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Monomial","stem":"On all R, f(x)=(-12)x^5, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.","answer":{"kind":"expression","value":"[0/1, 0/1, 0/1, 0/1, 0/1, 0/1, -2/1]","explanation":"Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 0/1, 0/1, 0/1, 0/1, 0/1, -2/1]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family."},"skills":["Zero-Constant Antiderivative of a Monomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0013","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89d86c520f9fc07e5dace8d0e90d5031ae8a3c02ae71147f5f24177f65872899","family_id":"qf-antiderivative-monomial","seed":202610086013,"semantic_template":"qf-antiderivative-monomial/{\"inputs\":[\"13\",0],\"operation\":\"monomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Monomial","stem":"On all R, f(x)=(13)x^0, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.","answer":{"kind":"expression","value":"[0/1, 13/1]","explanation":"Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 13/1]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family."},"skills":["Zero-Constant Antiderivative of a Monomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0014","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47b11e92151c459431d2bed5bc347c4b399f4563360e0cb096bcf887a461a022","family_id":"qf-antiderivative-monomial","seed":202610086014,"semantic_template":"qf-antiderivative-monomial/{\"inputs\":[\"-14\",1],\"operation\":\"monomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Monomial","stem":"On all R, f(x)=(-14)x^1, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.","answer":{"kind":"expression","value":"[0/1, 0/1, -7/1]","explanation":"Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 0/1, -7/1]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family."},"skills":["Zero-Constant Antiderivative of a Monomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0015","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"643d7daba9af6c744ee5d9a610dbda3089756c2f56303b577a905dca5af8f940","family_id":"qf-antiderivative-monomial","seed":202610086015,"semantic_template":"qf-antiderivative-monomial/{\"inputs\":[\"15\",2],\"operation\":\"monomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Monomial","stem":"On all R, f(x)=(15)x^2, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.","answer":{"kind":"expression","value":"[0/1, 0/1, 0/1, 5/1]","explanation":"Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 0/1, 0/1, 5/1]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family."},"skills":["Zero-Constant Antiderivative of a Monomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0016","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1274ae19a8fe6d3465df4c952423b391005fbef260613edc8d8a5b24389f6123","family_id":"qf-antiderivative-monomial","seed":202610086016,"semantic_template":"qf-antiderivative-monomial/{\"inputs\":[\"-16\",3],\"operation\":\"monomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Monomial","stem":"On all R, f(x)=(-16)x^3, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.","answer":{"kind":"expression","value":"[0/1, 0/1, 0/1, 0/1, -4/1]","explanation":"Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 0/1, 0/1, 0/1, -4/1]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family."},"skills":["Zero-Constant Antiderivative of a Monomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0017","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ada41c8bafc730015c116fb1a87c502f977b47a8857467f8cfbf016ccdb0769","family_id":"qf-antiderivative-monomial","seed":202610086017,"semantic_template":"qf-antiderivative-monomial/{\"inputs\":[\"17\",4],\"operation\":\"monomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Monomial","stem":"On all R, f(x)=(17)x^4, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.","answer":{"kind":"expression","value":"[0/1, 0/1, 0/1, 0/1, 0/1, 17/5]","explanation":"Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 0/1, 0/1, 0/1, 0/1, 17/5]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family."},"skills":["Zero-Constant Antiderivative of a Monomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0018","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c56215c556cc5794be163fa0bde3f54ac959f021d59cd93a92421f0299bf0656","family_id":"qf-antiderivative-monomial","seed":202610086018,"semantic_template":"qf-antiderivative-monomial/{\"inputs\":[\"-18\",5],\"operation\":\"monomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Monomial","stem":"On all R, f(x)=(-18)x^5, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.","answer":{"kind":"expression","value":"[0/1, 0/1, 0/1, 0/1, 0/1, 0/1, -3/1]","explanation":"Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 0/1, 0/1, 0/1, 0/1, 0/1, -3/1]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family."},"skills":["Zero-Constant Antiderivative of a Monomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0019","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe45be5de755711f4ac744d46bc4f981b0cad77732f29b21a5d591c177793508","family_id":"qf-antiderivative-monomial","seed":202610086019,"semantic_template":"qf-antiderivative-monomial/{\"inputs\":[\"19\",0],\"operation\":\"monomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Monomial","stem":"On all R, f(x)=(19)x^0, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.","answer":{"kind":"expression","value":"[0/1, 19/1]","explanation":"Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 19/1]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family."},"skills":["Zero-Constant Antiderivative of a Monomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0020","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e22cc8617338cf4468fe7b6c8b4a8dd291e4ec7c54932fd28d6fd9c6b3cb5aa","family_id":"qf-antiderivative-monomial","seed":202610086020,"semantic_template":"qf-antiderivative-monomial/{\"inputs\":[\"-20\",1],\"operation\":\"monomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Monomial","stem":"On all R, f(x)=(-20)x^1, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.","answer":{"kind":"expression","value":"[0/1, 0/1, -10/1]","explanation":"Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 0/1, -10/1]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family."},"skills":["Zero-Constant Antiderivative of a Monomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0021","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ff72915dbff83363d94b7db89e50dc106c7d4066856fe32c586fb5181c6bdc3","family_id":"qf-antiderivative-monomial","seed":202610086021,"semantic_template":"qf-antiderivative-monomial/{\"inputs\":[\"21\",2],\"operation\":\"monomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Monomial","stem":"On all R, f(x)=(21)x^2, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.","answer":{"kind":"expression","value":"[0/1, 0/1, 0/1, 7/1]","explanation":"Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 0/1, 0/1, 7/1]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family."},"skills":["Zero-Constant Antiderivative of a Monomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0022","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1b98b175869b31b9fc823ef34533823f7e3ed88eb2b98f1b2287ef4ddb7b0e9","family_id":"qf-antiderivative-monomial","seed":202610086022,"semantic_template":"qf-antiderivative-monomial/{\"inputs\":[\"-22\",3],\"operation\":\"monomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Monomial","stem":"On all R, f(x)=(-22)x^3, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.","answer":{"kind":"expression","value":"[0/1, 0/1, 0/1, 0/1, -11/2]","explanation":"Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 0/1, 0/1, 0/1, -11/2]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family."},"skills":["Zero-Constant Antiderivative of a Monomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0023","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d85b948e4656f09afa9bd95db5ddf6020abe2aafe637556cb9ede5606c8747d0","family_id":"qf-antiderivative-monomial","seed":202610086023,"semantic_template":"qf-antiderivative-monomial/{\"inputs\":[\"23\",4],\"operation\":\"monomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Monomial","stem":"On all R, f(x)=(23)x^4, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.","answer":{"kind":"expression","value":"[0/1, 0/1, 0/1, 0/1, 0/1, 23/5]","explanation":"Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 0/1, 0/1, 0/1, 0/1, 23/5]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family."},"skills":["Zero-Constant Antiderivative of a Monomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0024","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9018ad82cbdb88f313bf495adcf70b8f76e7883a917bf0cfb1b50b92f7510109","family_id":"qf-antiderivative-monomial","seed":202610086024,"semantic_template":"qf-antiderivative-monomial/{\"inputs\":[\"-24\",5],\"operation\":\"monomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Monomial","stem":"On all R, f(x)=(-24)x^5, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.","answer":{"kind":"expression","value":"[0/1, 0/1, 0/1, 0/1, 0/1, 0/1, -4/1]","explanation":"Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 0/1, 0/1, 0/1, 0/1, 0/1, -4/1]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family."},"skills":["Zero-Constant Antiderivative of a Monomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0025","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57d6b794cf1a5a2983b70534dd9c11ce16b2d02e8b5d93f761c9ecedd1264722","family_id":"qf-antiderivative-monomial","seed":202610086025,"semantic_template":"qf-antiderivative-monomial/{\"inputs\":[\"25\",0],\"operation\":\"monomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Monomial","stem":"On all R, f(x)=(25)x^0, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.","answer":{"kind":"expression","value":"[0/1, 25/1]","explanation":"Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 25/1]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family."},"skills":["Zero-Constant Antiderivative of a Monomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0026","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"09a0ae2256336c59ea22e0051a06203ae8c53a1cc74392d1a01db50fa54c7fcd","family_id":"qf-antiderivative-polynomial","seed":202610086026,"semantic_template":"qf-antiderivative-polynomial/{\"inputs\":[\"0\"],\"operation\":\"polynomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Polynomial","stem":"On all R, f has ascending coefficients [0], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.","answer":{"kind":"expression","value":"[0/1]","explanation":"Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1]. The actual antiderivative family also includes adding a constant."},"skills":["Zero-Constant Antiderivative of a Polynomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0027","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64fb681c9222bbd12ec3fbca5c4182373a3526213625784ccf1dba927a0e4cfd","family_id":"qf-antiderivative-polynomial","seed":202610086027,"semantic_template":"qf-antiderivative-polynomial/{\"inputs\":[\"3\"],\"operation\":\"polynomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Polynomial","stem":"On all R, f has ascending coefficients [3], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.","answer":{"kind":"expression","value":"[0/1, 3/1]","explanation":"Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, 3/1]. The actual antiderivative family also includes adding a constant."},"skills":["Zero-Constant Antiderivative of a Polynomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0028","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb3cd83eda0d60153451d1997f06abd31f49362b1cf0396d6bcf2d49882bc181","family_id":"qf-antiderivative-polynomial","seed":202610086028,"semantic_template":"qf-antiderivative-polynomial/{\"inputs\":[\"-10\",\"3\",\"1\",\"3\"],\"operation\":\"polynomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Polynomial","stem":"On all R, f has ascending coefficients [-10, 3, 1, 3], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.","answer":{"kind":"expression","value":"[0/1, -10/1, 3/2, 1/3, 3/4]","explanation":"Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, -10/1, 3/2, 1/3, 3/4]. The actual antiderivative family also includes adding a constant."},"skills":["Zero-Constant Antiderivative of a Polynomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0029","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b7b4fb304d6b3c4fb8baeb7049497362eade168a9825c8e14efeeec955bb515","family_id":"qf-antiderivative-polynomial","seed":202610086029,"semantic_template":"qf-antiderivative-polynomial/{\"inputs\":[\"-9\",\"4\",\"-1\",\"4\"],\"operation\":\"polynomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Polynomial","stem":"On all R, f has ascending coefficients [-9, 4, -1, 4], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.","answer":{"kind":"expression","value":"[0/1, -9/1, 2/1, -1/3, 1/1]","explanation":"Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, -9/1, 2/1, -1/3, 1/1]. The actual antiderivative family also includes adding a constant."},"skills":["Zero-Constant Antiderivative of a Polynomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0030","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a42acd29f3d34193f6c7286250c9366ae2be8fd296e67f417aad5e02fca1948a","family_id":"qf-antiderivative-polynomial","seed":202610086030,"semantic_template":"qf-antiderivative-polynomial/{\"inputs\":[\"-8\",\"5\",\"0\",\"1\"],\"operation\":\"polynomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Polynomial","stem":"On all R, f has ascending coefficients [-8, 5, 0, 1], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.","answer":{"kind":"expression","value":"[0/1, -8/1, 5/2, 0/1, 1/4]","explanation":"Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, -8/1, 5/2, 0/1, 1/4]. The actual antiderivative family also includes adding a constant."},"skills":["Zero-Constant Antiderivative of a Polynomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0031","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"878942ccca71ccc8bdbe53f38e064faf32f503ef3a1d66f8a2dc603b5870bb1f","family_id":"qf-antiderivative-polynomial","seed":202610086031,"semantic_template":"qf-antiderivative-polynomial/{\"inputs\":[\"-7\",\"6\",\"1\",\"2\"],\"operation\":\"polynomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Polynomial","stem":"On all R, f has ascending coefficients [-7, 6, 1, 2], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.","answer":{"kind":"expression","value":"[0/1, -7/1, 3/1, 1/3, 1/2]","explanation":"Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, -7/1, 3/1, 1/3, 1/2]. The actual antiderivative family also includes adding a constant."},"skills":["Zero-Constant Antiderivative of a Polynomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0032","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c3b34f399a25afceb39ef29809eabc7f792f287bb9b8fca0a90536f2ddd8180c","family_id":"qf-antiderivative-polynomial","seed":202610086032,"semantic_template":"qf-antiderivative-polynomial/{\"inputs\":[\"-6\",\"7\",\"-1\",\"3\"],\"operation\":\"polynomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Polynomial","stem":"On all R, f has ascending coefficients [-6, 7, -1, 3], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.","answer":{"kind":"expression","value":"[0/1, -6/1, 7/2, -1/3, 3/4]","explanation":"Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, -6/1, 7/2, -1/3, 3/4]. The actual antiderivative family also includes adding a constant."},"skills":["Zero-Constant Antiderivative of a Polynomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0033","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d7f732ce119c7421ae9f04bc3fdddabc5cd04ad403c65e02eb23f7254bc0b74","family_id":"qf-antiderivative-polynomial","seed":202610086033,"semantic_template":"qf-antiderivative-polynomial/{\"inputs\":[\"-5\",\"8\",\"0\",\"4\"],\"operation\":\"polynomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Polynomial","stem":"On all R, f has ascending coefficients [-5, 8, 0, 4], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.","answer":{"kind":"expression","value":"[0/1, -5/1, 4/1, 0/1, 1/1]","explanation":"Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, -5/1, 4/1, 0/1, 1/1]. The actual antiderivative family also includes adding a constant."},"skills":["Zero-Constant Antiderivative of a Polynomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0034","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6963a1221736a705242c21deb2d7aa54482c5a97b84d2f1155ab8d511939ea7","family_id":"qf-antiderivative-polynomial","seed":202610086034,"semantic_template":"qf-antiderivative-polynomial/{\"inputs\":[\"-4\",\"9\",\"1\",\"1\"],\"operation\":\"polynomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Polynomial","stem":"On all R, f has ascending coefficients [-4, 9, 1, 1], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.","answer":{"kind":"expression","value":"[0/1, -4/1, 9/2, 1/3, 1/4]","explanation":"Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, -4/1, 9/2, 1/3, 1/4]. The actual antiderivative family also includes adding a constant."},"skills":["Zero-Constant Antiderivative of a Polynomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0035","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba8b3b491086730d7e0bfc4cff1fa3ed7b7b26e73e7f3b6e52d032fd1166a5df","family_id":"qf-antiderivative-polynomial","seed":202610086035,"semantic_template":"qf-antiderivative-polynomial/{\"inputs\":[\"-3\",\"10\",\"-1\",\"2\"],\"operation\":\"polynomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Polynomial","stem":"On all R, f has ascending coefficients [-3, 10, -1, 2], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.","answer":{"kind":"expression","value":"[0/1, -3/1, 5/1, -1/3, 1/2]","explanation":"Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, -3/1, 5/1, -1/3, 1/2]. The actual antiderivative family also includes adding a constant."},"skills":["Zero-Constant Antiderivative of a Polynomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0036","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"500521ac72b0f8f5c5d278d49fb60f5e434a8b4b7f43926cb4e6723bdba269a1","family_id":"qf-antiderivative-polynomial","seed":202610086036,"semantic_template":"qf-antiderivative-polynomial/{\"inputs\":[\"-2\",\"11\",\"0\",\"3\"],\"operation\":\"polynomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Polynomial","stem":"On all R, f has ascending coefficients [-2, 11, 0, 3], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.","answer":{"kind":"expression","value":"[0/1, -2/1, 11/2, 0/1, 3/4]","explanation":"Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, -2/1, 11/2, 0/1, 3/4]. The actual antiderivative family also includes adding a constant."},"skills":["Zero-Constant Antiderivative of a Polynomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0037","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71d5cbf4e57b8eb56ea73f02fa33ba7eda4d596764ea969d2e56ce0399bae696","family_id":"qf-antiderivative-polynomial","seed":202610086037,"semantic_template":"qf-antiderivative-polynomial/{\"inputs\":[\"-1\",\"12\",\"1\",\"4\"],\"operation\":\"polynomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Polynomial","stem":"On all R, f has ascending coefficients [-1, 12, 1, 4], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.","answer":{"kind":"expression","value":"[0/1, -1/1, 6/1, 1/3, 1/1]","explanation":"Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, -1/1, 6/1, 1/3, 1/1]. The actual antiderivative family also includes adding a constant."},"skills":["Zero-Constant Antiderivative of a Polynomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0038","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f691a22fd8ab3220aace611102d39e28eb8dc20a4c8c9e2de4baa6d015992348","family_id":"qf-antiderivative-polynomial","seed":202610086038,"semantic_template":"qf-antiderivative-polynomial/{\"inputs\":[\"0\",\"13\",\"-1\",\"1\"],\"operation\":\"polynomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Polynomial","stem":"On all R, f has ascending coefficients [0, 13, -1, 1], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.","answer":{"kind":"expression","value":"[0/1, 0/1, 13/2, -1/3, 1/4]","explanation":"Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, 0/1, 13/2, -1/3, 1/4]. The actual antiderivative family also includes adding a constant."},"skills":["Zero-Constant Antiderivative of a Polynomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0039","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60d869882aa85c28fe39333ae43668e97c272b0424efd346c77c6cc1273a7a34","family_id":"qf-antiderivative-polynomial","seed":202610086039,"semantic_template":"qf-antiderivative-polynomial/{\"inputs\":[\"1\",\"14\",\"0\",\"2\"],\"operation\":\"polynomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Polynomial","stem":"On all R, f has ascending coefficients [1, 14, 0, 2], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.","answer":{"kind":"expression","value":"[0/1, 1/1, 7/1, 0/1, 1/2]","explanation":"Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, 1/1, 7/1, 0/1, 1/2]. The actual antiderivative family also includes adding a constant."},"skills":["Zero-Constant Antiderivative of a Polynomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0040","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52e10283ac1336bd71a9bf9e1afe6bbe04dc286a391d229a3e257a49e7aa136e","family_id":"qf-antiderivative-polynomial","seed":202610086040,"semantic_template":"qf-antiderivative-polynomial/{\"inputs\":[\"2\",\"15\",\"1\",\"3\"],\"operation\":\"polynomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Polynomial","stem":"On all R, f has ascending coefficients [2, 15, 1, 3], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.","answer":{"kind":"expression","value":"[0/1, 2/1, 15/2, 1/3, 3/4]","explanation":"Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, 2/1, 15/2, 1/3, 3/4]. The actual antiderivative family also includes adding a constant."},"skills":["Zero-Constant Antiderivative of a Polynomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0041","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9038b0d50139fa03d9dc00fcd491d69488762f05030c8ff12c24f835296d3e1","family_id":"qf-antiderivative-polynomial","seed":202610086041,"semantic_template":"qf-antiderivative-polynomial/{\"inputs\":[\"3\",\"16\",\"-1\",\"4\"],\"operation\":\"polynomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Polynomial","stem":"On all R, f has ascending coefficients [3, 16, -1, 4], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.","answer":{"kind":"expression","value":"[0/1, 3/1, 8/1, -1/3, 1/1]","explanation":"Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, 3/1, 8/1, -1/3, 1/1]. The actual antiderivative family also includes adding a constant."},"skills":["Zero-Constant Antiderivative of a Polynomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0042","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b37d655f731f98627a682bdf5914225731503ba64d69391e44b2585b4b2c842c","family_id":"qf-antiderivative-polynomial","seed":202610086042,"semantic_template":"qf-antiderivative-polynomial/{\"inputs\":[\"4\",\"17\",\"0\",\"1\"],\"operation\":\"polynomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Polynomial","stem":"On all R, f has ascending coefficients [4, 17, 0, 1], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.","answer":{"kind":"expression","value":"[0/1, 4/1, 17/2, 0/1, 1/4]","explanation":"Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, 4/1, 17/2, 0/1, 1/4]. The actual antiderivative family also includes adding a constant."},"skills":["Zero-Constant Antiderivative of a Polynomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0043","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a9320032b492d8c7f9c5d8cc4450976178ce739105f1e5771dcc109e0dbe61e","family_id":"qf-antiderivative-polynomial","seed":202610086043,"semantic_template":"qf-antiderivative-polynomial/{\"inputs\":[\"5\",\"18\",\"1\",\"2\"],\"operation\":\"polynomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Polynomial","stem":"On all R, f has ascending coefficients [5, 18, 1, 2], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.","answer":{"kind":"expression","value":"[0/1, 5/1, 9/1, 1/3, 1/2]","explanation":"Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, 5/1, 9/1, 1/3, 1/2]. The actual antiderivative family also includes adding a constant."},"skills":["Zero-Constant Antiderivative of a Polynomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0044","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a142e4fbc39f95ab4cf745a1380c4406c685d19a2b4477d9e31525500ee68896","family_id":"qf-antiderivative-polynomial","seed":202610086044,"semantic_template":"qf-antiderivative-polynomial/{\"inputs\":[\"6\",\"19\",\"-1\",\"3\"],\"operation\":\"polynomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Polynomial","stem":"On all R, f has ascending coefficients [6, 19, -1, 3], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.","answer":{"kind":"expression","value":"[0/1, 6/1, 19/2, -1/3, 3/4]","explanation":"Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, 6/1, 19/2, -1/3, 3/4]. The actual antiderivative family also includes adding a constant."},"skills":["Zero-Constant Antiderivative of a Polynomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0045","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa61a716e24cd547960d5be18f87f679a5b5a222a96b3afd6033a68bf29e8f23","family_id":"qf-antiderivative-polynomial","seed":202610086045,"semantic_template":"qf-antiderivative-polynomial/{\"inputs\":[\"7\",\"20\",\"0\",\"4\"],\"operation\":\"polynomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Polynomial","stem":"On all R, f has ascending coefficients [7, 20, 0, 4], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.","answer":{"kind":"expression","value":"[0/1, 7/1, 10/1, 0/1, 1/1]","explanation":"Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, 7/1, 10/1, 0/1, 1/1]. The actual antiderivative family also includes adding a constant."},"skills":["Zero-Constant Antiderivative of a Polynomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0046","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4759695aced87e81f20cd6e34279f63c6b8c486808899fe17bc028c1ddc4623a","family_id":"qf-antiderivative-polynomial","seed":202610086046,"semantic_template":"qf-antiderivative-polynomial/{\"inputs\":[\"8\",\"21\",\"1\",\"1\"],\"operation\":\"polynomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Polynomial","stem":"On all R, f has ascending coefficients [8, 21, 1, 1], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.","answer":{"kind":"expression","value":"[0/1, 8/1, 21/2, 1/3, 1/4]","explanation":"Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, 8/1, 21/2, 1/3, 1/4]. The actual antiderivative family also includes adding a constant."},"skills":["Zero-Constant Antiderivative of a Polynomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0047","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"437c8e9c51ca6e221203ad74cbe9fb8ec11efce5ed91b64d4314b3cae2781a83","family_id":"qf-antiderivative-polynomial","seed":202610086047,"semantic_template":"qf-antiderivative-polynomial/{\"inputs\":[\"9\",\"22\",\"-1\",\"2\"],\"operation\":\"polynomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Polynomial","stem":"On all R, f has ascending coefficients [9, 22, -1, 2], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.","answer":{"kind":"expression","value":"[0/1, 9/1, 11/1, -1/3, 1/2]","explanation":"Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, 9/1, 11/1, -1/3, 1/2]. The actual antiderivative family also includes adding a constant."},"skills":["Zero-Constant Antiderivative of a Polynomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0048","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b464e817268170e7e02a2ec123fcda553161ca44928e4056542f4403fba24d6","family_id":"qf-antiderivative-polynomial","seed":202610086048,"semantic_template":"qf-antiderivative-polynomial/{\"inputs\":[\"10\",\"23\",\"0\",\"3\"],\"operation\":\"polynomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Polynomial","stem":"On all R, f has ascending coefficients [10, 23, 0, 3], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.","answer":{"kind":"expression","value":"[0/1, 10/1, 23/2, 0/1, 3/4]","explanation":"Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, 10/1, 23/2, 0/1, 3/4]. The actual antiderivative family also includes adding a constant."},"skills":["Zero-Constant Antiderivative of a Polynomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0049","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c36149b33d9d93348eda150d4a8726814386d9a10ae7824f8700ee89b9a2d8f2","family_id":"qf-antiderivative-polynomial","seed":202610086049,"semantic_template":"qf-antiderivative-polynomial/{\"inputs\":[\"11\",\"24\",\"1\",\"4\"],\"operation\":\"polynomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Polynomial","stem":"On all R, f has ascending coefficients [11, 24, 1, 4], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.","answer":{"kind":"expression","value":"[0/1, 11/1, 12/1, 1/3, 1/1]","explanation":"Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, 11/1, 12/1, 1/3, 1/1]. The actual antiderivative family also includes adding a constant."},"skills":["Zero-Constant Antiderivative of a Polynomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0050","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d8585098514e722aec0c3cf29444422e626b60cc3fccdf6142eec075d0a570c","family_id":"qf-antiderivative-polynomial","seed":202610086050,"semantic_template":"qf-antiderivative-polynomial/{\"inputs\":[\"12\",\"25\",\"-1\",\"1\"],\"operation\":\"polynomial_primitive\"}"},"fields":{"title":"Zero-Constant Antiderivative of a Polynomial","stem":"On all R, f has ascending coefficients [12, 25, -1, 1], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.","answer":{"kind":"expression","value":"[0/1, 12/1, 25/2, -1/3, 1/4]","explanation":"Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, 12/1, 25/2, -1/3, 1/4]. The actual antiderivative family also includes adding a constant."},"skills":["Zero-Constant Antiderivative of a Polynomial"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0051","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e86559741f76e43bde2db6b00344014ee50fd1a01190cb442fcf6ed3f47fb960","family_id":"qf-antiderivative-candidate","seed":202610086051,"semantic_template":"qf-antiderivative-candidate/{\"inputs\":[[\"0\"],[]],\"operation\":\"primitive_candidate\"}"},"fields":{"title":"Coefficient Eligibility of a Candidate Antiderivative","stem":"On all R, f has ascending coefficients [0], starting with its constant term. The candidate F has ascending rational coefficient pairs [[7, 1]], with positive denominators. Is F′=f as a coefficient identity?","answer":{"kind":"short_text","value":"Yes","explanation":"Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving Yes. This is coefficient identity, not finite point sampling; the candidate constant is free."},"skills":["Coefficient Eligibility of a Candidate Antiderivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0052","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aef013f85f07e2d094698b09a4be4ea87ee156cd22b6696f85ebf4afc3e25dd9","family_id":"qf-antiderivative-candidate","seed":202610086052,"semantic_template":"qf-antiderivative-candidate/{\"inputs\":[[\"-11\",\"2\",\"0\",\"2\"],[\"-11/1\",\"1/1\",\"0/1\",\"3/2\"]],\"operation\":\"primitive_candidate\"}"},"fields":{"title":"Coefficient Eligibility of a Candidate Antiderivative","stem":"On all R, f has ascending coefficients [-11, 2, 0, 2], starting with its constant term. The candidate F has ascending rational coefficient pairs [[-12, 1], [-11, 1], [1, 1], [0, 1], [3, 2]], with positive denominators. Is F′=f as a coefficient identity?","answer":{"kind":"short_text","value":"No","explanation":"Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving No. This is coefficient identity, not finite point sampling; the candidate constant is free."},"skills":["Coefficient Eligibility of a Candidate Antiderivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0053","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"56d16778ed0b3a7f03591ad891cdc7b003249a665ac563d1b0419cc7ce60eb85","family_id":"qf-antiderivative-candidate","seed":202610086053,"semantic_template":"qf-antiderivative-candidate/{\"inputs\":[[\"-10\",\"3\",\"1\",\"3\"],[\"-10/1\",\"3/2\",\"1/3\",\"3/4\"]],\"operation\":\"primitive_candidate\"}"},"fields":{"title":"Coefficient Eligibility of a Candidate Antiderivative","stem":"On all R, f has ascending coefficients [-10, 3, 1, 3], starting with its constant term. The candidate F has ascending rational coefficient pairs [[-11, 1], [-10, 1], [3, 2], [1, 3], [3, 4]], with positive denominators. Is F′=f as a coefficient identity?","answer":{"kind":"short_text","value":"Yes","explanation":"Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving Yes. This is coefficient identity, not finite point sampling; the candidate constant is free."},"skills":["Coefficient Eligibility of a Candidate Antiderivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0054","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e1d550dbe308c5cd84543b828fe3911ddb03955bc4223155707f6876816ae9c4","family_id":"qf-antiderivative-candidate","seed":202610086054,"semantic_template":"qf-antiderivative-candidate/{\"inputs\":[[\"-9\",\"4\",\"-1\",\"4\"],[\"-9/1\",\"2/1\",\"-1/3\",\"2/1\"]],\"operation\":\"primitive_candidate\"}"},"fields":{"title":"Coefficient Eligibility of a Candidate Antiderivative","stem":"On all R, f has ascending coefficients [-9, 4, -1, 4], starting with its constant term. The candidate F has ascending rational coefficient pairs [[-10, 1], [-9, 1], [2, 1], [-1, 3], [2, 1]], with positive denominators. Is F′=f as a coefficient identity?","answer":{"kind":"short_text","value":"No","explanation":"Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving No. This is coefficient identity, not finite point sampling; the candidate constant is free."},"skills":["Coefficient Eligibility of a Candidate Antiderivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0055","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd5a0878e0d5a7830519b8b7df003c5f2b578d37da3c3ffe257204ad6b003bbd","family_id":"qf-antiderivative-candidate","seed":202610086055,"semantic_template":"qf-antiderivative-candidate/{\"inputs\":[[\"-8\",\"5\",\"0\",\"1\"],[\"-8/1\",\"5/2\",\"0/1\",\"1/4\"]],\"operation\":\"primitive_candidate\"}"},"fields":{"title":"Coefficient Eligibility of a Candidate Antiderivative","stem":"On all R, f has ascending coefficients [-8, 5, 0, 1], starting with its constant term. The candidate F has ascending rational coefficient pairs [[-9, 1], [-8, 1], [5, 2], [0, 1], [1, 4]], with positive denominators. Is F′=f as a coefficient identity?","answer":{"kind":"short_text","value":"Yes","explanation":"Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving Yes. This is coefficient identity, not finite point sampling; the candidate constant is free."},"skills":["Coefficient Eligibility of a Candidate Antiderivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0056","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa01e4a7980a3d78e4c6e8eb924556308143e0a08443b55f3bc888488d0293de","family_id":"qf-antiderivative-candidate","seed":202610086056,"semantic_template":"qf-antiderivative-candidate/{\"inputs\":[[\"-7\",\"6\",\"1\",\"2\"],[\"-7/1\",\"3/1\",\"1/3\",\"3/2\"]],\"operation\":\"primitive_candidate\"}"},"fields":{"title":"Coefficient Eligibility of a Candidate Antiderivative","stem":"On all R, f has ascending coefficients [-7, 6, 1, 2], starting with its constant term. The candidate F has ascending rational coefficient pairs [[-8, 1], [-7, 1], [3, 1], [1, 3], [3, 2]], with positive denominators. Is F′=f as a coefficient identity?","answer":{"kind":"short_text","value":"No","explanation":"Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving No. This is coefficient identity, not finite point sampling; the candidate constant is free."},"skills":["Coefficient Eligibility of a Candidate Antiderivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0057","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d92c8fed41f9ed0e42486ee68bba84c0c1fc05fdd339e35d3de62bb758560d4","family_id":"qf-antiderivative-candidate","seed":202610086057,"semantic_template":"qf-antiderivative-candidate/{\"inputs\":[[\"-6\",\"7\",\"-1\",\"3\"],[\"-6/1\",\"7/2\",\"-1/3\",\"3/4\"]],\"operation\":\"primitive_candidate\"}"},"fields":{"title":"Coefficient Eligibility of a Candidate Antiderivative","stem":"On all R, f has ascending coefficients [-6, 7, -1, 3], starting with its constant term. The candidate F has ascending rational coefficient pairs [[-7, 1], [-6, 1], [7, 2], [-1, 3], [3, 4]], with positive denominators. Is F′=f as a coefficient identity?","answer":{"kind":"short_text","value":"Yes","explanation":"Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving Yes. This is coefficient identity, not finite point sampling; the candidate constant is free."},"skills":["Coefficient Eligibility of a Candidate Antiderivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0058","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b32aaff36b6df144f647213b27dfde6bd7d7a0b8a0d89f5a11b8fb26c543af1","family_id":"qf-antiderivative-candidate","seed":202610086058,"semantic_template":"qf-antiderivative-candidate/{\"inputs\":[[\"-5\",\"8\",\"0\",\"4\"],[\"-5/1\",\"4/1\",\"0/1\",\"2/1\"]],\"operation\":\"primitive_candidate\"}"},"fields":{"title":"Coefficient Eligibility of a Candidate Antiderivative","stem":"On all R, f has ascending coefficients [-5, 8, 0, 4], starting with its constant term. The candidate F has ascending rational coefficient pairs [[-6, 1], [-5, 1], [4, 1], [0, 1], [2, 1]], with positive denominators. Is F′=f as a coefficient identity?","answer":{"kind":"short_text","value":"No","explanation":"Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving No. This is coefficient identity, not finite point sampling; the candidate constant is free."},"skills":["Coefficient Eligibility of a Candidate Antiderivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0059","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2033042f6d7485fcea728c2e9c0c4daec5e9072bf43ffa58f223e3ece1d712a6","family_id":"qf-antiderivative-candidate","seed":202610086059,"semantic_template":"qf-antiderivative-candidate/{\"inputs\":[[\"-4\",\"9\",\"1\",\"1\"],[\"-4/1\",\"9/2\",\"1/3\",\"1/4\"]],\"operation\":\"primitive_candidate\"}"},"fields":{"title":"Coefficient Eligibility of a Candidate Antiderivative","stem":"On all R, f has ascending coefficients [-4, 9, 1, 1], starting with its constant term. The candidate F has ascending rational coefficient pairs [[-5, 1], [-4, 1], [9, 2], [1, 3], [1, 4]], with positive denominators. Is F′=f as a coefficient identity?","answer":{"kind":"short_text","value":"Yes","explanation":"Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving Yes. This is coefficient identity, not finite point sampling; the candidate constant is free."},"skills":["Coefficient Eligibility of a Candidate Antiderivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0060","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5df6339d427e0902b2cba07fb8e9cb54e7b21b2d91a51c2a11ab1140cdb4fbc","family_id":"qf-antiderivative-candidate","seed":202610086060,"semantic_template":"qf-antiderivative-candidate/{\"inputs\":[[\"-3\",\"10\",\"-1\",\"2\"],[\"-3/1\",\"5/1\",\"-1/3\",\"3/2\"]],\"operation\":\"primitive_candidate\"}"},"fields":{"title":"Coefficient Eligibility of a Candidate Antiderivative","stem":"On all R, f has ascending coefficients [-3, 10, -1, 2], starting with its constant term. The candidate F has ascending rational coefficient pairs [[-4, 1], [-3, 1], [5, 1], [-1, 3], [3, 2]], with positive denominators. Is F′=f as a coefficient identity?","answer":{"kind":"short_text","value":"No","explanation":"Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving No. This is coefficient identity, not finite point sampling; the candidate constant is free."},"skills":["Coefficient Eligibility of a Candidate Antiderivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0061","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a02fa09d77edeb41a954ed747c9de7ecd855be0f32efa8cf57857d8234119692","family_id":"qf-antiderivative-candidate","seed":202610086061,"semantic_template":"qf-antiderivative-candidate/{\"inputs\":[[\"-2\",\"11\",\"0\",\"3\"],[\"-2/1\",\"11/2\",\"0/1\",\"3/4\"]],\"operation\":\"primitive_candidate\"}"},"fields":{"title":"Coefficient Eligibility of a Candidate Antiderivative","stem":"On all R, f has ascending coefficients [-2, 11, 0, 3], starting with its constant term. The candidate F has ascending rational coefficient pairs [[-3, 1], [-2, 1], [11, 2], [0, 1], [3, 4]], with positive denominators. Is F′=f as a coefficient identity?","answer":{"kind":"short_text","value":"Yes","explanation":"Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving Yes. This is coefficient identity, not finite point sampling; the candidate constant is free."},"skills":["Coefficient Eligibility of a Candidate Antiderivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0062","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8a48ada75ba23aa39936fd7a74868682702389c1c581c8a9d19dfe73c366a5b","family_id":"qf-antiderivative-candidate","seed":202610086062,"semantic_template":"qf-antiderivative-candidate/{\"inputs\":[[\"-1\",\"12\",\"1\",\"4\"],[\"-1/1\",\"6/1\",\"1/3\",\"2/1\"]],\"operation\":\"primitive_candidate\"}"},"fields":{"title":"Coefficient Eligibility of a Candidate Antiderivative","stem":"On all R, f has ascending coefficients [-1, 12, 1, 4], starting with its constant term. The candidate F has ascending rational coefficient pairs [[-2, 1], [-1, 1], [6, 1], [1, 3], [2, 1]], with positive denominators. Is F′=f as a coefficient identity?","answer":{"kind":"short_text","value":"No","explanation":"Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving No. This is coefficient identity, not finite point sampling; the candidate constant is free."},"skills":["Coefficient Eligibility of a Candidate Antiderivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0063","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a143da1bc41858d40b4b7dab6ba325352b988928963553e6427b547247634482","family_id":"qf-antiderivative-candidate","seed":202610086063,"semantic_template":"qf-antiderivative-candidate/{\"inputs\":[[\"0\",\"13\",\"-1\",\"1\"],[\"0/1\",\"13/2\",\"-1/3\",\"1/4\"]],\"operation\":\"primitive_candidate\"}"},"fields":{"title":"Coefficient Eligibility of a Candidate Antiderivative","stem":"On all R, f has ascending coefficients [0, 13, -1, 1], starting with its constant term. The candidate F has ascending rational coefficient pairs [[-1, 1], [0, 1], [13, 2], [-1, 3], [1, 4]], with positive denominators. Is F′=f as a coefficient identity?","answer":{"kind":"short_text","value":"Yes","explanation":"Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving Yes. This is coefficient identity, not finite point sampling; the candidate constant is free."},"skills":["Coefficient Eligibility of a Candidate Antiderivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0064","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3593e0774e7faba081f12d9e51a30ba55eaf7cbef0996d37679165ed98178af4","family_id":"qf-antiderivative-candidate","seed":202610086064,"semantic_template":"qf-antiderivative-candidate/{\"inputs\":[[\"1\",\"14\",\"0\",\"2\"],[\"1/1\",\"7/1\",\"0/1\",\"3/2\"]],\"operation\":\"primitive_candidate\"}"},"fields":{"title":"Coefficient Eligibility of a Candidate Antiderivative","stem":"On all R, f has ascending coefficients [1, 14, 0, 2], starting with its constant term. The candidate F has ascending rational coefficient pairs [[0, 1], [1, 1], [7, 1], [0, 1], [3, 2]], with positive denominators. Is F′=f as a coefficient identity?","answer":{"kind":"short_text","value":"No","explanation":"Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving No. This is coefficient identity, not finite point sampling; the candidate constant is free."},"skills":["Coefficient Eligibility of a Candidate Antiderivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0065","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3fae5e817d8417dfd245eba051ef20eb4a90280a599f2ec5d509a89ba34b6eac","family_id":"qf-antiderivative-candidate","seed":202610086065,"semantic_template":"qf-antiderivative-candidate/{\"inputs\":[[\"2\",\"15\",\"1\",\"3\"],[\"2/1\",\"15/2\",\"1/3\",\"3/4\"]],\"operation\":\"primitive_candidate\"}"},"fields":{"title":"Coefficient Eligibility of a Candidate Antiderivative","stem":"On all R, f has ascending coefficients [2, 15, 1, 3], starting with its constant term. The candidate F has ascending rational coefficient pairs [[1, 1], [2, 1], [15, 2], [1, 3], [3, 4]], with positive denominators. Is F′=f as a coefficient identity?","answer":{"kind":"short_text","value":"Yes","explanation":"Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving Yes. This is coefficient identity, not finite point sampling; the candidate constant is free."},"skills":["Coefficient Eligibility of a Candidate Antiderivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0066","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f03e08679726957d75cb3ecfbd5056844265addd3b3882d70b51c0cf18484d1","family_id":"qf-antiderivative-candidate","seed":202610086066,"semantic_template":"qf-antiderivative-candidate/{\"inputs\":[[\"3\",\"16\",\"-1\",\"4\"],[\"3/1\",\"8/1\",\"-1/3\",\"2/1\"]],\"operation\":\"primitive_candidate\"}"},"fields":{"title":"Coefficient Eligibility of a Candidate Antiderivative","stem":"On all R, f has ascending coefficients [3, 16, -1, 4], starting with its constant term. The candidate F has ascending rational coefficient pairs [[2, 1], [3, 1], [8, 1], [-1, 3], [2, 1]], with positive denominators. Is F′=f as a coefficient identity?","answer":{"kind":"short_text","value":"No","explanation":"Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving No. This is coefficient identity, not finite point sampling; the candidate constant is free."},"skills":["Coefficient Eligibility of a Candidate Antiderivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0067","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8cc27eabc6b3457fd49a20057aeb81cada2d94eccc7073e8806d5253dbb5c9b2","family_id":"qf-antiderivative-candidate","seed":202610086067,"semantic_template":"qf-antiderivative-candidate/{\"inputs\":[[\"4\",\"17\",\"0\",\"1\"],[\"4/1\",\"17/2\",\"0/1\",\"1/4\"]],\"operation\":\"primitive_candidate\"}"},"fields":{"title":"Coefficient Eligibility of a Candidate Antiderivative","stem":"On all R, f has ascending coefficients [4, 17, 0, 1], starting with its constant term. The candidate F has ascending rational coefficient pairs [[3, 1], [4, 1], [17, 2], [0, 1], [1, 4]], with positive denominators. Is F′=f as a coefficient identity?","answer":{"kind":"short_text","value":"Yes","explanation":"Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving Yes. This is coefficient identity, not finite point sampling; the candidate constant is free."},"skills":["Coefficient Eligibility of a Candidate Antiderivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0068","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6c3d761c6f03bbf13b2cb5f6cfef9b939a47dba45b893f3d2bc5942119adc2b","family_id":"qf-antiderivative-candidate","seed":202610086068,"semantic_template":"qf-antiderivative-candidate/{\"inputs\":[[\"5\",\"18\",\"1\",\"2\"],[\"5/1\",\"9/1\",\"1/3\",\"3/2\"]],\"operation\":\"primitive_candidate\"}"},"fields":{"title":"Coefficient Eligibility of a Candidate Antiderivative","stem":"On all R, f has ascending coefficients [5, 18, 1, 2], starting with its constant term. The candidate F has ascending rational coefficient pairs [[4, 1], [5, 1], [9, 1], [1, 3], [3, 2]], with positive denominators. Is F′=f as a coefficient identity?","answer":{"kind":"short_text","value":"No","explanation":"Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving No. This is coefficient identity, not finite point sampling; the candidate constant is free."},"skills":["Coefficient Eligibility of a Candidate Antiderivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0069","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5663dbd0ddc99b354441e8b1848318fc417817584de2a61b26a3953ccb2d150","family_id":"qf-antiderivative-candidate","seed":202610086069,"semantic_template":"qf-antiderivative-candidate/{\"inputs\":[[\"6\",\"19\",\"-1\",\"3\"],[\"6/1\",\"19/2\",\"-1/3\",\"3/4\"]],\"operation\":\"primitive_candidate\"}"},"fields":{"title":"Coefficient Eligibility of a Candidate Antiderivative","stem":"On all R, f has ascending coefficients [6, 19, -1, 3], starting with its constant term. The candidate F has ascending rational coefficient pairs [[5, 1], [6, 1], [19, 2], [-1, 3], [3, 4]], with positive denominators. Is F′=f as a coefficient identity?","answer":{"kind":"short_text","value":"Yes","explanation":"Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving Yes. This is coefficient identity, not finite point sampling; the candidate constant is free."},"skills":["Coefficient Eligibility of a Candidate Antiderivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0070","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16d15d827f51ae3d901f92e91b0e19b36f4250abc8b41c4e9d2bbe07f6ca0234","family_id":"qf-antiderivative-candidate","seed":202610086070,"semantic_template":"qf-antiderivative-candidate/{\"inputs\":[[\"7\",\"20\",\"0\",\"4\"],[\"7/1\",\"10/1\",\"0/1\",\"2/1\"]],\"operation\":\"primitive_candidate\"}"},"fields":{"title":"Coefficient Eligibility of a Candidate Antiderivative","stem":"On all R, f has ascending coefficients [7, 20, 0, 4], starting with its constant term. The candidate F has ascending rational coefficient pairs [[6, 1], [7, 1], [10, 1], [0, 1], [2, 1]], with positive denominators. Is F′=f as a coefficient identity?","answer":{"kind":"short_text","value":"No","explanation":"Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving No. This is coefficient identity, not finite point sampling; the candidate constant is free."},"skills":["Coefficient Eligibility of a Candidate Antiderivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0071","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a86c0c5c0852392729534725b4ed67f4df41097948560c9f015719a4307edb9a","family_id":"qf-antiderivative-candidate","seed":202610086071,"semantic_template":"qf-antiderivative-candidate/{\"inputs\":[[\"8\",\"21\",\"1\",\"1\"],[\"8/1\",\"21/2\",\"1/3\",\"1/4\"]],\"operation\":\"primitive_candidate\"}"},"fields":{"title":"Coefficient Eligibility of a Candidate Antiderivative","stem":"On all R, f has ascending coefficients [8, 21, 1, 1], starting with its constant term. The candidate F has ascending rational coefficient pairs [[7, 1], [8, 1], [21, 2], [1, 3], [1, 4]], with positive denominators. Is F′=f as a coefficient identity?","answer":{"kind":"short_text","value":"Yes","explanation":"Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving Yes. This is coefficient identity, not finite point sampling; the candidate constant is free."},"skills":["Coefficient Eligibility of a Candidate Antiderivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0072","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"904676b91177e4911fcfe05dfaf89299b1b077024ba5ade29604f2a5b900bdc9","family_id":"qf-antiderivative-candidate","seed":202610086072,"semantic_template":"qf-antiderivative-candidate/{\"inputs\":[[\"9\",\"22\",\"-1\",\"2\"],[\"9/1\",\"11/1\",\"-1/3\",\"3/2\"]],\"operation\":\"primitive_candidate\"}"},"fields":{"title":"Coefficient Eligibility of a Candidate Antiderivative","stem":"On all R, f has ascending coefficients [9, 22, -1, 2], starting with its constant term. The candidate F has ascending rational coefficient pairs [[8, 1], [9, 1], [11, 1], [-1, 3], [3, 2]], with positive denominators. Is F′=f as a coefficient identity?","answer":{"kind":"short_text","value":"No","explanation":"Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving No. This is coefficient identity, not finite point sampling; the candidate constant is free."},"skills":["Coefficient Eligibility of a Candidate Antiderivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0073","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e938719caa7ba6c639ebc7df830778ebf7e02f474daa43fdbe186194aab861d0","family_id":"qf-antiderivative-candidate","seed":202610086073,"semantic_template":"qf-antiderivative-candidate/{\"inputs\":[[\"10\",\"23\",\"0\",\"3\"],[\"10/1\",\"23/2\",\"0/1\",\"3/4\"]],\"operation\":\"primitive_candidate\"}"},"fields":{"title":"Coefficient Eligibility of a Candidate Antiderivative","stem":"On all R, f has ascending coefficients [10, 23, 0, 3], starting with its constant term. The candidate F has ascending rational coefficient pairs [[9, 1], [10, 1], [23, 2], [0, 1], [3, 4]], with positive denominators. Is F′=f as a coefficient identity?","answer":{"kind":"short_text","value":"Yes","explanation":"Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving Yes. This is coefficient identity, not finite point sampling; the candidate constant is free."},"skills":["Coefficient Eligibility of a Candidate Antiderivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0074","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6b0f96367ceb8a386cff904dce89fbd6e8bc3a2397f38bfbc44f40be8a3a76b","family_id":"qf-antiderivative-candidate","seed":202610086074,"semantic_template":"qf-antiderivative-candidate/{\"inputs\":[[\"11\",\"24\",\"1\",\"4\"],[\"11/1\",\"12/1\",\"1/3\",\"2/1\"]],\"operation\":\"primitive_candidate\"}"},"fields":{"title":"Coefficient Eligibility of a Candidate Antiderivative","stem":"On all R, f has ascending coefficients [11, 24, 1, 4], starting with its constant term. The candidate F has ascending rational coefficient pairs [[10, 1], [11, 1], [12, 1], [1, 3], [2, 1]], with positive denominators. Is F′=f as a coefficient identity?","answer":{"kind":"short_text","value":"No","explanation":"Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving No. This is coefficient identity, not finite point sampling; the candidate constant is free."},"skills":["Coefficient Eligibility of a Candidate Antiderivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0075","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e1a218935868ccabc54351c091052594939f9bdc084654010fdf66f1d3ed3ff","family_id":"qf-antiderivative-candidate","seed":202610086075,"semantic_template":"qf-antiderivative-candidate/{\"inputs\":[[\"12\",\"25\",\"-1\",\"1\"],[\"12/1\",\"25/2\",\"-1/3\",\"1/4\"]],\"operation\":\"primitive_candidate\"}"},"fields":{"title":"Coefficient Eligibility of a Candidate Antiderivative","stem":"On all R, f has ascending coefficients [12, 25, -1, 1], starting with its constant term. The candidate F has ascending rational coefficient pairs [[11, 1], [12, 1], [25, 2], [-1, 3], [1, 4]], with positive denominators. Is F′=f as a coefficient identity?","answer":{"kind":"short_text","value":"Yes","explanation":"Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving Yes. This is coefficient identity, not finite point sampling; the candidate constant is free."},"skills":["Coefficient Eligibility of a Candidate Antiderivative"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0076","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"204c92ab008d1f96061637de42852d45b42deab6c959d253a76fd3f4de25cb6f","family_id":"qf-antiderivative-shifted-value","seed":202610086076,"semantic_template":"qf-antiderivative-shifted-value/{\"inputs\":[\"constant_value\",\"0\"],\"operation\":\"shifted_primitive_value\"}"},"fields":{"title":"Antiderivative Value Retaining the Actual Constant","stem":"On all R, f has ascending coefficients [1, 2], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(0)+(0), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"0/1","explanation":"Evaluate the rational primitive exactly at 0, then add C=0, obtaining 0/1. A constant irrelevant to differentiation remains relevant to this value."},"skills":["Antiderivative Value Retaining the Actual Constant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0077","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"250c1e9af02bf6293930168762ef5b557b0477aaa706905f09eaaf319af5aaee","family_id":"qf-antiderivative-shifted-value","seed":202610086077,"semantic_template":"qf-antiderivative-shifted-value/{\"inputs\":[\"constant_value\",\"1\"],\"operation\":\"shifted_primitive_value\"}"},"fields":{"title":"Antiderivative Value Retaining the Actual Constant","stem":"On all R, f has ascending coefficients [0], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(3)+(1), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"1/1","explanation":"Evaluate the rational primitive exactly at 3, then add C=1, obtaining 1/1. A constant irrelevant to differentiation remains relevant to this value."},"skills":["Antiderivative Value Retaining the Actual Constant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0078","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dea04945e7f801e7a9fbe6c0a0293bfdc7aedd3dfade21a7b0b563af39e669a0","family_id":"qf-antiderivative-shifted-value","seed":202610086078,"semantic_template":"qf-antiderivative-shifted-value/{\"inputs\":[[\"-10\",\"3\",\"1\",\"3\"],\"-10\",\"3\"],\"operation\":\"shifted_primitive_value\"}"},"fields":{"title":"Antiderivative Value Retaining the Actual Constant","stem":"On all R, f has ascending coefficients [-10, 3, 1, 3], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(3)+(-10), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"173/4","explanation":"Evaluate the rational primitive exactly at 3, then add C=-10, obtaining 173/4. A constant irrelevant to differentiation remains relevant to this value."},"skills":["Antiderivative Value Retaining the Actual Constant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0079","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53efa0df479bde1579ab49793f89210a12075fbb5b693d2023b601473a37c825","family_id":"qf-antiderivative-shifted-value","seed":202610086079,"semantic_template":"qf-antiderivative-shifted-value/{\"inputs\":[[\"-9\",\"4\",\"-1\",\"4\"],\"-9\",\"-1\"],\"operation\":\"shifted_primitive_value\"}"},"fields":{"title":"Antiderivative Value Retaining the Actual Constant","stem":"On all R, f has ascending coefficients [-9, 4, -1, 4], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(-1)+(-9), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"10/3","explanation":"Evaluate the rational primitive exactly at -1, then add C=-9, obtaining 10/3. A constant irrelevant to differentiation remains relevant to this value."},"skills":["Antiderivative Value Retaining the Actual Constant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0080","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8d7d4df98f017df57bab18c503219e8d32567482179344c0a17f18f7a6061dc","family_id":"qf-antiderivative-shifted-value","seed":202610086080,"semantic_template":"qf-antiderivative-shifted-value/{\"inputs\":[[\"-8\",\"5\",\"0\",\"1\"],\"-8\",\"2\"],\"operation\":\"shifted_primitive_value\"}"},"fields":{"title":"Antiderivative Value Retaining the Actual Constant","stem":"On all R, f has ascending coefficients [-8, 5, 0, 1], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(2)+(-8), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-10/1","explanation":"Evaluate the rational primitive exactly at 2, then add C=-8, obtaining -10/1. A constant irrelevant to differentiation remains relevant to this value."},"skills":["Antiderivative Value Retaining the Actual Constant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0081","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"426dc9c3c3fc0652b71fc1a4514eb29400ff2c946f6ec040113290be552ed3a7","family_id":"qf-antiderivative-shifted-value","seed":202610086081,"semantic_template":"qf-antiderivative-shifted-value/{\"inputs\":[[\"-7\",\"6\",\"1\",\"2\"],\"-7\",\"-3\"],\"operation\":\"shifted_primitive_value\"}"},"fields":{"title":"Antiderivative Value Retaining the Actual Constant","stem":"On all R, f has ascending coefficients [-7, 6, 1, 2], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(-3)+(-7), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"145/2","explanation":"Evaluate the rational primitive exactly at -3, then add C=-7, obtaining 145/2. A constant irrelevant to differentiation remains relevant to this value."},"skills":["Antiderivative Value Retaining the Actual Constant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0082","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"82d67c97128c490b1efb81b72601df3313bfaeba07f6cac5f86755e989f46285","family_id":"qf-antiderivative-shifted-value","seed":202610086082,"semantic_template":"qf-antiderivative-shifted-value/{\"inputs\":[[\"-6\",\"7\",\"-1\",\"3\"],\"-6\",\"1\"],\"operation\":\"shifted_primitive_value\"}"},"fields":{"title":"Antiderivative Value Retaining the Actual Constant","stem":"On all R, f has ascending coefficients [-6, 7, -1, 3], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(1)+(-6), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"-97/12","explanation":"Evaluate the rational primitive exactly at 1, then add C=-6, obtaining -97/12. A constant irrelevant to differentiation remains relevant to this value."},"skills":["Antiderivative Value Retaining the Actual Constant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0083","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40e46b1d800b247e62d1f1738051e111f62a64ed1f0c0a7cf1ea7063a030077f","family_id":"qf-antiderivative-shifted-value","seed":202610086083,"semantic_template":"qf-antiderivative-shifted-value/{\"inputs\":[[\"-5\",\"8\",\"0\",\"4\"],\"-5\",\"-2\"],\"operation\":\"shifted_primitive_value\"}"},"fields":{"title":"Antiderivative Value Retaining the Actual Constant","stem":"On all R, f has ascending coefficients [-5, 8, 0, 4], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(-2)+(-5), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"37/1","explanation":"Evaluate the rational primitive exactly at -2, then add C=-5, obtaining 37/1. A constant irrelevant to differentiation remains relevant to this value."},"skills":["Antiderivative Value Retaining the Actual Constant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0084","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eba7ee84b8201720e1475457be57010d19dae19a15a5d66e35eef075dd18e765","family_id":"qf-antiderivative-shifted-value","seed":202610086084,"semantic_template":"qf-antiderivative-shifted-value/{\"inputs\":[[\"-4\",\"9\",\"1\",\"1\"],\"-4\",\"3\"],\"operation\":\"shifted_primitive_value\"}"},"fields":{"title":"Antiderivative Value Retaining the Actual Constant","stem":"On all R, f has ascending coefficients [-4, 9, 1, 1], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(3)+(-4), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"215/4","explanation":"Evaluate the rational primitive exactly at 3, then add C=-4, obtaining 215/4. A constant irrelevant to differentiation remains relevant to this value."},"skills":["Antiderivative Value Retaining the Actual Constant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0085","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a24b1e758524b38020314b21dab5ea512a81b9f63100b0e6841e6b87bfe8d235","family_id":"qf-antiderivative-shifted-value","seed":202610086085,"semantic_template":"qf-antiderivative-shifted-value/{\"inputs\":[[\"-3\",\"10\",\"-1\",\"2\"],\"-3\",\"-1\"],\"operation\":\"shifted_primitive_value\"}"},"fields":{"title":"Antiderivative Value Retaining the Actual Constant","stem":"On all R, f has ascending coefficients [-3, 10, -1, 2], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(-1)+(-3), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"35/6","explanation":"Evaluate the rational primitive exactly at -1, then add C=-3, obtaining 35/6. A constant irrelevant to differentiation remains relevant to this value."},"skills":["Antiderivative Value Retaining the Actual Constant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0086","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11a8f174212853043b1bd6ddc442b8c442d883e9d57bef9af008171df99665cc","family_id":"qf-antiderivative-shifted-value","seed":202610086086,"semantic_template":"qf-antiderivative-shifted-value/{\"inputs\":[[\"-2\",\"11\",\"0\",\"3\"],\"-2\",\"2\"],\"operation\":\"shifted_primitive_value\"}"},"fields":{"title":"Antiderivative Value Retaining the Actual Constant","stem":"On all R, f has ascending coefficients [-2, 11, 0, 3], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(2)+(-2), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"28/1","explanation":"Evaluate the rational primitive exactly at 2, then add C=-2, obtaining 28/1. A constant irrelevant to differentiation remains relevant to this value."},"skills":["Antiderivative Value Retaining the Actual Constant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0087","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b48b3d095ef9222df7a2d7db9c70398c58dc3b8cbc967d3a30bcf21db312db7","family_id":"qf-antiderivative-shifted-value","seed":202610086087,"semantic_template":"qf-antiderivative-shifted-value/{\"inputs\":[[\"-1\",\"12\",\"1\",\"4\"],\"-1\",\"-3\"],\"operation\":\"shifted_primitive_value\"}"},"fields":{"title":"Antiderivative Value Retaining the Actual Constant","stem":"On all R, f has ascending coefficients [-1, 12, 1, 4], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(-3)+(-1), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"128/1","explanation":"Evaluate the rational primitive exactly at -3, then add C=-1, obtaining 128/1. A constant irrelevant to differentiation remains relevant to this value."},"skills":["Antiderivative Value Retaining the Actual Constant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0088","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53cf624ef31bc5244aaf41883cc62f584c839e60196b010643a0c490815d5b3e","family_id":"qf-antiderivative-shifted-value","seed":202610086088,"semantic_template":"qf-antiderivative-shifted-value/{\"inputs\":[[\"0\",\"13\",\"-1\",\"1\"],\"0\",\"1\"],\"operation\":\"shifted_primitive_value\"}"},"fields":{"title":"Antiderivative Value Retaining the Actual Constant","stem":"On all R, f has ascending coefficients [0, 13, -1, 1], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(1)+(0), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"77/12","explanation":"Evaluate the rational primitive exactly at 1, then add C=0, obtaining 77/12. A constant irrelevant to differentiation remains relevant to this value."},"skills":["Antiderivative Value Retaining the Actual Constant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0089","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ddd6eca7b6cb99b371ade1f5fd480c344909ff182ea095521bff76b2c0a69864","family_id":"qf-antiderivative-shifted-value","seed":202610086089,"semantic_template":"qf-antiderivative-shifted-value/{\"inputs\":[[\"1\",\"14\",\"0\",\"2\"],\"1\",\"-2\"],\"operation\":\"shifted_primitive_value\"}"},"fields":{"title":"Antiderivative Value Retaining the Actual Constant","stem":"On all R, f has ascending coefficients [1, 14, 0, 2], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(-2)+(1), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"35/1","explanation":"Evaluate the rational primitive exactly at -2, then add C=1, obtaining 35/1. A constant irrelevant to differentiation remains relevant to this value."},"skills":["Antiderivative Value Retaining the Actual Constant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0090","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7c35045b58285d110d6e2150b3da117ea98d11eb35a7ac4ec50c76eb2d73f384","family_id":"qf-antiderivative-shifted-value","seed":202610086090,"semantic_template":"qf-antiderivative-shifted-value/{\"inputs\":[[\"2\",\"15\",\"1\",\"3\"],\"2\",\"3\"],\"operation\":\"shifted_primitive_value\"}"},"fields":{"title":"Antiderivative Value Retaining the Actual Constant","stem":"On all R, f has ascending coefficients [2, 15, 1, 3], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(3)+(2), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"581/4","explanation":"Evaluate the rational primitive exactly at 3, then add C=2, obtaining 581/4. A constant irrelevant to differentiation remains relevant to this value."},"skills":["Antiderivative Value Retaining the Actual Constant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0091","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19c1e849c6020d4e0b8453963351a92a1649ef4f0f2b927385c61c95a87738cd","family_id":"qf-antiderivative-shifted-value","seed":202610086091,"semantic_template":"qf-antiderivative-shifted-value/{\"inputs\":[[\"3\",\"16\",\"-1\",\"4\"],\"3\",\"-1\"],\"operation\":\"shifted_primitive_value\"}"},"fields":{"title":"Antiderivative Value Retaining the Actual Constant","stem":"On all R, f has ascending coefficients [3, 16, -1, 4], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(-1)+(3), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"28/3","explanation":"Evaluate the rational primitive exactly at -1, then add C=3, obtaining 28/3. A constant irrelevant to differentiation remains relevant to this value."},"skills":["Antiderivative Value Retaining the Actual Constant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0092","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34c653d58f916a971aa28f8c732ef684e8f022f634ca078134b8f2acb28fd2b7","family_id":"qf-antiderivative-shifted-value","seed":202610086092,"semantic_template":"qf-antiderivative-shifted-value/{\"inputs\":[[\"4\",\"17\",\"0\",\"1\"],\"4\",\"2\"],\"operation\":\"shifted_primitive_value\"}"},"fields":{"title":"Antiderivative Value Retaining the Actual Constant","stem":"On all R, f has ascending coefficients [4, 17, 0, 1], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(2)+(4), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"50/1","explanation":"Evaluate the rational primitive exactly at 2, then add C=4, obtaining 50/1. A constant irrelevant to differentiation remains relevant to this value."},"skills":["Antiderivative Value Retaining the Actual Constant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0093","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3445d4528082672572c0983a78ad5cfd7a00814f80fd1568908d1cb49e7ffabe","family_id":"qf-antiderivative-shifted-value","seed":202610086093,"semantic_template":"qf-antiderivative-shifted-value/{\"inputs\":[[\"5\",\"18\",\"1\",\"2\"],\"5\",\"-3\"],\"operation\":\"shifted_primitive_value\"}"},"fields":{"title":"Antiderivative Value Retaining the Actual Constant","stem":"On all R, f has ascending coefficients [5, 18, 1, 2], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(-3)+(5), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"205/2","explanation":"Evaluate the rational primitive exactly at -3, then add C=5, obtaining 205/2. A constant irrelevant to differentiation remains relevant to this value."},"skills":["Antiderivative Value Retaining the Actual Constant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0094","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3cd8dc7511ab800273314d78766cf148c5c43dde79b8585ce59487cd1cf42b87","family_id":"qf-antiderivative-shifted-value","seed":202610086094,"semantic_template":"qf-antiderivative-shifted-value/{\"inputs\":[[\"6\",\"19\",\"-1\",\"3\"],\"6\",\"1\"],\"operation\":\"shifted_primitive_value\"}"},"fields":{"title":"Antiderivative Value Retaining the Actual Constant","stem":"On all R, f has ascending coefficients [6, 19, -1, 3], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(1)+(6), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"263/12","explanation":"Evaluate the rational primitive exactly at 1, then add C=6, obtaining 263/12. A constant irrelevant to differentiation remains relevant to this value."},"skills":["Antiderivative Value Retaining the Actual Constant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0095","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c66ccf2d30233e83a84730033e70b1151b03b2b914700ad4147252874c5dac4","family_id":"qf-antiderivative-shifted-value","seed":202610086095,"semantic_template":"qf-antiderivative-shifted-value/{\"inputs\":[[\"7\",\"20\",\"0\",\"4\"],\"7\",\"-2\"],\"operation\":\"shifted_primitive_value\"}"},"fields":{"title":"Antiderivative Value Retaining the Actual Constant","stem":"On all R, f has ascending coefficients [7, 20, 0, 4], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(-2)+(7), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"49/1","explanation":"Evaluate the rational primitive exactly at -2, then add C=7, obtaining 49/1. A constant irrelevant to differentiation remains relevant to this value."},"skills":["Antiderivative Value Retaining the Actual Constant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0096","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d30807c264cb2ac34e100240589d80429749366615965a90b93575990ba116ee","family_id":"qf-antiderivative-shifted-value","seed":202610086096,"semantic_template":"qf-antiderivative-shifted-value/{\"inputs\":[[\"8\",\"21\",\"1\",\"1\"],\"8\",\"3\"],\"operation\":\"shifted_primitive_value\"}"},"fields":{"title":"Antiderivative Value Retaining the Actual Constant","stem":"On all R, f has ascending coefficients [8, 21, 1, 1], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(3)+(8), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"623/4","explanation":"Evaluate the rational primitive exactly at 3, then add C=8, obtaining 623/4. A constant irrelevant to differentiation remains relevant to this value."},"skills":["Antiderivative Value Retaining the Actual Constant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0097","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"921976fff1d9b7e904df5b741a8fcb01a8bc7fff7e8b7211cd10b4f10538e5b2","family_id":"qf-antiderivative-shifted-value","seed":202610086097,"semantic_template":"qf-antiderivative-shifted-value/{\"inputs\":[[\"9\",\"22\",\"-1\",\"2\"],\"9\",\"-1\"],\"operation\":\"shifted_primitive_value\"}"},"fields":{"title":"Antiderivative Value Retaining the Actual Constant","stem":"On all R, f has ascending coefficients [9, 22, -1, 2], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(-1)+(9), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"71/6","explanation":"Evaluate the rational primitive exactly at -1, then add C=9, obtaining 71/6. A constant irrelevant to differentiation remains relevant to this value."},"skills":["Antiderivative Value Retaining the Actual Constant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0098","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84f2b244d12f93bf08897a94c0972bb264ee9c36670a94402fc441b82c760fb3","family_id":"qf-antiderivative-shifted-value","seed":202610086098,"semantic_template":"qf-antiderivative-shifted-value/{\"inputs\":[[\"10\",\"23\",\"0\",\"3\"],\"10\",\"2\"],\"operation\":\"shifted_primitive_value\"}"},"fields":{"title":"Antiderivative Value Retaining the Actual Constant","stem":"On all R, f has ascending coefficients [10, 23, 0, 3], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(2)+(10), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"88/1","explanation":"Evaluate the rational primitive exactly at 2, then add C=10, obtaining 88/1. A constant irrelevant to differentiation remains relevant to this value."},"skills":["Antiderivative Value Retaining the Actual Constant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0099","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8865cc04427f6411b61c81ae4144732ed146ae54cc33dd0bed6980b8aa101d90","family_id":"qf-antiderivative-shifted-value","seed":202610086099,"semantic_template":"qf-antiderivative-shifted-value/{\"inputs\":[[\"11\",\"24\",\"1\",\"4\"],\"11\",\"-3\"],\"operation\":\"shifted_primitive_value\"}"},"fields":{"title":"Antiderivative Value Retaining the Actual Constant","stem":"On all R, f has ascending coefficients [11, 24, 1, 4], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(-3)+(11), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"158/1","explanation":"Evaluate the rational primitive exactly at -3, then add C=11, obtaining 158/1. A constant irrelevant to differentiation remains relevant to this value."},"skills":["Antiderivative Value Retaining the Actual Constant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261008-r06-antiderivative-0100","batch_id":"batch-20261007-advance-20261008-r06-antiderivative-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6fb8c5ff1e36a8c31d256b85c8594db16ac71e05e38c8434f2f1494104ca6d6","family_id":"qf-antiderivative-shifted-value","seed":202610086100,"semantic_template":"qf-antiderivative-shifted-value/{\"inputs\":[[\"12\",\"25\",\"-1\",\"1\"],\"12\",\"1\"],\"operation\":\"shifted_primitive_value\"}"},"fields":{"title":"Antiderivative Value Retaining the Actual Constant","stem":"On all R, f has ascending coefficients [12, 25, -1, 1], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(1)+(12), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.","answer":{"kind":"expression","value":"437/12","explanation":"Evaluate the rational primitive exactly at 1, then add C=12, obtaining 437/12. A constant irrelevant to differentiation remains relevant to this value."},"skills":["Antiderivative Value Retaining the Actual Constant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}