{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0001","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a6d1b9229b26be8df2fddfc75eff7be538cc54f4e3d3325fe20aa1c62226462","family_id":"qf-app20-dimension-product","seed":202610205001,"semantic_template":"qf-app20-dimension-product/{\"left\":[-3,-1,-3],\"right\":[-3,-2,-2]}"},"fields":{"title":"Product Dimension by Componentwise Exponent Addition","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-3, -1, -3], right=[-3, -2, -2]. Give (left, right, product exponent vector).","answer":{"kind":"expression","value":"([-3, -1, -3], [-3, -2, -2], [-6, -3, -5])","explanation":"Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, -1, -3], [-3, -2, -2], [-6, -3, -5])."},"skills":["Product Dimension by Componentwise Exponent Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0002","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f4cb9386659287ef115b05b829cf7d5c789287d4d6db45d9fa7dd39364da3c2","family_id":"qf-app20-dimension-product","seed":202610205002,"semantic_template":"qf-app20-dimension-product/{\"left\":[-2,-1,-1],\"right\":[0,-1,-2]}"},"fields":{"title":"Product Dimension by Componentwise Exponent Addition","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-2, -1, -1], right=[0, -1, -2]. Give (left, right, product exponent vector).","answer":{"kind":"expression","value":"([-2, -1, -1], [0, -1, -2], [-2, -2, -3])","explanation":"Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, -1, -1], [0, -1, -2], [-2, -2, -3])."},"skills":["Product Dimension by Componentwise Exponent Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0003","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a62548bbae229462e59d9cc383e817fd5878e1a3cdf1cc6697b9a915e6b08dd0","family_id":"qf-app20-dimension-product","seed":202610205003,"semantic_template":"qf-app20-dimension-product/{\"left\":[-1,-1,1],\"right\":[3,0,-2]}"},"fields":{"title":"Product Dimension by Componentwise Exponent Addition","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-1, -1, 1], right=[3, 0, -2]. Give (left, right, product exponent vector).","answer":{"kind":"expression","value":"([-1, -1, 1], [3, 0, -2], [2, -1, -1])","explanation":"Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, -1, 1], [3, 0, -2], [2, -1, -1])."},"skills":["Product Dimension by Componentwise Exponent Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0004","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab1bc0cdb2e8db30fed08513478f76e4bd4f1e75d49300792bcfbda136aff06c","family_id":"qf-app20-dimension-product","seed":202610205004,"semantic_template":"qf-app20-dimension-product/{\"left\":[0,-1,3],\"right\":[-1,1,-2]}"},"fields":{"title":"Product Dimension by Componentwise Exponent Addition","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[0, -1, 3], right=[-1, 1, -2]. Give (left, right, product exponent vector).","answer":{"kind":"expression","value":"([0, -1, 3], [-1, 1, -2], [-1, 0, 1])","explanation":"Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, -1, 3], [-1, 1, -2], [-1, 0, 1])."},"skills":["Product Dimension by Componentwise Exponent Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0005","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a2202508b35f60a761758635a072a757dee709ef2a4de616120c33998d34079","family_id":"qf-app20-dimension-product","seed":202610205005,"semantic_template":"qf-app20-dimension-product/{\"left\":[1,-1,-2],\"right\":[2,2,-2]}"},"fields":{"title":"Product Dimension by Componentwise Exponent Addition","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[1, -1, -2], right=[2, 2, -2]. Give (left, right, product exponent vector).","answer":{"kind":"expression","value":"([1, -1, -2], [2, 2, -2], [3, 1, -4])","explanation":"Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, -1, -2], [2, 2, -2], [3, 1, -4])."},"skills":["Product Dimension by Componentwise Exponent Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0006","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5c41aac4099690f57f1ba67d521155d406cfde3957a62841d6b91ec204454c8","family_id":"qf-app20-dimension-product","seed":202610205006,"semantic_template":"qf-app20-dimension-product/{\"left\":[2,-1,0],\"right\":[-2,-2,-1]}"},"fields":{"title":"Product Dimension by Componentwise Exponent Addition","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[2, -1, 0], right=[-2, -2, -1]. Give (left, right, product exponent vector).","answer":{"kind":"expression","value":"([2, -1, 0], [-2, -2, -1], [0, -3, -1])","explanation":"Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, -1, 0], [-2, -2, -1], [0, -3, -1])."},"skills":["Product Dimension by Componentwise Exponent Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0007","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd6f7b61aa7190838ea4d8252cbfa2f93540f7b487b59aebf41fa656646c8dea","family_id":"qf-app20-dimension-product","seed":202610205007,"semantic_template":"qf-app20-dimension-product/{\"left\":[3,-1,2],\"right\":[1,-1,-1]}"},"fields":{"title":"Product Dimension by Componentwise Exponent Addition","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[3, -1, 2], right=[1, -1, -1]. Give (left, right, product exponent vector).","answer":{"kind":"expression","value":"([3, -1, 2], [1, -1, -1], [4, -2, 1])","explanation":"Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, -1, 2], [1, -1, -1], [4, -2, 1])."},"skills":["Product Dimension by Componentwise Exponent Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0008","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3abe1aa8465435a0e3f243215126664ec134ffb068f2102d74b668a1882d033a","family_id":"qf-app20-dimension-product","seed":202610205008,"semantic_template":"qf-app20-dimension-product/{\"left\":[-3,0,-3],\"right\":[-3,0,-1]}"},"fields":{"title":"Product Dimension by Componentwise Exponent Addition","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-3, 0, -3], right=[-3, 0, -1]. Give (left, right, product exponent vector).","answer":{"kind":"expression","value":"([-3, 0, -3], [-3, 0, -1], [-6, 0, -4])","explanation":"Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, 0, -3], [-3, 0, -1], [-6, 0, -4])."},"skills":["Product Dimension by Componentwise Exponent Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0009","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02629507aa46b803f2216d6b4d91e26976f653154e096880c6672d8be3f7daa1","family_id":"qf-app20-dimension-product","seed":202610205009,"semantic_template":"qf-app20-dimension-product/{\"left\":[-2,0,-1],\"right\":[0,1,-1]}"},"fields":{"title":"Product Dimension by Componentwise Exponent Addition","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-2, 0, -1], right=[0, 1, -1]. Give (left, right, product exponent vector).","answer":{"kind":"expression","value":"([-2, 0, -1], [0, 1, -1], [-2, 1, -2])","explanation":"Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, 0, -1], [0, 1, -1], [-2, 1, -2])."},"skills":["Product Dimension by Componentwise Exponent Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0010","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0eeab988141d5a57a048525ecce67cf31936f466e61337e602d014fbec80169f","family_id":"qf-app20-dimension-product","seed":202610205010,"semantic_template":"qf-app20-dimension-product/{\"left\":[-1,0,1],\"right\":[3,2,-1]}"},"fields":{"title":"Product Dimension by Componentwise Exponent Addition","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-1, 0, 1], right=[3, 2, -1]. Give (left, right, product exponent vector).","answer":{"kind":"expression","value":"([-1, 0, 1], [3, 2, -1], [2, 2, 0])","explanation":"Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, 0, 1], [3, 2, -1], [2, 2, 0])."},"skills":["Product Dimension by Componentwise Exponent Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0011","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d41aef0102d58191200d9e6041fd6484084ef311c04b29eebbb6ceadb6444812","family_id":"qf-app20-dimension-product","seed":202610205011,"semantic_template":"qf-app20-dimension-product/{\"left\":[0,0,3],\"right\":[-1,-2,0]}"},"fields":{"title":"Product Dimension by Componentwise Exponent Addition","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[0, 0, 3], right=[-1, -2, 0]. Give (left, right, product exponent vector).","answer":{"kind":"expression","value":"([0, 0, 3], [-1, -2, 0], [-1, -2, 3])","explanation":"Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 0, 3], [-1, -2, 0], [-1, -2, 3])."},"skills":["Product Dimension by Componentwise Exponent Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0012","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b72710054754ac776a54170662038fec0f1576581291b5d7d5e328e8fe13aad","family_id":"qf-app20-dimension-product","seed":202610205012,"semantic_template":"qf-app20-dimension-product/{\"left\":[1,0,-2],\"right\":[2,-1,0]}"},"fields":{"title":"Product Dimension by Componentwise Exponent Addition","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[1, 0, -2], right=[2, -1, 0]. Give (left, right, product exponent vector).","answer":{"kind":"expression","value":"([1, 0, -2], [2, -1, 0], [3, -1, -2])","explanation":"Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 0, -2], [2, -1, 0], [3, -1, -2])."},"skills":["Product Dimension by Componentwise Exponent Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0013","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7861438e10cc29ebb8af5e184421843ee75dcb7b5a13b0767f3cd2bf1c748702","family_id":"qf-app20-dimension-product","seed":202610205013,"semantic_template":"qf-app20-dimension-product/{\"left\":[2,0,0],\"right\":[-2,0,0]}"},"fields":{"title":"Product Dimension by Componentwise Exponent Addition","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[2, 0, 0], right=[-2, 0, 0]. Give (left, right, product exponent vector).","answer":{"kind":"expression","value":"([2, 0, 0], [-2, 0, 0], [0, 0, 0])","explanation":"Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, 0, 0], [-2, 0, 0], [0, 0, 0])."},"skills":["Product Dimension by Componentwise Exponent Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0014","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aac127522cccdba7753fd257a830263e2ef19bbbf3a2358a54abf36cc140b51b","family_id":"qf-app20-dimension-product","seed":202610205014,"semantic_template":"qf-app20-dimension-product/{\"left\":[3,0,2],\"right\":[1,1,0]}"},"fields":{"title":"Product Dimension by Componentwise Exponent Addition","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[3, 0, 2], right=[1, 1, 0]. Give (left, right, product exponent vector).","answer":{"kind":"expression","value":"([3, 0, 2], [1, 1, 0], [4, 1, 2])","explanation":"Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, 0, 2], [1, 1, 0], [4, 1, 2])."},"skills":["Product Dimension by Componentwise Exponent Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0015","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5018b3b9be39d8526fb14abc71fed908e2135877d5bd0ce936692d59368a170e","family_id":"qf-app20-dimension-product","seed":202610205015,"semantic_template":"qf-app20-dimension-product/{\"left\":[-3,1,-3],\"right\":[-3,2,0]}"},"fields":{"title":"Product Dimension by Componentwise Exponent Addition","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-3, 1, -3], right=[-3, 2, 0]. Give (left, right, product exponent vector).","answer":{"kind":"expression","value":"([-3, 1, -3], [-3, 2, 0], [-6, 3, -3])","explanation":"Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, 1, -3], [-3, 2, 0], [-6, 3, -3])."},"skills":["Product Dimension by Componentwise Exponent Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0016","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f336d1a5486cd6d59708b72cd55e6c0ca63b4186ab1c0e791289d623b133d0a7","family_id":"qf-app20-dimension-product","seed":202610205016,"semantic_template":"qf-app20-dimension-product/{\"left\":[-2,1,-1],\"right\":[0,-2,1]}"},"fields":{"title":"Product Dimension by Componentwise Exponent Addition","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-2, 1, -1], right=[0, -2, 1]. Give (left, right, product exponent vector).","answer":{"kind":"expression","value":"([-2, 1, -1], [0, -2, 1], [-2, -1, 0])","explanation":"Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, 1, -1], [0, -2, 1], [-2, -1, 0])."},"skills":["Product Dimension by Componentwise Exponent Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0017","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8664aec6c9e28d6dc1be7385381d2bbe9b3c68a2d41b71156304aaa7dfa44be","family_id":"qf-app20-dimension-product","seed":202610205017,"semantic_template":"qf-app20-dimension-product/{\"left\":[-1,1,1],\"right\":[3,-1,1]}"},"fields":{"title":"Product Dimension by Componentwise Exponent Addition","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-1, 1, 1], right=[3, -1, 1]. Give (left, right, product exponent vector).","answer":{"kind":"expression","value":"([-1, 1, 1], [3, -1, 1], [2, 0, 2])","explanation":"Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, 1, 1], [3, -1, 1], [2, 0, 2])."},"skills":["Product Dimension by Componentwise Exponent Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0018","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e401d0abef6a8daaf259079d6f9776c78622886795ecbb7a3d53073b4d88bc0","family_id":"qf-app20-dimension-product","seed":202610205018,"semantic_template":"qf-app20-dimension-product/{\"left\":[0,1,3],\"right\":[-1,0,1]}"},"fields":{"title":"Product Dimension by Componentwise Exponent Addition","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[0, 1, 3], right=[-1, 0, 1]. Give (left, right, product exponent vector).","answer":{"kind":"expression","value":"([0, 1, 3], [-1, 0, 1], [-1, 1, 4])","explanation":"Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 1, 3], [-1, 0, 1], [-1, 1, 4])."},"skills":["Product Dimension by Componentwise Exponent Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0019","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a86eae59ad932475e1202e3cf4504b8954862de404bfa5930699ba14af235d37","family_id":"qf-app20-dimension-product","seed":202610205019,"semantic_template":"qf-app20-dimension-product/{\"left\":[1,1,-2],\"right\":[2,1,1]}"},"fields":{"title":"Product Dimension by Componentwise Exponent Addition","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[1, 1, -2], right=[2, 1, 1]. Give (left, right, product exponent vector).","answer":{"kind":"expression","value":"([1, 1, -2], [2, 1, 1], [3, 2, -1])","explanation":"Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 1, -2], [2, 1, 1], [3, 2, -1])."},"skills":["Product Dimension by Componentwise Exponent Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0020","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7fb3cb07bd4b5a7b612bf24fd934321b902212188fe189886145c5cdd1d7723","family_id":"qf-app20-dimension-product","seed":202610205020,"semantic_template":"qf-app20-dimension-product/{\"left\":[2,1,0],\"right\":[-2,2,1]}"},"fields":{"title":"Product Dimension by Componentwise Exponent Addition","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[2, 1, 0], right=[-2, 2, 1]. Give (left, right, product exponent vector).","answer":{"kind":"expression","value":"([2, 1, 0], [-2, 2, 1], [0, 3, 1])","explanation":"Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, 1, 0], [-2, 2, 1], [0, 3, 1])."},"skills":["Product Dimension by Componentwise Exponent Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0021","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7556076bc806c762150063c262c91a700fc71c8d1d013484ace99b38935ef247","family_id":"qf-app20-dimension-product","seed":202610205021,"semantic_template":"qf-app20-dimension-product/{\"left\":[3,1,2],\"right\":[1,-2,2]}"},"fields":{"title":"Product Dimension by Componentwise Exponent Addition","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[3, 1, 2], right=[1, -2, 2]. Give (left, right, product exponent vector).","answer":{"kind":"expression","value":"([3, 1, 2], [1, -2, 2], [4, -1, 4])","explanation":"Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, 1, 2], [1, -2, 2], [4, -1, 4])."},"skills":["Product Dimension by Componentwise Exponent Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0022","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3dfdb049b0d18e412e7761cd020e7a076c1509af1452937a62b0312017a46d5","family_id":"qf-app20-dimension-product","seed":202610205022,"semantic_template":"qf-app20-dimension-product/{\"left\":[-3,2,-3],\"right\":[-3,-1,2]}"},"fields":{"title":"Product Dimension by Componentwise Exponent Addition","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-3, 2, -3], right=[-3, -1, 2]. Give (left, right, product exponent vector).","answer":{"kind":"expression","value":"([-3, 2, -3], [-3, -1, 2], [-6, 1, -1])","explanation":"Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, 2, -3], [-3, -1, 2], [-6, 1, -1])."},"skills":["Product Dimension by Componentwise Exponent Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0023","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a4702aad543bdcafe56e968fb201af9a7ba2d2dece22d66d90b9df903f32503","family_id":"qf-app20-dimension-product","seed":202610205023,"semantic_template":"qf-app20-dimension-product/{\"left\":[-2,2,-1],\"right\":[0,0,2]}"},"fields":{"title":"Product Dimension by Componentwise Exponent Addition","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-2, 2, -1], right=[0, 0, 2]. Give (left, right, product exponent vector).","answer":{"kind":"expression","value":"([-2, 2, -1], [0, 0, 2], [-2, 2, 1])","explanation":"Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, 2, -1], [0, 0, 2], [-2, 2, 1])."},"skills":["Product Dimension by Componentwise Exponent Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0024","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30d59bce20e7d8089ffd1a4fb9d7c307ce5ed233b54c13d863f9c9be49db63bd","family_id":"qf-app20-dimension-product","seed":202610205024,"semantic_template":"qf-app20-dimension-product/{\"left\":[-1,2,1],\"right\":[3,1,2]}"},"fields":{"title":"Product Dimension by Componentwise Exponent Addition","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-1, 2, 1], right=[3, 1, 2]. Give (left, right, product exponent vector).","answer":{"kind":"expression","value":"([-1, 2, 1], [3, 1, 2], [2, 3, 3])","explanation":"Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, 2, 1], [3, 1, 2], [2, 3, 3])."},"skills":["Product Dimension by Componentwise Exponent Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0025","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c373bb8fb8e0c842a2ee5beeaf4f219b76b45a948b44eefeb844e443952ad8f","family_id":"qf-app20-dimension-product","seed":202610205025,"semantic_template":"qf-app20-dimension-product/{\"left\":[0,2,3],\"right\":[-1,2,2]}"},"fields":{"title":"Product Dimension by Componentwise Exponent Addition","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[0, 2, 3], right=[-1, 2, 2]. Give (left, right, product exponent vector).","answer":{"kind":"expression","value":"([0, 2, 3], [-1, 2, 2], [-1, 4, 5])","explanation":"Add corresponding exponents. Necessary dimensional consistency is not sufficient evidence of a physical model. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 2, 3], [-1, 2, 2], [-1, 4, 5])."},"skills":["Product Dimension by Componentwise Exponent Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0026","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ddd7e66419bd497c13eeba7160fbdf83a18126101f036cd9c990ecbe3c5c22b","family_id":"qf-app20-dimension-quotient","seed":202610205026,"semantic_template":"qf-app20-dimension-quotient/{\"left\":[-3,-1,-3],\"right\":[-3,-2,-2]}"},"fields":{"title":"Quotient Dimension with Ordered Exponent Subtraction","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[-3, -1, -3], denominator=[-3, -2, -2]. Give (numerator vector, denominator vector, quotient exponent vector).","answer":{"kind":"expression","value":"([-3, -1, -3], [-3, -2, -2], [0, 1, -1])","explanation":"Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, -1, -3], [-3, -2, -2], [0, 1, -1])."},"skills":["Quotient Dimension with Ordered Exponent Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0027","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f94d700c6501e791fd651a3705e8f4df781c332054817c6bbaaa1f3919d8c38","family_id":"qf-app20-dimension-quotient","seed":202610205027,"semantic_template":"qf-app20-dimension-quotient/{\"left\":[-2,-1,-1],\"right\":[0,-1,-2]}"},"fields":{"title":"Quotient Dimension with Ordered Exponent Subtraction","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[-2, -1, -1], denominator=[0, -1, -2]. Give (numerator vector, denominator vector, quotient exponent vector).","answer":{"kind":"expression","value":"([-2, -1, -1], [0, -1, -2], [-2, 0, 1])","explanation":"Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, -1, -1], [0, -1, -2], [-2, 0, 1])."},"skills":["Quotient Dimension with Ordered Exponent Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0028","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a7bb2684c54697f1d2e822936b2e4807caf0718137ac8e010749c3b82bc21be","family_id":"qf-app20-dimension-quotient","seed":202610205028,"semantic_template":"qf-app20-dimension-quotient/{\"left\":[-1,-1,1],\"right\":[3,0,-2]}"},"fields":{"title":"Quotient Dimension with Ordered Exponent Subtraction","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[-1, -1, 1], denominator=[3, 0, -2]. Give (numerator vector, denominator vector, quotient exponent vector).","answer":{"kind":"expression","value":"([-1, -1, 1], [3, 0, -2], [-4, -1, 3])","explanation":"Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, -1, 1], [3, 0, -2], [-4, -1, 3])."},"skills":["Quotient Dimension with Ordered Exponent Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0029","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70bb576978bb2c7649cf596f16805e039adbb3374758024ca51d59b7c9427905","family_id":"qf-app20-dimension-quotient","seed":202610205029,"semantic_template":"qf-app20-dimension-quotient/{\"left\":[0,-1,3],\"right\":[-1,1,-2]}"},"fields":{"title":"Quotient Dimension with Ordered Exponent Subtraction","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[0, -1, 3], denominator=[-1, 1, -2]. Give (numerator vector, denominator vector, quotient exponent vector).","answer":{"kind":"expression","value":"([0, -1, 3], [-1, 1, -2], [1, -2, 5])","explanation":"Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, -1, 3], [-1, 1, -2], [1, -2, 5])."},"skills":["Quotient Dimension with Ordered Exponent Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0030","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5f6de801383c0ddfa508c0ab2937e1783530317c58a2428fa9e0b21b454339b","family_id":"qf-app20-dimension-quotient","seed":202610205030,"semantic_template":"qf-app20-dimension-quotient/{\"left\":[1,-1,-2],\"right\":[2,2,-2]}"},"fields":{"title":"Quotient Dimension with Ordered Exponent Subtraction","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[1, -1, -2], denominator=[2, 2, -2]. Give (numerator vector, denominator vector, quotient exponent vector).","answer":{"kind":"expression","value":"([1, -1, -2], [2, 2, -2], [-1, -3, 0])","explanation":"Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, -1, -2], [2, 2, -2], [-1, -3, 0])."},"skills":["Quotient Dimension with Ordered Exponent Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0031","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5cebc3232dec88c390a36cbfc49fb676688298217ab5c89f5b706ccc1b5b0609","family_id":"qf-app20-dimension-quotient","seed":202610205031,"semantic_template":"qf-app20-dimension-quotient/{\"left\":[2,-1,0],\"right\":[-2,-2,-1]}"},"fields":{"title":"Quotient Dimension with Ordered Exponent Subtraction","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[2, -1, 0], denominator=[-2, -2, -1]. Give (numerator vector, denominator vector, quotient exponent vector).","answer":{"kind":"expression","value":"([2, -1, 0], [-2, -2, -1], [4, 1, 1])","explanation":"Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, -1, 0], [-2, -2, -1], [4, 1, 1])."},"skills":["Quotient Dimension with Ordered Exponent Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0032","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6cfa11c9bbaf2cf5177da03e032838a669a24c132bb912904d14d265c2c2976c","family_id":"qf-app20-dimension-quotient","seed":202610205032,"semantic_template":"qf-app20-dimension-quotient/{\"left\":[3,-1,2],\"right\":[1,-1,-1]}"},"fields":{"title":"Quotient Dimension with Ordered Exponent Subtraction","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[3, -1, 2], denominator=[1, -1, -1]. Give (numerator vector, denominator vector, quotient exponent vector).","answer":{"kind":"expression","value":"([3, -1, 2], [1, -1, -1], [2, 0, 3])","explanation":"Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, -1, 2], [1, -1, -1], [2, 0, 3])."},"skills":["Quotient Dimension with Ordered Exponent Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0033","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65c536a77c5d08fbbbe201056529d0c475ac04a141af229f44d19dd50d4cfd2a","family_id":"qf-app20-dimension-quotient","seed":202610205033,"semantic_template":"qf-app20-dimension-quotient/{\"left\":[-3,0,-3],\"right\":[-3,0,-1]}"},"fields":{"title":"Quotient Dimension with Ordered Exponent Subtraction","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[-3, 0, -3], denominator=[-3, 0, -1]. Give (numerator vector, denominator vector, quotient exponent vector).","answer":{"kind":"expression","value":"([-3, 0, -3], [-3, 0, -1], [0, 0, -2])","explanation":"Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, 0, -3], [-3, 0, -1], [0, 0, -2])."},"skills":["Quotient Dimension with Ordered Exponent Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0034","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93837baaf9e37277eff0f8b928392542776233bcb1e32b91fac03e9d28922417","family_id":"qf-app20-dimension-quotient","seed":202610205034,"semantic_template":"qf-app20-dimension-quotient/{\"left\":[-2,0,-1],\"right\":[0,1,-1]}"},"fields":{"title":"Quotient Dimension with Ordered Exponent Subtraction","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[-2, 0, -1], denominator=[0, 1, -1]. Give (numerator vector, denominator vector, quotient exponent vector).","answer":{"kind":"expression","value":"([-2, 0, -1], [0, 1, -1], [-2, -1, 0])","explanation":"Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, 0, -1], [0, 1, -1], [-2, -1, 0])."},"skills":["Quotient Dimension with Ordered Exponent Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0035","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f66bdc44492532cb4af9af03206958d0e49c03aa97d65004dbec4d4b53f4a94c","family_id":"qf-app20-dimension-quotient","seed":202610205035,"semantic_template":"qf-app20-dimension-quotient/{\"left\":[-1,0,1],\"right\":[3,2,-1]}"},"fields":{"title":"Quotient Dimension with Ordered Exponent Subtraction","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[-1, 0, 1], denominator=[3, 2, -1]. Give (numerator vector, denominator vector, quotient exponent vector).","answer":{"kind":"expression","value":"([-1, 0, 1], [3, 2, -1], [-4, -2, 2])","explanation":"Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, 0, 1], [3, 2, -1], [-4, -2, 2])."},"skills":["Quotient Dimension with Ordered Exponent Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0036","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b959d88bb095dfd73742bdad2cd9dacf060abce8ac204ede3104a0d6e7265b8f","family_id":"qf-app20-dimension-quotient","seed":202610205036,"semantic_template":"qf-app20-dimension-quotient/{\"left\":[0,0,3],\"right\":[-1,-2,0]}"},"fields":{"title":"Quotient Dimension with Ordered Exponent Subtraction","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[0, 0, 3], denominator=[-1, -2, 0]. Give (numerator vector, denominator vector, quotient exponent vector).","answer":{"kind":"expression","value":"([0, 0, 3], [-1, -2, 0], [1, 2, 3])","explanation":"Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 0, 3], [-1, -2, 0], [1, 2, 3])."},"skills":["Quotient Dimension with Ordered Exponent Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0037","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5dedc92ab8c250dfb2e54308c93bf7fd9a0f740273b3f5e9f4ce1b88874817c0","family_id":"qf-app20-dimension-quotient","seed":202610205037,"semantic_template":"qf-app20-dimension-quotient/{\"left\":[1,0,-2],\"right\":[2,-1,0]}"},"fields":{"title":"Quotient Dimension with Ordered Exponent Subtraction","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[1, 0, -2], denominator=[2, -1, 0]. Give (numerator vector, denominator vector, quotient exponent vector).","answer":{"kind":"expression","value":"([1, 0, -2], [2, -1, 0], [-1, 1, -2])","explanation":"Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 0, -2], [2, -1, 0], [-1, 1, -2])."},"skills":["Quotient Dimension with Ordered Exponent Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0038","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7c67aca77f37afdf6c48d999cda27bc00b02186f59b0d1dbcdc2a6d2eb20ef6b","family_id":"qf-app20-dimension-quotient","seed":202610205038,"semantic_template":"qf-app20-dimension-quotient/{\"left\":[2,0,0],\"right\":[-2,0,0]}"},"fields":{"title":"Quotient Dimension with Ordered Exponent Subtraction","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[2, 0, 0], denominator=[-2, 0, 0]. Give (numerator vector, denominator vector, quotient exponent vector).","answer":{"kind":"expression","value":"([2, 0, 0], [-2, 0, 0], [4, 0, 0])","explanation":"Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, 0, 0], [-2, 0, 0], [4, 0, 0])."},"skills":["Quotient Dimension with Ordered Exponent Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0039","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9014279f8d862ca9e68cc707bfd61ad71e5ada98a6f0d34da31dfe0a5acda11e","family_id":"qf-app20-dimension-quotient","seed":202610205039,"semantic_template":"qf-app20-dimension-quotient/{\"left\":[3,0,2],\"right\":[1,1,0]}"},"fields":{"title":"Quotient Dimension with Ordered Exponent Subtraction","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[3, 0, 2], denominator=[1, 1, 0]. Give (numerator vector, denominator vector, quotient exponent vector).","answer":{"kind":"expression","value":"([3, 0, 2], [1, 1, 0], [2, -1, 2])","explanation":"Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, 0, 2], [1, 1, 0], [2, -1, 2])."},"skills":["Quotient Dimension with Ordered Exponent Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0040","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9a6ef339b4ee96e13e0ca288a2eaa949de293635587d0b1010e881919445d66","family_id":"qf-app20-dimension-quotient","seed":202610205040,"semantic_template":"qf-app20-dimension-quotient/{\"left\":[-3,1,-3],\"right\":[-3,2,0]}"},"fields":{"title":"Quotient Dimension with Ordered Exponent Subtraction","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[-3, 1, -3], denominator=[-3, 2, 0]. Give (numerator vector, denominator vector, quotient exponent vector).","answer":{"kind":"expression","value":"([-3, 1, -3], [-3, 2, 0], [0, -1, -3])","explanation":"Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, 1, -3], [-3, 2, 0], [0, -1, -3])."},"skills":["Quotient Dimension with Ordered Exponent Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0041","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"184c9f1dbb8ed7d7985e52b63710d1d8f76a72f340c230f56985a1278f46ba47","family_id":"qf-app20-dimension-quotient","seed":202610205041,"semantic_template":"qf-app20-dimension-quotient/{\"left\":[-2,1,-1],\"right\":[0,-2,1]}"},"fields":{"title":"Quotient Dimension with Ordered Exponent Subtraction","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[-2, 1, -1], denominator=[0, -2, 1]. Give (numerator vector, denominator vector, quotient exponent vector).","answer":{"kind":"expression","value":"([-2, 1, -1], [0, -2, 1], [-2, 3, -2])","explanation":"Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, 1, -1], [0, -2, 1], [-2, 3, -2])."},"skills":["Quotient Dimension with Ordered Exponent Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0042","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef2b11913562b2536c31a8d79f24a6ed544f77676f3db34f3e6d7ce7925a1d99","family_id":"qf-app20-dimension-quotient","seed":202610205042,"semantic_template":"qf-app20-dimension-quotient/{\"left\":[-1,1,1],\"right\":[3,-1,1]}"},"fields":{"title":"Quotient Dimension with Ordered Exponent Subtraction","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[-1, 1, 1], denominator=[3, -1, 1]. Give (numerator vector, denominator vector, quotient exponent vector).","answer":{"kind":"expression","value":"([-1, 1, 1], [3, -1, 1], [-4, 2, 0])","explanation":"Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, 1, 1], [3, -1, 1], [-4, 2, 0])."},"skills":["Quotient Dimension with Ordered Exponent Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0043","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce39909bb4b901f5355be8bed021553d0792c947ee993cbfc098177f63118d53","family_id":"qf-app20-dimension-quotient","seed":202610205043,"semantic_template":"qf-app20-dimension-quotient/{\"left\":[0,1,3],\"right\":[-1,0,1]}"},"fields":{"title":"Quotient Dimension with Ordered Exponent Subtraction","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[0, 1, 3], denominator=[-1, 0, 1]. Give (numerator vector, denominator vector, quotient exponent vector).","answer":{"kind":"expression","value":"([0, 1, 3], [-1, 0, 1], [1, 1, 2])","explanation":"Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 1, 3], [-1, 0, 1], [1, 1, 2])."},"skills":["Quotient Dimension with Ordered Exponent Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0044","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14662b7b97564d271bc8c2e60de58b0f0555e58030ae571df024d57167b1dd3e","family_id":"qf-app20-dimension-quotient","seed":202610205044,"semantic_template":"qf-app20-dimension-quotient/{\"left\":[1,1,-2],\"right\":[2,1,1]}"},"fields":{"title":"Quotient Dimension with Ordered Exponent Subtraction","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[1, 1, -2], denominator=[2, 1, 1]. Give (numerator vector, denominator vector, quotient exponent vector).","answer":{"kind":"expression","value":"([1, 1, -2], [2, 1, 1], [-1, 0, -3])","explanation":"Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 1, -2], [2, 1, 1], [-1, 0, -3])."},"skills":["Quotient Dimension with Ordered Exponent Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0045","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"caa404389136103bdee2147748883f634696a3f30d84440e2330be1c00f391a4","family_id":"qf-app20-dimension-quotient","seed":202610205045,"semantic_template":"qf-app20-dimension-quotient/{\"left\":[2,1,0],\"right\":[-2,2,1]}"},"fields":{"title":"Quotient Dimension with Ordered Exponent Subtraction","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[2, 1, 0], denominator=[-2, 2, 1]. Give (numerator vector, denominator vector, quotient exponent vector).","answer":{"kind":"expression","value":"([2, 1, 0], [-2, 2, 1], [4, -1, -1])","explanation":"Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, 1, 0], [-2, 2, 1], [4, -1, -1])."},"skills":["Quotient Dimension with Ordered Exponent Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0046","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eae79e74f7c9fcd0186b690254627fd7af921c651fd03734296f95a957f10a2f","family_id":"qf-app20-dimension-quotient","seed":202610205046,"semantic_template":"qf-app20-dimension-quotient/{\"left\":[3,1,2],\"right\":[1,-2,2]}"},"fields":{"title":"Quotient Dimension with Ordered Exponent Subtraction","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[3, 1, 2], denominator=[1, -2, 2]. Give (numerator vector, denominator vector, quotient exponent vector).","answer":{"kind":"expression","value":"([3, 1, 2], [1, -2, 2], [2, 3, 0])","explanation":"Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, 1, 2], [1, -2, 2], [2, 3, 0])."},"skills":["Quotient Dimension with Ordered Exponent Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0047","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4385b4dd1cf1e2902859f4eb30aedb018add3324afa4c81ce8478694b1f8b61","family_id":"qf-app20-dimension-quotient","seed":202610205047,"semantic_template":"qf-app20-dimension-quotient/{\"left\":[-3,2,-3],\"right\":[-3,-1,2]}"},"fields":{"title":"Quotient Dimension with Ordered Exponent Subtraction","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[-3, 2, -3], denominator=[-3, -1, 2]. Give (numerator vector, denominator vector, quotient exponent vector).","answer":{"kind":"expression","value":"([-3, 2, -3], [-3, -1, 2], [0, 3, -5])","explanation":"Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, 2, -3], [-3, -1, 2], [0, 3, -5])."},"skills":["Quotient Dimension with Ordered Exponent Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0048","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d0bba95da5f4701f7885e5fc2331e847ed6d56e4bcbb8965fcb9d50c3ed1624","family_id":"qf-app20-dimension-quotient","seed":202610205048,"semantic_template":"qf-app20-dimension-quotient/{\"left\":[-2,2,-1],\"right\":[0,0,2]}"},"fields":{"title":"Quotient Dimension with Ordered Exponent Subtraction","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[-2, 2, -1], denominator=[0, 0, 2]. Give (numerator vector, denominator vector, quotient exponent vector).","answer":{"kind":"expression","value":"([-2, 2, -1], [0, 0, 2], [-2, 2, -3])","explanation":"Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, 2, -1], [0, 0, 2], [-2, 2, -3])."},"skills":["Quotient Dimension with Ordered Exponent Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0049","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57c33212e32859aeeeca0bd05332010a2ffda1ebb8310e2b422ea5340406afbb","family_id":"qf-app20-dimension-quotient","seed":202610205049,"semantic_template":"qf-app20-dimension-quotient/{\"left\":[-1,2,1],\"right\":[3,1,2]}"},"fields":{"title":"Quotient Dimension with Ordered Exponent Subtraction","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[-1, 2, 1], denominator=[3, 1, 2]. Give (numerator vector, denominator vector, quotient exponent vector).","answer":{"kind":"expression","value":"([-1, 2, 1], [3, 1, 2], [-4, 1, -1])","explanation":"Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, 2, 1], [3, 1, 2], [-4, 1, -1])."},"skills":["Quotient Dimension with Ordered Exponent Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0050","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fda939ebc85d003dec5f80f148664b3506c871afacf6b6faf8b45c7205133522","family_id":"qf-app20-dimension-quotient","seed":202610205050,"semantic_template":"qf-app20-dimension-quotient/{\"left\":[0,2,3],\"right\":[-1,2,2]}"},"fields":{"title":"Quotient Dimension with Ordered Exponent Subtraction","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. numerator=[0, 2, 3], denominator=[-1, 2, 2]. Give (numerator vector, denominator vector, quotient exponent vector).","answer":{"kind":"expression","value":"([0, 2, 3], [-1, 2, 2], [1, 0, 1])","explanation":"Subtract denominator exponents from numerator exponents, preserving [L,M,T] order. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 2, 3], [-1, 2, 2], [1, 0, 1])."},"skills":["Quotient Dimension with Ordered Exponent Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0051","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d450a93d68ee24264027df3b1c79a6471ac3007fa7c991ec57dc68f939d3fe5","family_id":"qf-app20-dimension-power","seed":202610205051,"semantic_template":"qf-app20-dimension-power/{\"base\":[-3,-1,-3],\"exponent\":-3}"},"fields":{"title":"Integer Power of a Dimension Vector","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[-3, -1, -3], exponent=-3. Give (base vector, exponent, exponent-scaled vector).","answer":{"kind":"expression","value":"([-3, -1, -3], -3, [9, 3, 9])","explanation":"Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, -1, -3], -3, [9, 3, 9])."},"skills":["Integer Power of a Dimension Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0052","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61775869935f695d20c2bd4ba05f6edc01873eee427ea14f5666aa88d3beed24","family_id":"qf-app20-dimension-power","seed":202610205052,"semantic_template":"qf-app20-dimension-power/{\"base\":[-2,-1,-1],\"exponent\":-2}"},"fields":{"title":"Integer Power of a Dimension Vector","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[-2, -1, -1], exponent=-2. Give (base vector, exponent, exponent-scaled vector).","answer":{"kind":"expression","value":"([-2, -1, -1], -2, [4, 2, 2])","explanation":"Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, -1, -1], -2, [4, 2, 2])."},"skills":["Integer Power of a Dimension Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0053","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a1f2927b4d466056382cd9771f872d90603c2b203d2949eb496985c24dbc7e1","family_id":"qf-app20-dimension-power","seed":202610205053,"semantic_template":"qf-app20-dimension-power/{\"base\":[-1,-1,1],\"exponent\":-1}"},"fields":{"title":"Integer Power of a Dimension Vector","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[-1, -1, 1], exponent=-1. Give (base vector, exponent, exponent-scaled vector).","answer":{"kind":"expression","value":"([-1, -1, 1], -1, [1, 1, -1])","explanation":"Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, -1, 1], -1, [1, 1, -1])."},"skills":["Integer Power of a Dimension Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0054","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad58567071d5eb3256b6c337cb565af088de996a8694a8f8725f609f2d51af77","family_id":"qf-app20-dimension-power","seed":202610205054,"semantic_template":"qf-app20-dimension-power/{\"base\":[0,-1,3],\"exponent\":0}"},"fields":{"title":"Integer Power of a Dimension Vector","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[0, -1, 3], exponent=0. Give (base vector, exponent, exponent-scaled vector).","answer":{"kind":"expression","value":"([0, -1, 3], 0, [0, 0, 0])","explanation":"Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, -1, 3], 0, [0, 0, 0])."},"skills":["Integer Power of a Dimension Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0055","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e59686f0d31697744dad814a59d9de942f20d40bc47c9f855d0993d0af95ca97","family_id":"qf-app20-dimension-power","seed":202610205055,"semantic_template":"qf-app20-dimension-power/{\"base\":[1,-1,-2],\"exponent\":1}"},"fields":{"title":"Integer Power of a Dimension Vector","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[1, -1, -2], exponent=1. Give (base vector, exponent, exponent-scaled vector).","answer":{"kind":"expression","value":"([1, -1, -2], 1, [1, -1, -2])","explanation":"Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, -1, -2], 1, [1, -1, -2])."},"skills":["Integer Power of a Dimension Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0056","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1023da3526b9cc935674cff039cd207a4e04cdeb52205de6da4abfc5034299b","family_id":"qf-app20-dimension-power","seed":202610205056,"semantic_template":"qf-app20-dimension-power/{\"base\":[2,-1,0],\"exponent\":2}"},"fields":{"title":"Integer Power of a Dimension Vector","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[2, -1, 0], exponent=2. Give (base vector, exponent, exponent-scaled vector).","answer":{"kind":"expression","value":"([2, -1, 0], 2, [4, -2, 0])","explanation":"Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, -1, 0], 2, [4, -2, 0])."},"skills":["Integer Power of a Dimension Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0057","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ee560a155d97ab78a3e10f56f2d667e50e3ada00811dd233deb937e6dd349a9","family_id":"qf-app20-dimension-power","seed":202610205057,"semantic_template":"qf-app20-dimension-power/{\"base\":[3,-1,2],\"exponent\":3}"},"fields":{"title":"Integer Power of a Dimension Vector","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[3, -1, 2], exponent=3. Give (base vector, exponent, exponent-scaled vector).","answer":{"kind":"expression","value":"([3, -1, 2], 3, [9, -3, 6])","explanation":"Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, -1, 2], 3, [9, -3, 6])."},"skills":["Integer Power of a Dimension Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0058","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6434e5b6227c32b67556f919d9b741cce240096e6b549b4ab0ed9d115723d3a2","family_id":"qf-app20-dimension-power","seed":202610205058,"semantic_template":"qf-app20-dimension-power/{\"base\":[-3,0,-3],\"exponent\":-3}"},"fields":{"title":"Integer Power of a Dimension Vector","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[-3, 0, -3], exponent=-3. Give (base vector, exponent, exponent-scaled vector).","answer":{"kind":"expression","value":"([-3, 0, -3], -3, [9, 0, 9])","explanation":"Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, 0, -3], -3, [9, 0, 9])."},"skills":["Integer Power of a Dimension Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0059","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"add2f52676d0a0eb69f5403df7450f20c224fda46605ba5253fad6d9ef8c3c2b","family_id":"qf-app20-dimension-power","seed":202610205059,"semantic_template":"qf-app20-dimension-power/{\"base\":[-2,0,-1],\"exponent\":-2}"},"fields":{"title":"Integer Power of a Dimension Vector","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[-2, 0, -1], exponent=-2. Give (base vector, exponent, exponent-scaled vector).","answer":{"kind":"expression","value":"([-2, 0, -1], -2, [4, 0, 2])","explanation":"Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, 0, -1], -2, [4, 0, 2])."},"skills":["Integer Power of a Dimension Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0060","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22d9e26c9f533641f88eba34af7647b93e66715c18f555a972b783c83c307bee","family_id":"qf-app20-dimension-power","seed":202610205060,"semantic_template":"qf-app20-dimension-power/{\"base\":[-1,0,1],\"exponent\":-1}"},"fields":{"title":"Integer Power of a Dimension Vector","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[-1, 0, 1], exponent=-1. Give (base vector, exponent, exponent-scaled vector).","answer":{"kind":"expression","value":"([-1, 0, 1], -1, [1, 0, -1])","explanation":"Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, 0, 1], -1, [1, 0, -1])."},"skills":["Integer Power of a Dimension Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0061","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8cbbd633d24b2e887a6b5af884e5e9dd124977854328154dfe4ad40304abd7be","family_id":"qf-app20-dimension-power","seed":202610205061,"semantic_template":"qf-app20-dimension-power/{\"base\":[0,0,3],\"exponent\":0}"},"fields":{"title":"Integer Power of a Dimension Vector","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[0, 0, 3], exponent=0. Give (base vector, exponent, exponent-scaled vector).","answer":{"kind":"expression","value":"([0, 0, 3], 0, [0, 0, 0])","explanation":"Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 0, 3], 0, [0, 0, 0])."},"skills":["Integer Power of a Dimension Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0062","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e60a9458c75cdaad3c206279453aad617199cbf3b088cabb901b17c128a24b15","family_id":"qf-app20-dimension-power","seed":202610205062,"semantic_template":"qf-app20-dimension-power/{\"base\":[1,0,-2],\"exponent\":1}"},"fields":{"title":"Integer Power of a Dimension Vector","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[1, 0, -2], exponent=1. Give (base vector, exponent, exponent-scaled vector).","answer":{"kind":"expression","value":"([1, 0, -2], 1, [1, 0, -2])","explanation":"Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 0, -2], 1, [1, 0, -2])."},"skills":["Integer Power of a Dimension Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0063","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a95976ba5d84d4ced2eca33ce9061aa816ddf480f9de765684b8020c68744a0a","family_id":"qf-app20-dimension-power","seed":202610205063,"semantic_template":"qf-app20-dimension-power/{\"base\":[2,0,0],\"exponent\":2}"},"fields":{"title":"Integer Power of a Dimension Vector","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[2, 0, 0], exponent=2. Give (base vector, exponent, exponent-scaled vector).","answer":{"kind":"expression","value":"([2, 0, 0], 2, [4, 0, 0])","explanation":"Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, 0, 0], 2, [4, 0, 0])."},"skills":["Integer Power of a Dimension Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0064","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2458b705caa4e00bb55d03882338263583ad8da3fe0a0f7cdbc7e14489b15bcb","family_id":"qf-app20-dimension-power","seed":202610205064,"semantic_template":"qf-app20-dimension-power/{\"base\":[3,0,2],\"exponent\":3}"},"fields":{"title":"Integer Power of a Dimension Vector","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[3, 0, 2], exponent=3. Give (base vector, exponent, exponent-scaled vector).","answer":{"kind":"expression","value":"([3, 0, 2], 3, [9, 0, 6])","explanation":"Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, 0, 2], 3, [9, 0, 6])."},"skills":["Integer Power of a Dimension Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0065","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d407530794c1565f3cfda7ce307301bb721528d83ea1c5f72eda1b252eed7ea","family_id":"qf-app20-dimension-power","seed":202610205065,"semantic_template":"qf-app20-dimension-power/{\"base\":[-3,1,-3],\"exponent\":-3}"},"fields":{"title":"Integer Power of a Dimension Vector","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[-3, 1, -3], exponent=-3. Give (base vector, exponent, exponent-scaled vector).","answer":{"kind":"expression","value":"([-3, 1, -3], -3, [9, -3, 9])","explanation":"Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, 1, -3], -3, [9, -3, 9])."},"skills":["Integer Power of a Dimension Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0066","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e52158c7031dd60bad2872a683a4e6d3b4d43181f4d8427ad153ce315ef44ad2","family_id":"qf-app20-dimension-power","seed":202610205066,"semantic_template":"qf-app20-dimension-power/{\"base\":[-2,1,-1],\"exponent\":-2}"},"fields":{"title":"Integer Power of a Dimension Vector","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[-2, 1, -1], exponent=-2. Give (base vector, exponent, exponent-scaled vector).","answer":{"kind":"expression","value":"([-2, 1, -1], -2, [4, -2, 2])","explanation":"Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, 1, -1], -2, [4, -2, 2])."},"skills":["Integer Power of a Dimension Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0067","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2825bda8af0190e6fd253593e371ab8c20ce797caa1cd944915b2afa72627c31","family_id":"qf-app20-dimension-power","seed":202610205067,"semantic_template":"qf-app20-dimension-power/{\"base\":[-1,1,1],\"exponent\":-1}"},"fields":{"title":"Integer Power of a Dimension Vector","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[-1, 1, 1], exponent=-1. Give (base vector, exponent, exponent-scaled vector).","answer":{"kind":"expression","value":"([-1, 1, 1], -1, [1, -1, -1])","explanation":"Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, 1, 1], -1, [1, -1, -1])."},"skills":["Integer Power of a Dimension Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0068","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"747a969838777256ffb3b02bb578bd36e50e9d09497c8bd1a46bc166d80079ef","family_id":"qf-app20-dimension-power","seed":202610205068,"semantic_template":"qf-app20-dimension-power/{\"base\":[0,1,3],\"exponent\":0}"},"fields":{"title":"Integer Power of a Dimension Vector","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[0, 1, 3], exponent=0. Give (base vector, exponent, exponent-scaled vector).","answer":{"kind":"expression","value":"([0, 1, 3], 0, [0, 0, 0])","explanation":"Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 1, 3], 0, [0, 0, 0])."},"skills":["Integer Power of a Dimension Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0069","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"564cbbdb32565edfabf470601269d1200f4095c0478b7c1838c9414fdf862b14","family_id":"qf-app20-dimension-power","seed":202610205069,"semantic_template":"qf-app20-dimension-power/{\"base\":[1,1,-2],\"exponent\":1}"},"fields":{"title":"Integer Power of a Dimension Vector","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[1, 1, -2], exponent=1. Give (base vector, exponent, exponent-scaled vector).","answer":{"kind":"expression","value":"([1, 1, -2], 1, [1, 1, -2])","explanation":"Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 1, -2], 1, [1, 1, -2])."},"skills":["Integer Power of a Dimension Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0070","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a91a01c0a6457614392b58c74b035917b1d98e2316d783acade47bb8286a718","family_id":"qf-app20-dimension-power","seed":202610205070,"semantic_template":"qf-app20-dimension-power/{\"base\":[2,1,0],\"exponent\":2}"},"fields":{"title":"Integer Power of a Dimension Vector","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[2, 1, 0], exponent=2. Give (base vector, exponent, exponent-scaled vector).","answer":{"kind":"expression","value":"([2, 1, 0], 2, [4, 2, 0])","explanation":"Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, 1, 0], 2, [4, 2, 0])."},"skills":["Integer Power of a Dimension Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0071","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6cf8d3a6cfd5d60c0b453133b843fd6ae5e2ea3e8af2c038801118be0944dccb","family_id":"qf-app20-dimension-power","seed":202610205071,"semantic_template":"qf-app20-dimension-power/{\"base\":[3,1,2],\"exponent\":3}"},"fields":{"title":"Integer Power of a Dimension Vector","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[3, 1, 2], exponent=3. Give (base vector, exponent, exponent-scaled vector).","answer":{"kind":"expression","value":"([3, 1, 2], 3, [9, 3, 6])","explanation":"Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, 1, 2], 3, [9, 3, 6])."},"skills":["Integer Power of a Dimension Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0072","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e76fbbafc45d038609ac22910acd706cdecef5e4db096bdf694718d94215901","family_id":"qf-app20-dimension-power","seed":202610205072,"semantic_template":"qf-app20-dimension-power/{\"base\":[-3,2,-3],\"exponent\":-3}"},"fields":{"title":"Integer Power of a Dimension Vector","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[-3, 2, -3], exponent=-3. Give (base vector, exponent, exponent-scaled vector).","answer":{"kind":"expression","value":"([-3, 2, -3], -3, [9, -6, 9])","explanation":"Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, 2, -3], -3, [9, -6, 9])."},"skills":["Integer Power of a Dimension Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0073","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2708b78a0760d7a07e6b2af75923d32da70f4df3cc0070ef4e3def58cfa70be8","family_id":"qf-app20-dimension-power","seed":202610205073,"semantic_template":"qf-app20-dimension-power/{\"base\":[-2,2,-1],\"exponent\":-2}"},"fields":{"title":"Integer Power of a Dimension Vector","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[-2, 2, -1], exponent=-2. Give (base vector, exponent, exponent-scaled vector).","answer":{"kind":"expression","value":"([-2, 2, -1], -2, [4, -4, 2])","explanation":"Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, 2, -1], -2, [4, -4, 2])."},"skills":["Integer Power of a Dimension Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0074","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83a4f327df2771fcdb26731ce70e779f90ea3017bdff9369eff3ab1f9ad1e61f","family_id":"qf-app20-dimension-power","seed":202610205074,"semantic_template":"qf-app20-dimension-power/{\"base\":[-1,2,1],\"exponent\":-1}"},"fields":{"title":"Integer Power of a Dimension Vector","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[-1, 2, 1], exponent=-1. Give (base vector, exponent, exponent-scaled vector).","answer":{"kind":"expression","value":"([-1, 2, 1], -1, [1, -2, -1])","explanation":"Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, 2, 1], -1, [1, -2, -1])."},"skills":["Integer Power of a Dimension Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0075","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92513be03c10262763c55af1e6caac419127a3203cc3c2ea51b7bba81e9e83b2","family_id":"qf-app20-dimension-power","seed":202610205075,"semantic_template":"qf-app20-dimension-power/{\"base\":[0,2,3],\"exponent\":0}"},"fields":{"title":"Integer Power of a Dimension Vector","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. base=[0, 2, 3], exponent=0. Give (base vector, exponent, exponent-scaled vector).","answer":{"kind":"expression","value":"([0, 2, 3], 0, [0, 0, 0])","explanation":"Multiply each exponent by the declared integer power; do not clamp valid computed exponents to the input bounds. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 2, 3], 0, [0, 0, 0])."},"skills":["Integer Power of a Dimension Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0076","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50e9ff451ab81ca850e2a136200f4c39527756f1f14dcf715770d74d1d2f86ce","family_id":"qf-app20-dimension-equation-consistency","seed":202610205076,"semantic_template":"qf-app20-dimension-equation-consistency/{\"left\":[-3,-1,-3],\"right\":[-2,-1,-3]}"},"fields":{"title":"Equal Dimension Vectors Are Not a Physical Proof","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-3, -1, -3], right=[-2, -1, -3]. Give (left vector, right vector, identical?).","answer":{"kind":"expression","value":"([-3, -1, -3], [-2, -1, -3], No)","explanation":"Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, -1, -3], [-2, -1, -3], No)."},"skills":["Equal Dimension Vectors Are Not a Physical Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0077","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"10af7be23d44dc21046e559c06a35718e06f9464092286a4ba6d62f5063b68b5","family_id":"qf-app20-dimension-equation-consistency","seed":202610205077,"semantic_template":"qf-app20-dimension-equation-consistency/{\"left\":[-2,-1,-1],\"right\":[-2,-1,-1]}"},"fields":{"title":"Equal Dimension Vectors Are Not a Physical Proof","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-2, -1, -1], right=[-2, -1, -1]. Give (left vector, right vector, identical?).","answer":{"kind":"expression","value":"([-2, -1, -1], [-2, -1, -1], Yes)","explanation":"Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, -1, -1], [-2, -1, -1], Yes)."},"skills":["Equal Dimension Vectors Are Not a Physical Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0078","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b6f9d806987cb322d8d446c1feb95a811ea016f20d181d3facfc0c291914951","family_id":"qf-app20-dimension-equation-consistency","seed":202610205078,"semantic_template":"qf-app20-dimension-equation-consistency/{\"left\":[-1,-1,1],\"right\":[-1,-1,1]}"},"fields":{"title":"Equal Dimension Vectors Are Not a Physical Proof","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-1, -1, 1], right=[-1, -1, 1]. Give (left vector, right vector, identical?).","answer":{"kind":"expression","value":"([-1, -1, 1], [-1, -1, 1], Yes)","explanation":"Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, -1, 1], [-1, -1, 1], Yes)."},"skills":["Equal Dimension Vectors Are Not a Physical Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0079","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba251b27a4689eaa0a5fae9ecb0ed2bb40aae27d4d58cb3d03fbcbd7cab79f3e","family_id":"qf-app20-dimension-equation-consistency","seed":202610205079,"semantic_template":"qf-app20-dimension-equation-consistency/{\"left\":[0,-1,3],\"right\":[1,-1,3]}"},"fields":{"title":"Equal Dimension Vectors Are Not a Physical Proof","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[0, -1, 3], right=[1, -1, 3]. Give (left vector, right vector, identical?).","answer":{"kind":"expression","value":"([0, -1, 3], [1, -1, 3], No)","explanation":"Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, -1, 3], [1, -1, 3], No)."},"skills":["Equal Dimension Vectors Are Not a Physical Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0080","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d646b575e146ac26655f4be4e5b6efbda0b9da35c02df666fc5462eef766abc","family_id":"qf-app20-dimension-equation-consistency","seed":202610205080,"semantic_template":"qf-app20-dimension-equation-consistency/{\"left\":[1,-1,-2],\"right\":[1,-1,-2]}"},"fields":{"title":"Equal Dimension Vectors Are Not a Physical Proof","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[1, -1, -2], right=[1, -1, -2]. Give (left vector, right vector, identical?).","answer":{"kind":"expression","value":"([1, -1, -2], [1, -1, -2], Yes)","explanation":"Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, -1, -2], [1, -1, -2], Yes)."},"skills":["Equal Dimension Vectors Are Not a Physical Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0081","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dfe17f4d0d9e0a708d0ca8557a8edae56c9512faa57eb732de7093baa46f6c1f","family_id":"qf-app20-dimension-equation-consistency","seed":202610205081,"semantic_template":"qf-app20-dimension-equation-consistency/{\"left\":[2,-1,0],\"right\":[2,-1,0]}"},"fields":{"title":"Equal Dimension Vectors Are Not a Physical Proof","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[2, -1, 0], right=[2, -1, 0]. Give (left vector, right vector, identical?).","answer":{"kind":"expression","value":"([2, -1, 0], [2, -1, 0], Yes)","explanation":"Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, -1, 0], [2, -1, 0], Yes)."},"skills":["Equal Dimension Vectors Are Not a Physical Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0082","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8afc7224dbf9acbd4a6df8f0ebf4125c2fd241eccf5d378532419557906747e4","family_id":"qf-app20-dimension-equation-consistency","seed":202610205082,"semantic_template":"qf-app20-dimension-equation-consistency/{\"left\":[3,-1,2],\"right\":[4,-1,2]}"},"fields":{"title":"Equal Dimension Vectors Are Not a Physical Proof","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[3, -1, 2], right=[4, -1, 2]. Give (left vector, right vector, identical?).","answer":{"kind":"expression","value":"([3, -1, 2], [4, -1, 2], No)","explanation":"Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, -1, 2], [4, -1, 2], No)."},"skills":["Equal Dimension Vectors Are Not a Physical Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0083","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1b2a6b98821a45ebaba774c0768764daac56e0fe8874e89245d3aa4b8d97c20","family_id":"qf-app20-dimension-equation-consistency","seed":202610205083,"semantic_template":"qf-app20-dimension-equation-consistency/{\"left\":[-3,0,-3],\"right\":[-3,0,-3]}"},"fields":{"title":"Equal Dimension Vectors Are Not a Physical Proof","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-3, 0, -3], right=[-3, 0, -3]. Give (left vector, right vector, identical?).","answer":{"kind":"expression","value":"([-3, 0, -3], [-3, 0, -3], Yes)","explanation":"Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, 0, -3], [-3, 0, -3], Yes)."},"skills":["Equal Dimension Vectors Are Not a Physical Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0084","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b322730569643dcd906ed8c37afb5fdacd11f01d10073cc92b29746d2bb5fb4e","family_id":"qf-app20-dimension-equation-consistency","seed":202610205084,"semantic_template":"qf-app20-dimension-equation-consistency/{\"left\":[-2,0,-1],\"right\":[-2,0,-1]}"},"fields":{"title":"Equal Dimension Vectors Are Not a Physical Proof","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-2, 0, -1], right=[-2, 0, -1]. Give (left vector, right vector, identical?).","answer":{"kind":"expression","value":"([-2, 0, -1], [-2, 0, -1], Yes)","explanation":"Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, 0, -1], [-2, 0, -1], Yes)."},"skills":["Equal Dimension Vectors Are Not a Physical Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0085","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73ad6d8bfc93fc29ffd2584f2a9b61c4be804de8d65936453b9251821459ce7c","family_id":"qf-app20-dimension-equation-consistency","seed":202610205085,"semantic_template":"qf-app20-dimension-equation-consistency/{\"left\":[-1,0,1],\"right\":[0,0,1]}"},"fields":{"title":"Equal Dimension Vectors Are Not a Physical Proof","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-1, 0, 1], right=[0, 0, 1]. Give (left vector, right vector, identical?).","answer":{"kind":"expression","value":"([-1, 0, 1], [0, 0, 1], No)","explanation":"Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, 0, 1], [0, 0, 1], No)."},"skills":["Equal Dimension Vectors Are Not a Physical Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0086","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a9e8293f107a4f171af07b244faa23c36f9ae792982d9dddca58f20d75d8953","family_id":"qf-app20-dimension-equation-consistency","seed":202610205086,"semantic_template":"qf-app20-dimension-equation-consistency/{\"left\":[0,0,3],\"right\":[0,0,3]}"},"fields":{"title":"Equal Dimension Vectors Are Not a Physical Proof","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[0, 0, 3], right=[0, 0, 3]. Give (left vector, right vector, identical?).","answer":{"kind":"expression","value":"([0, 0, 3], [0, 0, 3], Yes)","explanation":"Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 0, 3], [0, 0, 3], Yes)."},"skills":["Equal Dimension Vectors Are Not a Physical Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0087","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89f219cf5f611e936661d6597d9198a971fda1ad816c1194549584ae064c5407","family_id":"qf-app20-dimension-equation-consistency","seed":202610205087,"semantic_template":"qf-app20-dimension-equation-consistency/{\"left\":[1,0,-2],\"right\":[1,0,-2]}"},"fields":{"title":"Equal Dimension Vectors Are Not a Physical Proof","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[1, 0, -2], right=[1, 0, -2]. Give (left vector, right vector, identical?).","answer":{"kind":"expression","value":"([1, 0, -2], [1, 0, -2], Yes)","explanation":"Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 0, -2], [1, 0, -2], Yes)."},"skills":["Equal Dimension Vectors Are Not a Physical Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0088","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28dd804a43d975ef92696a7cf31424b33a3656c9e27b857743f8b6a969c31cdb","family_id":"qf-app20-dimension-equation-consistency","seed":202610205088,"semantic_template":"qf-app20-dimension-equation-consistency/{\"left\":[2,0,0],\"right\":[3,0,0]}"},"fields":{"title":"Equal Dimension Vectors Are Not a Physical Proof","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[2, 0, 0], right=[3, 0, 0]. Give (left vector, right vector, identical?).","answer":{"kind":"expression","value":"([2, 0, 0], [3, 0, 0], No)","explanation":"Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, 0, 0], [3, 0, 0], No)."},"skills":["Equal Dimension Vectors Are Not a Physical Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0089","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4f5fb75aa5ed3f89861b9a151809e6ee7db7254ad3fe91196624a14bced58dc","family_id":"qf-app20-dimension-equation-consistency","seed":202610205089,"semantic_template":"qf-app20-dimension-equation-consistency/{\"left\":[3,0,2],\"right\":[3,0,2]}"},"fields":{"title":"Equal Dimension Vectors Are Not a Physical Proof","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[3, 0, 2], right=[3, 0, 2]. Give (left vector, right vector, identical?).","answer":{"kind":"expression","value":"([3, 0, 2], [3, 0, 2], Yes)","explanation":"Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, 0, 2], [3, 0, 2], Yes)."},"skills":["Equal Dimension Vectors Are Not a Physical Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0090","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e8151ca902dfda9b4d27e97159253e688af8dd02ce5527f17b8b76b632d0103","family_id":"qf-app20-dimension-equation-consistency","seed":202610205090,"semantic_template":"qf-app20-dimension-equation-consistency/{\"left\":[-3,1,-3],\"right\":[-3,1,-3]}"},"fields":{"title":"Equal Dimension Vectors Are Not a Physical Proof","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-3, 1, -3], right=[-3, 1, -3]. Give (left vector, right vector, identical?).","answer":{"kind":"expression","value":"([-3, 1, -3], [-3, 1, -3], Yes)","explanation":"Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, 1, -3], [-3, 1, -3], Yes)."},"skills":["Equal Dimension Vectors Are Not a Physical Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0091","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66756303549f7655c4d6629c46ff2ee2b8bae2577e8cdf3d2a99b132e0b585e1","family_id":"qf-app20-dimension-equation-consistency","seed":202610205091,"semantic_template":"qf-app20-dimension-equation-consistency/{\"left\":[-2,1,-1],\"right\":[-1,1,-1]}"},"fields":{"title":"Equal Dimension Vectors Are Not a Physical Proof","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-2, 1, -1], right=[-1, 1, -1]. Give (left vector, right vector, identical?).","answer":{"kind":"expression","value":"([-2, 1, -1], [-1, 1, -1], No)","explanation":"Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, 1, -1], [-1, 1, -1], No)."},"skills":["Equal Dimension Vectors Are Not a Physical Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0092","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51c87d0e05ada25f744178192a7e49ee37dd118d068778465637c9406a1c9786","family_id":"qf-app20-dimension-equation-consistency","seed":202610205092,"semantic_template":"qf-app20-dimension-equation-consistency/{\"left\":[-1,1,1],\"right\":[-1,1,1]}"},"fields":{"title":"Equal Dimension Vectors Are Not a Physical Proof","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-1, 1, 1], right=[-1, 1, 1]. Give (left vector, right vector, identical?).","answer":{"kind":"expression","value":"([-1, 1, 1], [-1, 1, 1], Yes)","explanation":"Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, 1, 1], [-1, 1, 1], Yes)."},"skills":["Equal Dimension Vectors Are Not a Physical Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0093","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e49d26eff71959d2f17f76b8e917e858443dd53d52f43394656760385e4c643","family_id":"qf-app20-dimension-equation-consistency","seed":202610205093,"semantic_template":"qf-app20-dimension-equation-consistency/{\"left\":[0,1,3],\"right\":[0,1,3]}"},"fields":{"title":"Equal Dimension Vectors Are Not a Physical Proof","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[0, 1, 3], right=[0, 1, 3]. Give (left vector, right vector, identical?).","answer":{"kind":"expression","value":"([0, 1, 3], [0, 1, 3], Yes)","explanation":"Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 1, 3], [0, 1, 3], Yes)."},"skills":["Equal Dimension Vectors Are Not a Physical Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0094","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec8c26f6f3eacf1743a88097888509c38c7f56f46fbbc7c57d08d36c57b168ec","family_id":"qf-app20-dimension-equation-consistency","seed":202610205094,"semantic_template":"qf-app20-dimension-equation-consistency/{\"left\":[1,1,-2],\"right\":[2,1,-2]}"},"fields":{"title":"Equal Dimension Vectors Are Not a Physical Proof","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[1, 1, -2], right=[2, 1, -2]. Give (left vector, right vector, identical?).","answer":{"kind":"expression","value":"([1, 1, -2], [2, 1, -2], No)","explanation":"Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([1, 1, -2], [2, 1, -2], No)."},"skills":["Equal Dimension Vectors Are Not a Physical Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0095","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"abac6a2789f40c8417e56a8b06ddb16a5f2fc6394e5910be20de60737e2c3bb9","family_id":"qf-app20-dimension-equation-consistency","seed":202610205095,"semantic_template":"qf-app20-dimension-equation-consistency/{\"left\":[2,1,0],\"right\":[2,1,0]}"},"fields":{"title":"Equal Dimension Vectors Are Not a Physical Proof","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[2, 1, 0], right=[2, 1, 0]. Give (left vector, right vector, identical?).","answer":{"kind":"expression","value":"([2, 1, 0], [2, 1, 0], Yes)","explanation":"Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([2, 1, 0], [2, 1, 0], Yes)."},"skills":["Equal Dimension Vectors Are Not a Physical Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0096","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6f8029a7179f2129aabfe7ef881c6c2ccc0222cf51b54c044ed4146f4a67a1b","family_id":"qf-app20-dimension-equation-consistency","seed":202610205096,"semantic_template":"qf-app20-dimension-equation-consistency/{\"left\":[3,1,2],\"right\":[3,1,2]}"},"fields":{"title":"Equal Dimension Vectors Are Not a Physical Proof","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[3, 1, 2], right=[3, 1, 2]. Give (left vector, right vector, identical?).","answer":{"kind":"expression","value":"([3, 1, 2], [3, 1, 2], Yes)","explanation":"Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([3, 1, 2], [3, 1, 2], Yes)."},"skills":["Equal Dimension Vectors Are Not a Physical Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0097","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b17a9314b3de273ecc1d5da5c0de0510b4f26acd1f2fb3fba013d27cbe01356","family_id":"qf-app20-dimension-equation-consistency","seed":202610205097,"semantic_template":"qf-app20-dimension-equation-consistency/{\"left\":[-3,2,-3],\"right\":[-2,2,-3]}"},"fields":{"title":"Equal Dimension Vectors Are Not a Physical Proof","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-3, 2, -3], right=[-2, 2, -3]. Give (left vector, right vector, identical?).","answer":{"kind":"expression","value":"([-3, 2, -3], [-2, 2, -3], No)","explanation":"Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-3, 2, -3], [-2, 2, -3], No)."},"skills":["Equal Dimension Vectors Are Not a Physical Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0098","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d562404166232be36ade765f3baeb2f1e557a115e833c1dcb00518ffa32d6654","family_id":"qf-app20-dimension-equation-consistency","seed":202610205098,"semantic_template":"qf-app20-dimension-equation-consistency/{\"left\":[-2,2,-1],\"right\":[-2,2,-1]}"},"fields":{"title":"Equal Dimension Vectors Are Not a Physical Proof","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-2, 2, -1], right=[-2, 2, -1]. Give (left vector, right vector, identical?).","answer":{"kind":"expression","value":"([-2, 2, -1], [-2, 2, -1], Yes)","explanation":"Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-2, 2, -1], [-2, 2, -1], Yes)."},"skills":["Equal Dimension Vectors Are Not a Physical Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0099","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91ac47316fdd61321baf3f658432e490ab18b3af7b2e9edb5fad426e0ab1d200","family_id":"qf-app20-dimension-equation-consistency","seed":202610205099,"semantic_template":"qf-app20-dimension-equation-consistency/{\"left\":[-1,2,1],\"right\":[-1,2,1]}"},"fields":{"title":"Equal Dimension Vectors Are Not a Physical Proof","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[-1, 2, 1], right=[-1, 2, 1]. Give (left vector, right vector, identical?).","answer":{"kind":"expression","value":"([-1, 2, 1], [-1, 2, 1], Yes)","explanation":"Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([-1, 2, 1], [-1, 2, 1], Yes)."},"skills":["Equal Dimension Vectors Are Not a Physical Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r05-dimensional-analysis-0100","batch_id":"batch-20261013-advance-20261020-r05-dimensional-analysis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e985de058b4153140c071b91f45b495252edc9e0741a08032e7eef5e682ddd14","family_id":"qf-app20-dimension-equation-consistency","seed":202610205100,"semantic_template":"qf-app20-dimension-equation-consistency/{\"left\":[0,2,3],\"right\":[1,2,3]}"},"fields":{"title":"Equal Dimension Vectors Are Not a Physical Proof","stem":"Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[0, 2, 3], right=[1, 2, 3]. Give (left vector, right vector, identical?).","answer":{"kind":"expression","value":"([0, 2, 3], [1, 2, 3], No)","explanation":"Vector equality is necessary but not sufficient for physical correctness; it does not establish numerical coefficients, direction or physical assumptions. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is ([0, 2, 3], [1, 2, 3], No)."},"skills":["Equal Dimension Vectors Are Not a Physical Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}