{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0001","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79009ed5e78b0e8ae3fb27287482d8e365e34a53ce254097294732137935a87b","family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162001,"semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[0]]}"},"fields":{"title":"Matrix Shape and Complete Transpose Round Trip","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[0]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).","answer":{"kind":"expression","value":"([1, 1], [[0]], [[0]], Yes)","explanation":"Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1], [[0]], [[0]], Yes)."},"skills":["Matrix Shape and Complete Transpose Round Trip"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0002","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3271bb3e5e8bc9c538c7b86f3d9c66695bffb78a7b47f5f1b0dd16aedcb155d8","family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162002,"semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[-11,-12]]}"},"fields":{"title":"Matrix Shape and Complete Transpose Round Trip","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-11, -12]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).","answer":{"kind":"expression","value":"([1, 2], [[-11], [-12]], [[-11, -12]], Yes)","explanation":"Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2], [[-11], [-12]], [[-11, -12]], Yes)."},"skills":["Matrix Shape and Complete Transpose Round Trip"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0003","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9263554242a88296cc65dc0b405217e9ab5ca5e9d73708f25b6d27e75d24ea78","family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162003,"semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[-10,-11,-12]]}"},"fields":{"title":"Matrix Shape and Complete Transpose Round Trip","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-10, -11, -12]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).","answer":{"kind":"expression","value":"([1, 3], [[-10], [-11], [-12]], [[-10, -11, -12]], Yes)","explanation":"Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 3], [[-10], [-11], [-12]], [[-10, -11, -12]], Yes)."},"skills":["Matrix Shape and Complete Transpose Round Trip"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0004","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"928bd48da522883fb6c38db1a96ed86c552e38ac869efb32a0f84d7be50a0702","family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162004,"semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[-9,-10,-11,-12]]}"},"fields":{"title":"Matrix Shape and Complete Transpose Round Trip","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-9, -10, -11, -12]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).","answer":{"kind":"expression","value":"([1, 4], [[-9], [-10], [-11], [-12]], [[-9, -10, -11, -12]], Yes)","explanation":"Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 4], [[-9], [-10], [-11], [-12]], [[-9, -10, -11, -12]], Yes)."},"skills":["Matrix Shape and Complete Transpose Round Trip"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0005","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd1cdd05c743fa3fd7e547f295fe60f30481395df094a757937b91c8350c5034","family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162005,"semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[-8,-9,-10,-11,-12]]}"},"fields":{"title":"Matrix Shape and Complete Transpose Round Trip","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-8, -9, -10, -11, -12]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).","answer":{"kind":"expression","value":"([1, 5], [[-8], [-9], [-10], [-11], [-12]], [[-8, -9, -10, -11, -12]], Yes)","explanation":"Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 5], [[-8], [-9], [-10], [-11], [-12]], [[-8, -9, -10, -11, -12]], Yes)."},"skills":["Matrix Shape and Complete Transpose Round Trip"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0006","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b71814e632955da36801ed1f3ef33e61e98482fb3b9ae2bed04f4f52623b241a","family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162006,"semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[-7],[-6]]}"},"fields":{"title":"Matrix Shape and Complete Transpose Round Trip","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-7], [-6]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).","answer":{"kind":"expression","value":"([2, 1], [[-7, -6]], [[-7], [-6]], Yes)","explanation":"Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1], [[-7, -6]], [[-7], [-6]], Yes)."},"skills":["Matrix Shape and Complete Transpose Round Trip"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0007","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4840c7c943cc07c07e532b472c0efd979e35cba3530b459dabd8be6993cd1335","family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162007,"semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[-6,-7],[-5,-6]]}"},"fields":{"title":"Matrix Shape and Complete Transpose Round Trip","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-6, -7], [-5, -6]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).","answer":{"kind":"expression","value":"([2, 2], [[-6, -5], [-7, -6]], [[-6, -7], [-5, -6]], Yes)","explanation":"Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2], [[-6, -5], [-7, -6]], [[-6, -7], [-5, -6]], Yes)."},"skills":["Matrix Shape and Complete Transpose Round Trip"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0008","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af7632bb6e9b05e82619672d5140009620d4a6c250de9a01b10a7168901caa24","family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162008,"semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[-5,-6,-7],[-4,-5,-6]]}"},"fields":{"title":"Matrix Shape and Complete Transpose Round Trip","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-5, -6, -7], [-4, -5, -6]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).","answer":{"kind":"expression","value":"([2, 3], [[-5, -4], [-6, -5], [-7, -6]], [[-5, -6, -7], [-4, -5, -6]], Yes)","explanation":"Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 3], [[-5, -4], [-6, -5], [-7, -6]], [[-5, -6, -7], [-4, -5, -6]], Yes)."},"skills":["Matrix Shape and Complete Transpose Round Trip"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0009","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb33f369e4d2e6c70233fbd68bd091b2f8399bc26fce1eccdb2feec9291c9775","family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162009,"semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[-4,-5,-6,-7],[-3,-4,-5,-6]]}"},"fields":{"title":"Matrix Shape and Complete Transpose Round Trip","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-4, -5, -6, -7], [-3, -4, -5, -6]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).","answer":{"kind":"expression","value":"([2, 4], [[-4, -3], [-5, -4], [-6, -5], [-7, -6]], [[-4, -5, -6, -7], [-3, -4, -5, -6]], Yes)","explanation":"Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 4], [[-4, -3], [-5, -4], [-6, -5], [-7, -6]], [[-4, -5, -6, -7], [-3, -4, -5, -6]], Yes)."},"skills":["Matrix Shape and Complete Transpose Round Trip"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0010","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a0c7d5f41f0cff127df8e92c4a1f3316798599dce7721e17e6090dec19b8a6cc","family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162010,"semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[-3,-4,-5,-6,-7],[-2,-3,-4,-5,-6]]}"},"fields":{"title":"Matrix Shape and Complete Transpose Round Trip","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-3, -4, -5, -6, -7], [-2, -3, -4, -5, -6]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).","answer":{"kind":"expression","value":"([2, 5], [[-3, -2], [-4, -3], [-5, -4], [-6, -5], [-7, -6]], [[-3, -4, -5, -6, -7], [-2, -3, -4, -5, -6]], Yes)","explanation":"Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 5], [[-3, -2], [-4, -3], [-5, -4], [-6, -5], [-7, -6]], [[-3, -4, -5, -6, -7], [-2, -3, -4, -5, -6]], Yes)."},"skills":["Matrix Shape and Complete Transpose Round Trip"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0011","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9120a9d30c0b4f81abda8611b759f00e93dcb4010b8d99cb987e8fc9c420f2db","family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162011,"semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[-2],[-1],[0]]}"},"fields":{"title":"Matrix Shape and Complete Transpose Round Trip","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-2], [-1], [0]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).","answer":{"kind":"expression","value":"([3, 1], [[-2, -1, 0]], [[-2], [-1], [0]], Yes)","explanation":"Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 1], [[-2, -1, 0]], [[-2], [-1], [0]], Yes)."},"skills":["Matrix Shape and Complete Transpose Round Trip"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0012","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f661d3d80b40521813fabff958e48f278f60fdaeb1c4bb20c6f598f63329242a","family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162012,"semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[-1,-2],[0,-1],[1,0]]}"},"fields":{"title":"Matrix Shape and Complete Transpose Round Trip","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-1, -2], [0, -1], [1, 0]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).","answer":{"kind":"expression","value":"([3, 2], [[-1, 0, 1], [-2, -1, 0]], [[-1, -2], [0, -1], [1, 0]], Yes)","explanation":"Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 2], [[-1, 0, 1], [-2, -1, 0]], [[-1, -2], [0, -1], [1, 0]], Yes)."},"skills":["Matrix Shape and Complete Transpose Round Trip"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0013","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6253771526391d967260cd26dfeb96232c05b1beba40cab6408d11736a171c0b","family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162013,"semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[0,-1,-2],[1,0,-1],[2,1,0]]}"},"fields":{"title":"Matrix Shape and Complete Transpose Round Trip","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[0, -1, -2], [1, 0, -1], [2, 1, 0]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).","answer":{"kind":"expression","value":"([3, 3], [[0, 1, 2], [-1, 0, 1], [-2, -1, 0]], [[0, -1, -2], [1, 0, -1], [2, 1, 0]], Yes)","explanation":"Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 1, 2], [-1, 0, 1], [-2, -1, 0]], [[0, -1, -2], [1, 0, -1], [2, 1, 0]], Yes)."},"skills":["Matrix Shape and Complete Transpose Round Trip"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0014","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"abd47c1fb893a5c253178343bb1af18a3e72dcf2cf78af79b5cd8ec7c17ca773","family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162014,"semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[1,0,-1,-2],[2,1,0,-1],[3,2,1,0]]}"},"fields":{"title":"Matrix Shape and Complete Transpose Round Trip","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[1, 0, -1, -2], [2, 1, 0, -1], [3, 2, 1, 0]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).","answer":{"kind":"expression","value":"([3, 4], [[1, 2, 3], [0, 1, 2], [-1, 0, 1], [-2, -1, 0]], [[1, 0, -1, -2], [2, 1, 0, -1], [3, 2, 1, 0]], Yes)","explanation":"Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 4], [[1, 2, 3], [0, 1, 2], [-1, 0, 1], [-2, -1, 0]], [[1, 0, -1, -2], [2, 1, 0, -1], [3, 2, 1, 0]], Yes)."},"skills":["Matrix Shape and Complete Transpose Round Trip"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0015","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4c803ad3864558d6804848968bc94ad8ab700eccd42c26e5c693d79f1c36dd6","family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162015,"semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[2,1,0,-1,-2],[3,2,1,0,-1],[4,3,2,1,0]]}"},"fields":{"title":"Matrix Shape and Complete Transpose Round Trip","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[2, 1, 0, -1, -2], [3, 2, 1, 0, -1], [4, 3, 2, 1, 0]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).","answer":{"kind":"expression","value":"([3, 5], [[2, 3, 4], [1, 2, 3], [0, 1, 2], [-1, 0, 1], [-2, -1, 0]], [[2, 1, 0, -1, -2], [3, 2, 1, 0, -1], [4, 3, 2, 1, 0]], Yes)","explanation":"Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 5], [[2, 3, 4], [1, 2, 3], [0, 1, 2], [-1, 0, 1], [-2, -1, 0]], [[2, 1, 0, -1, -2], [3, 2, 1, 0, -1], [4, 3, 2, 1, 0]], Yes)."},"skills":["Matrix Shape and Complete Transpose Round Trip"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0016","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a2598bbbbfd83c8657f5e73f320ae9371c690edb86b1b1e6de17f969f4167a3","family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162016,"semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[3],[4],[5],[6]]}"},"fields":{"title":"Matrix Shape and Complete Transpose Round Trip","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[3], [4], [5], [6]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).","answer":{"kind":"expression","value":"([4, 1], [[3, 4, 5, 6]], [[3], [4], [5], [6]], Yes)","explanation":"Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 1], [[3, 4, 5, 6]], [[3], [4], [5], [6]], Yes)."},"skills":["Matrix Shape and Complete Transpose Round Trip"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0017","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7a44bba1d6c83c8087eb6776b82c0d8dc8e3a23fee241c06587d2a48989a57b","family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162017,"semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[4,3],[5,4],[6,5],[7,6]]}"},"fields":{"title":"Matrix Shape and Complete Transpose Round Trip","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[4, 3], [5, 4], [6, 5], [7, 6]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).","answer":{"kind":"expression","value":"([4, 2], [[4, 5, 6, 7], [3, 4, 5, 6]], [[4, 3], [5, 4], [6, 5], [7, 6]], Yes)","explanation":"Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 2], [[4, 5, 6, 7], [3, 4, 5, 6]], [[4, 3], [5, 4], [6, 5], [7, 6]], Yes)."},"skills":["Matrix Shape and Complete Transpose Round Trip"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0018","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c3328fea4d4efcce29e6e146ccdff4de6fdfd79ce2a26f20dba1ba7605c1f25","family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162018,"semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[5,4,3],[6,5,4],[7,6,5],[8,7,6]]}"},"fields":{"title":"Matrix Shape and Complete Transpose Round Trip","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[5, 4, 3], [6, 5, 4], [7, 6, 5], [8, 7, 6]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).","answer":{"kind":"expression","value":"([4, 3], [[5, 6, 7, 8], [4, 5, 6, 7], [3, 4, 5, 6]], [[5, 4, 3], [6, 5, 4], [7, 6, 5], [8, 7, 6]], Yes)","explanation":"Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 3], [[5, 6, 7, 8], [4, 5, 6, 7], [3, 4, 5, 6]], [[5, 4, 3], [6, 5, 4], [7, 6, 5], [8, 7, 6]], Yes)."},"skills":["Matrix Shape and Complete Transpose Round Trip"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0019","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f44d4d8da2e22d5f39be4f3dbf9632de3efedaae2ea6a2d7dfab4913c01c54d6","family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162019,"semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[6,5,4,3],[7,6,5,4],[8,7,6,5],[9,8,7,6]]}"},"fields":{"title":"Matrix Shape and Complete Transpose Round Trip","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[6, 5, 4, 3], [7, 6, 5, 4], [8, 7, 6, 5], [9, 8, 7, 6]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).","answer":{"kind":"expression","value":"([4, 4], [[6, 7, 8, 9], [5, 6, 7, 8], [4, 5, 6, 7], [3, 4, 5, 6]], [[6, 5, 4, 3], [7, 6, 5, 4], [8, 7, 6, 5], [9, 8, 7, 6]], Yes)","explanation":"Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 4], [[6, 7, 8, 9], [5, 6, 7, 8], [4, 5, 6, 7], [3, 4, 5, 6]], [[6, 5, 4, 3], [7, 6, 5, 4], [8, 7, 6, 5], [9, 8, 7, 6]], Yes)."},"skills":["Matrix Shape and Complete Transpose Round Trip"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0020","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01216aba1090e0f517e491fadee352448e0a40b299bf6b6112d9c40474ab8a97","family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162020,"semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[7,6,5,4,3],[8,7,6,5,4],[9,8,7,6,5],[10,9,8,7,6]]}"},"fields":{"title":"Matrix Shape and Complete Transpose Round Trip","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[7, 6, 5, 4, 3], [8, 7, 6, 5, 4], [9, 8, 7, 6, 5], [10, 9, 8, 7, 6]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).","answer":{"kind":"expression","value":"([4, 5], [[7, 8, 9, 10], [6, 7, 8, 9], [5, 6, 7, 8], [4, 5, 6, 7], [3, 4, 5, 6]], [[7, 6, 5, 4, 3], [8, 7, 6, 5, 4], [9, 8, 7, 6, 5], [10, 9, 8, 7, 6]], Yes)","explanation":"Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 5], [[7, 8, 9, 10], [6, 7, 8, 9], [5, 6, 7, 8], [4, 5, 6, 7], [3, 4, 5, 6]], [[7, 6, 5, 4, 3], [8, 7, 6, 5, 4], [9, 8, 7, 6, 5], [10, 9, 8, 7, 6]], Yes)."},"skills":["Matrix Shape and Complete Transpose Round Trip"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0021","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"901ced40645038ae48b0ec0c9acec6f5d0c210642c7e503621b0cebb696c43eb","family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162021,"semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[8],[9],[10],[11],[12]]}"},"fields":{"title":"Matrix Shape and Complete Transpose Round Trip","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[8], [9], [10], [11], [12]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).","answer":{"kind":"expression","value":"([5, 1], [[8, 9, 10, 11, 12]], [[8], [9], [10], [11], [12]], Yes)","explanation":"Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 1], [[8, 9, 10, 11, 12]], [[8], [9], [10], [11], [12]], Yes)."},"skills":["Matrix Shape and Complete Transpose Round Trip"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0022","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa8491ccfd56e0943f93acbf4b0c12f6b97a95a1f4897798f0e5033c3411b972","family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162022,"semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[9,8],[10,9],[11,10],[12,11],[13,12]]}"},"fields":{"title":"Matrix Shape and Complete Transpose Round Trip","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[9, 8], [10, 9], [11, 10], [12, 11], [13, 12]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).","answer":{"kind":"expression","value":"([5, 2], [[9, 10, 11, 12, 13], [8, 9, 10, 11, 12]], [[9, 8], [10, 9], [11, 10], [12, 11], [13, 12]], Yes)","explanation":"Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 2], [[9, 10, 11, 12, 13], [8, 9, 10, 11, 12]], [[9, 8], [10, 9], [11, 10], [12, 11], [13, 12]], Yes)."},"skills":["Matrix Shape and Complete Transpose Round Trip"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0023","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5bbbfd0b4877a44bb829769f8e209226be8cd164e5a4fe6e2b00e9cdd299d6d","family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162023,"semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[10,9,8],[11,10,9],[12,11,10],[13,12,11],[14,13,12]]}"},"fields":{"title":"Matrix Shape and Complete Transpose Round Trip","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[10, 9, 8], [11, 10, 9], [12, 11, 10], [13, 12, 11], [14, 13, 12]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).","answer":{"kind":"expression","value":"([5, 3], [[10, 11, 12, 13, 14], [9, 10, 11, 12, 13], [8, 9, 10, 11, 12]], [[10, 9, 8], [11, 10, 9], [12, 11, 10], [13, 12, 11], [14, 13, 12]], Yes)","explanation":"Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 3], [[10, 11, 12, 13, 14], [9, 10, 11, 12, 13], [8, 9, 10, 11, 12]], [[10, 9, 8], [11, 10, 9], [12, 11, 10], [13, 12, 11], [14, 13, 12]], Yes)."},"skills":["Matrix Shape and Complete Transpose Round Trip"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0024","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5dc9cb5e4e77d939d447335e8413d31ce18aae2665fc7241e6c426537ddddd7","family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162024,"semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[11,10,9,8],[12,11,10,9],[13,12,11,10],[14,13,12,11],[15,14,13,12]]}"},"fields":{"title":"Matrix Shape and Complete Transpose Round Trip","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[11, 10, 9, 8], [12, 11, 10, 9], [13, 12, 11, 10], [14, 13, 12, 11], [15, 14, 13, 12]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).","answer":{"kind":"expression","value":"([5, 4], [[11, 12, 13, 14, 15], [10, 11, 12, 13, 14], [9, 10, 11, 12, 13], [8, 9, 10, 11, 12]], [[11, 10, 9, 8], [12, 11, 10, 9], [13, 12, 11, 10], [14, 13, 12, 11], [15, 14, 13, 12]], Yes)","explanation":"Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 4], [[11, 12, 13, 14, 15], [10, 11, 12, 13, 14], [9, 10, 11, 12, 13], [8, 9, 10, 11, 12]], [[11, 10, 9, 8], [12, 11, 10, 9], [13, 12, 11, 10], [14, 13, 12, 11], [15, 14, 13, 12]], Yes)."},"skills":["Matrix Shape and Complete Transpose Round Trip"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0025","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9626b340194021ca25d4a9996df9d3b846534ca2b6679039ecc52aaac489abe","family_id":"qf-density16-matrix-transpose-roundtrip","seed":202610162025,"semantic_template":"qf-density16-matrix-transpose-roundtrip/{\"M\":[[12,11,10,9,8],[13,12,11,10,9],[14,13,12,11,10],[15,14,13,12,11],[16,15,14,13,12]]}"},"fields":{"title":"Matrix Shape and Complete Transpose Round Trip","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[12, 11, 10, 9, 8], [13, 12, 11, 10, 9], [14, 13, 12, 11, 10], [15, 14, 13, 12, 11], [16, 15, 14, 13, 12]]. Give (shape of M, Mᵀ, (Mᵀ)ᵀ, does the round trip equal M?).","answer":{"kind":"expression","value":"([5, 5], [[12, 13, 14, 15, 16], [11, 12, 13, 14, 15], [10, 11, 12, 13, 14], [9, 10, 11, 12, 13], [8, 9, 10, 11, 12]], [[12, 11, 10, 9, 8], [13, 12, 11, 10, 9], [14, 13, 12, 11, 10], [15, 14, 13, 12, 11], [16, 15, 14, 13, 12]], Yes)","explanation":"Exchange row and column positions twice and compare every entry; do not reverse the lists or call this an inverse. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 5], [[12, 13, 14, 15, 16], [11, 12, 13, 14, 15], [10, 11, 12, 13, 14], [9, 10, 11, 12, 13], [8, 9, 10, 11, 12]], [[12, 11, 10, 9, 8], [13, 12, 11, 10, 9], [14, 13, 12, 11, 10], [15, 14, 13, 12, 11], [16, 15, 14, 13, 12]], Yes)."},"skills":["Matrix Shape and Complete Transpose Round Trip"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0026","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3915f6326a16b73010ce11b2fc232461d6703fad37019c137d7460c3a25928bf","family_id":"qf-density16-matrix-sum-transpose","seed":202610162026,"semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[0]],\"N\":[[0]]}"},"fields":{"title":"Same-Shape Matrix Sum and Both Transpose Routes","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[0]], N=[[0]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).","answer":{"kind":"expression","value":"([1, 1], [[0]], [[0]], [[0]])","explanation":"Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1], [[0]], [[0]], [[0]])."},"skills":["Same-Shape Matrix Sum and Both Transpose Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0027","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e535ff1f9f59b7b199302f8c834952ae770759352b11f7aa027b44a8543f1f11","family_id":"qf-density16-matrix-sum-transpose","seed":202610162027,"semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[-11,-12]],\"N\":[[0,1]]}"},"fields":{"title":"Same-Shape Matrix Sum and Both Transpose Routes","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[-11, -12]], N=[[0, 1]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).","answer":{"kind":"expression","value":"([1, 2], [[-11, -11]], [[-11], [-11]], [[-11], [-11]])","explanation":"Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2], [[-11, -11]], [[-11], [-11]], [[-11], [-11]])."},"skills":["Same-Shape Matrix Sum and Both Transpose Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0028","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78bb71392d3ace738379ee476fdc5574735f7a25763e97992eca5844cff0b60a","family_id":"qf-density16-matrix-sum-transpose","seed":202610162028,"semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[-10,-11,-12]],\"N\":[[0,1,2]]}"},"fields":{"title":"Same-Shape Matrix Sum and Both Transpose Routes","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[-10, -11, -12]], N=[[0, 1, 2]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).","answer":{"kind":"expression","value":"([1, 3], [[-10, -10, -10]], [[-10], [-10], [-10]], [[-10], [-10], [-10]])","explanation":"Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 3], [[-10, -10, -10]], [[-10], [-10], [-10]], [[-10], [-10], [-10]])."},"skills":["Same-Shape Matrix Sum and Both Transpose Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0029","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8681d5f39b379892b88211fa5d15ee364f737b47dc1099b19e54f1a2821a0b82","family_id":"qf-density16-matrix-sum-transpose","seed":202610162029,"semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[-9,-10,-11,-12]],\"N\":[[0,1,2,3]]}"},"fields":{"title":"Same-Shape Matrix Sum and Both Transpose Routes","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[-9, -10, -11, -12]], N=[[0, 1, 2, 3]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).","answer":{"kind":"expression","value":"([1, 4], [[-9, -9, -9, -9]], [[-9], [-9], [-9], [-9]], [[-9], [-9], [-9], [-9]])","explanation":"Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 4], [[-9, -9, -9, -9]], [[-9], [-9], [-9], [-9]], [[-9], [-9], [-9], [-9]])."},"skills":["Same-Shape Matrix Sum and Both Transpose Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0030","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1c9c4302bdca09862cd4723d4fb09f87cc1df3840873370dd5901b9ef165a98","family_id":"qf-density16-matrix-sum-transpose","seed":202610162030,"semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[-8,-9,-10,-11,-12]],\"N\":[[0,1,2,3,4]]}"},"fields":{"title":"Same-Shape Matrix Sum and Both Transpose Routes","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[-8, -9, -10, -11, -12]], N=[[0, 1, 2, 3, 4]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).","answer":{"kind":"expression","value":"([1, 5], [[-8, -8, -8, -8, -8]], [[-8], [-8], [-8], [-8], [-8]], [[-8], [-8], [-8], [-8], [-8]])","explanation":"Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 5], [[-8, -8, -8, -8, -8]], [[-8], [-8], [-8], [-8], [-8]], [[-8], [-8], [-8], [-8], [-8]])."},"skills":["Same-Shape Matrix Sum and Both Transpose Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0031","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51769811826a9d2d252de3e02a6679f32bb5b01db61ab697d9a6855b2fc5d9e1","family_id":"qf-density16-matrix-sum-transpose","seed":202610162031,"semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[-7],[-6]],\"N\":[[0],[-1]]}"},"fields":{"title":"Same-Shape Matrix Sum and Both Transpose Routes","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[-7], [-6]], N=[[0], [-1]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).","answer":{"kind":"expression","value":"([2, 1], [[-7], [-7]], [[-7, -7]], [[-7, -7]])","explanation":"Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1], [[-7], [-7]], [[-7, -7]], [[-7, -7]])."},"skills":["Same-Shape Matrix Sum and Both Transpose Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0032","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43781c9882231e8ad1169740224cbe3644e90ccfa7404c71fda2870daeb5bc4e","family_id":"qf-density16-matrix-sum-transpose","seed":202610162032,"semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[-6,-7],[-5,-6]],\"N\":[[0,1],[-1,0]]}"},"fields":{"title":"Same-Shape Matrix Sum and Both Transpose Routes","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[-6, -7], [-5, -6]], N=[[0, 1], [-1, 0]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).","answer":{"kind":"expression","value":"([2, 2], [[-6, -6], [-6, -6]], [[-6, -6], [-6, -6]], [[-6, -6], [-6, -6]])","explanation":"Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2], [[-6, -6], [-6, -6]], [[-6, -6], [-6, -6]], [[-6, -6], [-6, -6]])."},"skills":["Same-Shape Matrix Sum and Both Transpose Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0033","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8703f745335566981142128a006ca95d9b6760a661c4867607cb77cbe0ebf7d","family_id":"qf-density16-matrix-sum-transpose","seed":202610162033,"semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[-5,-6,-7],[-4,-5,-6]],\"N\":[[0,1,2],[-1,0,1]]}"},"fields":{"title":"Same-Shape Matrix Sum and Both Transpose Routes","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[-5, -6, -7], [-4, -5, -6]], N=[[0, 1, 2], [-1, 0, 1]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).","answer":{"kind":"expression","value":"([2, 3], [[-5, -5, -5], [-5, -5, -5]], [[-5, -5], [-5, -5], [-5, -5]], [[-5, -5], [-5, -5], [-5, -5]])","explanation":"Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 3], [[-5, -5, -5], [-5, -5, -5]], [[-5, -5], [-5, -5], [-5, -5]], [[-5, -5], [-5, -5], [-5, -5]])."},"skills":["Same-Shape Matrix Sum and Both Transpose Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0034","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8ff436d9052dc41abc05c5575a85f935592ba5c33959284e86931b322f3b70c","family_id":"qf-density16-matrix-sum-transpose","seed":202610162034,"semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[-4,-5,-6,-7],[-3,-4,-5,-6]],\"N\":[[0,1,2,3],[-1,0,1,2]]}"},"fields":{"title":"Same-Shape Matrix Sum and Both Transpose Routes","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[-4, -5, -6, -7], [-3, -4, -5, -6]], N=[[0, 1, 2, 3], [-1, 0, 1, 2]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).","answer":{"kind":"expression","value":"([2, 4], [[-4, -4, -4, -4], [-4, -4, -4, -4]], [[-4, -4], [-4, -4], [-4, -4], [-4, -4]], [[-4, -4], [-4, -4], [-4, -4], [-4, -4]])","explanation":"Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 4], [[-4, -4, -4, -4], [-4, -4, -4, -4]], [[-4, -4], [-4, -4], [-4, -4], [-4, -4]], [[-4, -4], [-4, -4], [-4, -4], [-4, -4]])."},"skills":["Same-Shape Matrix Sum and Both Transpose Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0035","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"737f14e175d2b4dc672870423d65f52dba7aa839d4777d409649b5a1ddd13984","family_id":"qf-density16-matrix-sum-transpose","seed":202610162035,"semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[-3,-4,-5,-6,-7],[-2,-3,-4,-5,-6]],\"N\":[[0,1,2,3,4],[-1,0,1,2,3]]}"},"fields":{"title":"Same-Shape Matrix Sum and Both Transpose Routes","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[-3, -4, -5, -6, -7], [-2, -3, -4, -5, -6]], N=[[0, 1, 2, 3, 4], [-1, 0, 1, 2, 3]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).","answer":{"kind":"expression","value":"([2, 5], [[-3, -3, -3, -3, -3], [-3, -3, -3, -3, -3]], [[-3, -3], [-3, -3], [-3, -3], [-3, -3], [-3, -3]], [[-3, -3], [-3, -3], [-3, -3], [-3, -3], [-3, -3]])","explanation":"Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 5], [[-3, -3, -3, -3, -3], [-3, -3, -3, -3, -3]], [[-3, -3], [-3, -3], [-3, -3], [-3, -3], [-3, -3]], [[-3, -3], [-3, -3], [-3, -3], [-3, -3], [-3, -3]])."},"skills":["Same-Shape Matrix Sum and Both Transpose Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0036","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"970b1498cc9105e4ef32445197fedbd1821a850c35f7e128bba9553f98755e72","family_id":"qf-density16-matrix-sum-transpose","seed":202610162036,"semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[-2],[-1],[0]],\"N\":[[0],[-1],[-2]]}"},"fields":{"title":"Same-Shape Matrix Sum and Both Transpose Routes","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[-2], [-1], [0]], N=[[0], [-1], [-2]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).","answer":{"kind":"expression","value":"([3, 1], [[-2], [-2], [-2]], [[-2, -2, -2]], [[-2, -2, -2]])","explanation":"Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 1], [[-2], [-2], [-2]], [[-2, -2, -2]], [[-2, -2, -2]])."},"skills":["Same-Shape Matrix Sum and Both Transpose Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0037","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a64e648f5353b445548d26c3bb50a39776ef3cf0553bd69196f7c023091fc8e5","family_id":"qf-density16-matrix-sum-transpose","seed":202610162037,"semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[-1,-2],[0,-1],[1,0]],\"N\":[[0,1],[-1,0],[-2,-1]]}"},"fields":{"title":"Same-Shape Matrix Sum and Both Transpose Routes","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[-1, -2], [0, -1], [1, 0]], N=[[0, 1], [-1, 0], [-2, -1]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).","answer":{"kind":"expression","value":"([3, 2], [[-1, -1], [-1, -1], [-1, -1]], [[-1, -1, -1], [-1, -1, -1]], [[-1, -1, -1], [-1, -1, -1]])","explanation":"Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 2], [[-1, -1], [-1, -1], [-1, -1]], [[-1, -1, -1], [-1, -1, -1]], [[-1, -1, -1], [-1, -1, -1]])."},"skills":["Same-Shape Matrix Sum and Both Transpose Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0038","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"032b8cd2dd7a0820db7a416bba8050e318b489da0146c18669bb9512c69f4ba7","family_id":"qf-density16-matrix-sum-transpose","seed":202610162038,"semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[0,-1,-2],[1,0,-1],[2,1,0]],\"N\":[[0,1,2],[-1,0,1],[-2,-1,0]]}"},"fields":{"title":"Same-Shape Matrix Sum and Both Transpose Routes","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[0, -1, -2], [1, 0, -1], [2, 1, 0]], N=[[0, 1, 2], [-1, 0, 1], [-2, -1, 0]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).","answer":{"kind":"expression","value":"([3, 3], [[0, 0, 0], [0, 0, 0], [0, 0, 0]], [[0, 0, 0], [0, 0, 0], [0, 0, 0]], [[0, 0, 0], [0, 0, 0], [0, 0, 0]])","explanation":"Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0, 0], [0, 0, 0], [0, 0, 0]], [[0, 0, 0], [0, 0, 0], [0, 0, 0]], [[0, 0, 0], [0, 0, 0], [0, 0, 0]])."},"skills":["Same-Shape Matrix Sum and Both Transpose Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0039","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20b0d8f2c23e3360e646b9cd25a9efa5246bc88313ff01b05d2480b305fd48ae","family_id":"qf-density16-matrix-sum-transpose","seed":202610162039,"semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[1,0,-1,-2],[2,1,0,-1],[3,2,1,0]],\"N\":[[0,1,2,3],[-1,0,1,2],[-2,-1,0,1]]}"},"fields":{"title":"Same-Shape Matrix Sum and Both Transpose Routes","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[1, 0, -1, -2], [2, 1, 0, -1], [3, 2, 1, 0]], N=[[0, 1, 2, 3], [-1, 0, 1, 2], [-2, -1, 0, 1]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).","answer":{"kind":"expression","value":"([3, 4], [[1, 1, 1, 1], [1, 1, 1, 1], [1, 1, 1, 1]], [[1, 1, 1], [1, 1, 1], [1, 1, 1], [1, 1, 1]], [[1, 1, 1], [1, 1, 1], [1, 1, 1], [1, 1, 1]])","explanation":"Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 4], [[1, 1, 1, 1], [1, 1, 1, 1], [1, 1, 1, 1]], [[1, 1, 1], [1, 1, 1], [1, 1, 1], [1, 1, 1]], [[1, 1, 1], [1, 1, 1], [1, 1, 1], [1, 1, 1]])."},"skills":["Same-Shape Matrix Sum and Both Transpose Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0040","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5843a8cb1ed130fa0fe7f0c9f95bf9872dda1385118af50bd0208fe1c39dfa86","family_id":"qf-density16-matrix-sum-transpose","seed":202610162040,"semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[2,1,0,-1,-2],[3,2,1,0,-1],[4,3,2,1,0]],\"N\":[[0,1,2,3,4],[-1,0,1,2,3],[-2,-1,0,1,2]]}"},"fields":{"title":"Same-Shape Matrix Sum and Both Transpose Routes","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[2, 1, 0, -1, -2], [3, 2, 1, 0, -1], [4, 3, 2, 1, 0]], N=[[0, 1, 2, 3, 4], [-1, 0, 1, 2, 3], [-2, -1, 0, 1, 2]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).","answer":{"kind":"expression","value":"([3, 5], [[2, 2, 2, 2, 2], [2, 2, 2, 2, 2], [2, 2, 2, 2, 2]], [[2, 2, 2], [2, 2, 2], [2, 2, 2], [2, 2, 2], [2, 2, 2]], [[2, 2, 2], [2, 2, 2], [2, 2, 2], [2, 2, 2], [2, 2, 2]])","explanation":"Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 5], [[2, 2, 2, 2, 2], [2, 2, 2, 2, 2], [2, 2, 2, 2, 2]], [[2, 2, 2], [2, 2, 2], [2, 2, 2], [2, 2, 2], [2, 2, 2]], [[2, 2, 2], [2, 2, 2], [2, 2, 2], [2, 2, 2], [2, 2, 2]])."},"skills":["Same-Shape Matrix Sum and Both Transpose Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0041","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a722cc829aaca3feaa282dad80b14cf067969315f776b2df94018e5644c7822e","family_id":"qf-density16-matrix-sum-transpose","seed":202610162041,"semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[3],[4],[5],[6]],\"N\":[[0],[-1],[-2],[-3]]}"},"fields":{"title":"Same-Shape Matrix Sum and Both Transpose Routes","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[3], [4], [5], [6]], N=[[0], [-1], [-2], [-3]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).","answer":{"kind":"expression","value":"([4, 1], [[3], [3], [3], [3]], [[3, 3, 3, 3]], [[3, 3, 3, 3]])","explanation":"Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 1], [[3], [3], [3], [3]], [[3, 3, 3, 3]], [[3, 3, 3, 3]])."},"skills":["Same-Shape Matrix Sum and Both Transpose Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0042","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6444e595f5dbae49cf886b6809b6ca64d5eca5cd36b7261688962ac56f6104e","family_id":"qf-density16-matrix-sum-transpose","seed":202610162042,"semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[4,3],[5,4],[6,5],[7,6]],\"N\":[[0,1],[-1,0],[-2,-1],[-3,-2]]}"},"fields":{"title":"Same-Shape Matrix Sum and Both Transpose Routes","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[4, 3], [5, 4], [6, 5], [7, 6]], N=[[0, 1], [-1, 0], [-2, -1], [-3, -2]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).","answer":{"kind":"expression","value":"([4, 2], [[4, 4], [4, 4], [4, 4], [4, 4]], [[4, 4, 4, 4], [4, 4, 4, 4]], [[4, 4, 4, 4], [4, 4, 4, 4]])","explanation":"Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 2], [[4, 4], [4, 4], [4, 4], [4, 4]], [[4, 4, 4, 4], [4, 4, 4, 4]], [[4, 4, 4, 4], [4, 4, 4, 4]])."},"skills":["Same-Shape Matrix Sum and Both Transpose Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0043","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"674b06651e69ab09eb229a75e63c8e0c0cd4f02daf9fdcbc4c133b3f433b083c","family_id":"qf-density16-matrix-sum-transpose","seed":202610162043,"semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[5,4,3],[6,5,4],[7,6,5],[8,7,6]],\"N\":[[0,1,2],[-1,0,1],[-2,-1,0],[-3,-2,-1]]}"},"fields":{"title":"Same-Shape Matrix Sum and Both Transpose Routes","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[5, 4, 3], [6, 5, 4], [7, 6, 5], [8, 7, 6]], N=[[0, 1, 2], [-1, 0, 1], [-2, -1, 0], [-3, -2, -1]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).","answer":{"kind":"expression","value":"([4, 3], [[5, 5, 5], [5, 5, 5], [5, 5, 5], [5, 5, 5]], [[5, 5, 5, 5], [5, 5, 5, 5], [5, 5, 5, 5]], [[5, 5, 5, 5], [5, 5, 5, 5], [5, 5, 5, 5]])","explanation":"Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 3], [[5, 5, 5], [5, 5, 5], [5, 5, 5], [5, 5, 5]], [[5, 5, 5, 5], [5, 5, 5, 5], [5, 5, 5, 5]], [[5, 5, 5, 5], [5, 5, 5, 5], [5, 5, 5, 5]])."},"skills":["Same-Shape Matrix Sum and Both Transpose Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0044","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d684371dd5bca510bbcbcbb60832e450a5b6a3850b46a6f84c289bb123a5742b","family_id":"qf-density16-matrix-sum-transpose","seed":202610162044,"semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[6,5,4,3],[7,6,5,4],[8,7,6,5],[9,8,7,6]],\"N\":[[0,1,2,3],[-1,0,1,2],[-2,-1,0,1],[-3,-2,-1,0]]}"},"fields":{"title":"Same-Shape Matrix Sum and Both Transpose Routes","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[6, 5, 4, 3], [7, 6, 5, 4], [8, 7, 6, 5], [9, 8, 7, 6]], N=[[0, 1, 2, 3], [-1, 0, 1, 2], [-2, -1, 0, 1], [-3, -2, -1, 0]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).","answer":{"kind":"expression","value":"([4, 4], [[6, 6, 6, 6], [6, 6, 6, 6], [6, 6, 6, 6], [6, 6, 6, 6]], [[6, 6, 6, 6], [6, 6, 6, 6], [6, 6, 6, 6], [6, 6, 6, 6]], [[6, 6, 6, 6], [6, 6, 6, 6], [6, 6, 6, 6], [6, 6, 6, 6]])","explanation":"Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 4], [[6, 6, 6, 6], [6, 6, 6, 6], [6, 6, 6, 6], [6, 6, 6, 6]], [[6, 6, 6, 6], [6, 6, 6, 6], [6, 6, 6, 6], [6, 6, 6, 6]], [[6, 6, 6, 6], [6, 6, 6, 6], [6, 6, 6, 6], [6, 6, 6, 6]])."},"skills":["Same-Shape Matrix Sum and Both Transpose Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0045","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4acd765dd649f77bc4c5ca4ce299abe58ef252f6912fce27a821d88d454c7c48","family_id":"qf-density16-matrix-sum-transpose","seed":202610162045,"semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[7,6,5,4,3],[8,7,6,5,4],[9,8,7,6,5],[10,9,8,7,6]],\"N\":[[0,1,2,3,4],[-1,0,1,2,3],[-2,-1,0,1,2],[-3,-2,-1,0,1]]}"},"fields":{"title":"Same-Shape Matrix Sum and Both Transpose Routes","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[7, 6, 5, 4, 3], [8, 7, 6, 5, 4], [9, 8, 7, 6, 5], [10, 9, 8, 7, 6]], N=[[0, 1, 2, 3, 4], [-1, 0, 1, 2, 3], [-2, -1, 0, 1, 2], [-3, -2, -1, 0, 1]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).","answer":{"kind":"expression","value":"([4, 5], [[7, 7, 7, 7, 7], [7, 7, 7, 7, 7], [7, 7, 7, 7, 7], [7, 7, 7, 7, 7]], [[7, 7, 7, 7], [7, 7, 7, 7], [7, 7, 7, 7], [7, 7, 7, 7], [7, 7, 7, 7]], [[7, 7, 7, 7], [7, 7, 7, 7], [7, 7, 7, 7], [7, 7, 7, 7], [7, 7, 7, 7]])","explanation":"Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 5], [[7, 7, 7, 7, 7], [7, 7, 7, 7, 7], [7, 7, 7, 7, 7], [7, 7, 7, 7, 7]], [[7, 7, 7, 7], [7, 7, 7, 7], [7, 7, 7, 7], [7, 7, 7, 7], [7, 7, 7, 7]], [[7, 7, 7, 7], [7, 7, 7, 7], [7, 7, 7, 7], [7, 7, 7, 7], [7, 7, 7, 7]])."},"skills":["Same-Shape Matrix Sum and Both Transpose Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0046","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d759db5fce60bb223709f6bbcdd333b507a5b2798322a2011cf0be955d51211","family_id":"qf-density16-matrix-sum-transpose","seed":202610162046,"semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[8],[9],[10],[11],[12]],\"N\":[[0],[-1],[-2],[-3],[-4]]}"},"fields":{"title":"Same-Shape Matrix Sum and Both Transpose Routes","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[8], [9], [10], [11], [12]], N=[[0], [-1], [-2], [-3], [-4]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).","answer":{"kind":"expression","value":"([5, 1], [[8], [8], [8], [8], [8]], [[8, 8, 8, 8, 8]], [[8, 8, 8, 8, 8]])","explanation":"Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 1], [[8], [8], [8], [8], [8]], [[8, 8, 8, 8, 8]], [[8, 8, 8, 8, 8]])."},"skills":["Same-Shape Matrix Sum and Both Transpose Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0047","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86e2b9b0709134619497cdcb1a045eaf998ea66e874db281a761e22f731676b9","family_id":"qf-density16-matrix-sum-transpose","seed":202610162047,"semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[9,8],[10,9],[11,10],[12,11],[13,12]],\"N\":[[0,1],[-1,0],[-2,-1],[-3,-2],[-4,-3]]}"},"fields":{"title":"Same-Shape Matrix Sum and Both Transpose Routes","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[9, 8], [10, 9], [11, 10], [12, 11], [13, 12]], N=[[0, 1], [-1, 0], [-2, -1], [-3, -2], [-4, -3]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).","answer":{"kind":"expression","value":"([5, 2], [[9, 9], [9, 9], [9, 9], [9, 9], [9, 9]], [[9, 9, 9, 9, 9], [9, 9, 9, 9, 9]], [[9, 9, 9, 9, 9], [9, 9, 9, 9, 9]])","explanation":"Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 2], [[9, 9], [9, 9], [9, 9], [9, 9], [9, 9]], [[9, 9, 9, 9, 9], [9, 9, 9, 9, 9]], [[9, 9, 9, 9, 9], [9, 9, 9, 9, 9]])."},"skills":["Same-Shape Matrix Sum and Both Transpose Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0048","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e33281f3500b8644e5eaa0d93607eb78a772af90be2a1ddc7e39e0b69e6667c","family_id":"qf-density16-matrix-sum-transpose","seed":202610162048,"semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[10,9,8],[11,10,9],[12,11,10],[13,12,11],[14,13,12]],\"N\":[[0,1,2],[-1,0,1],[-2,-1,0],[-3,-2,-1],[-4,-3,-2]]}"},"fields":{"title":"Same-Shape Matrix Sum and Both Transpose Routes","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[10, 9, 8], [11, 10, 9], [12, 11, 10], [13, 12, 11], [14, 13, 12]], N=[[0, 1, 2], [-1, 0, 1], [-2, -1, 0], [-3, -2, -1], [-4, -3, -2]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).","answer":{"kind":"expression","value":"([5, 3], [[10, 10, 10], [10, 10, 10], [10, 10, 10], [10, 10, 10], [10, 10, 10]], [[10, 10, 10, 10, 10], [10, 10, 10, 10, 10], [10, 10, 10, 10, 10]], [[10, 10, 10, 10, 10], [10, 10, 10, 10, 10], [10, 10, 10, 10, 10]])","explanation":"Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 3], [[10, 10, 10], [10, 10, 10], [10, 10, 10], [10, 10, 10], [10, 10, 10]], [[10, 10, 10, 10, 10], [10, 10, 10, 10, 10], [10, 10, 10, 10, 10]], [[10, 10, 10, 10, 10], [10, 10, 10, 10, 10], [10, 10, 10, 10, 10]])."},"skills":["Same-Shape Matrix Sum and Both Transpose Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0049","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47b2b91fd7cb3c306bd391d217b66ae4a2c32e1d27b44f79c71d598dddc767a0","family_id":"qf-density16-matrix-sum-transpose","seed":202610162049,"semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[11,10,9,8],[12,11,10,9],[13,12,11,10],[14,13,12,11],[15,14,13,12]],\"N\":[[0,1,2,3],[-1,0,1,2],[-2,-1,0,1],[-3,-2,-1,0],[-4,-3,-2,-1]]}"},"fields":{"title":"Same-Shape Matrix Sum and Both Transpose Routes","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[11, 10, 9, 8], [12, 11, 10, 9], [13, 12, 11, 10], [14, 13, 12, 11], [15, 14, 13, 12]], N=[[0, 1, 2, 3], [-1, 0, 1, 2], [-2, -1, 0, 1], [-3, -2, -1, 0], [-4, -3, -2, -1]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).","answer":{"kind":"expression","value":"([5, 4], [[11, 11, 11, 11], [11, 11, 11, 11], [11, 11, 11, 11], [11, 11, 11, 11], [11, 11, 11, 11]], [[11, 11, 11, 11, 11], [11, 11, 11, 11, 11], [11, 11, 11, 11, 11], [11, 11, 11, 11, 11]], [[11, 11, 11, 11, 11], [11, 11, 11, 11, 11], [11, 11, 11, 11, 11], [11, 11, 11, 11, 11]])","explanation":"Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 4], [[11, 11, 11, 11], [11, 11, 11, 11], [11, 11, 11, 11], [11, 11, 11, 11], [11, 11, 11, 11]], [[11, 11, 11, 11, 11], [11, 11, 11, 11, 11], [11, 11, 11, 11, 11], [11, 11, 11, 11, 11]], [[11, 11, 11, 11, 11], [11, 11, 11, 11, 11], [11, 11, 11, 11, 11], [11, 11, 11, 11, 11]])."},"skills":["Same-Shape Matrix Sum and Both Transpose Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0050","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b270ac25ba8245a1331a19c83809b31c9c3a736fcb7971c66d97a9e4008b3223","family_id":"qf-density16-matrix-sum-transpose","seed":202610162050,"semantic_template":"qf-density16-matrix-sum-transpose/{\"M\":[[12,11,10,9,8],[13,12,11,10,9],[14,13,12,11,10],[15,14,13,12,11],[16,15,14,13,12]],\"N\":[[0,1,2,3,4],[-1,0,1,2,3],[-2,-1,0,1,2],[-3,-2,-1,0,1],[-4,-3,-2,-1,0]]}"},"fields":{"title":"Same-Shape Matrix Sum and Both Transpose Routes","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[12, 11, 10, 9, 8], [13, 12, 11, 10, 9], [14, 13, 12, 11, 10], [15, 14, 13, 12, 11], [16, 15, 14, 13, 12]], N=[[0, 1, 2, 3, 4], [-1, 0, 1, 2, 3], [-2, -1, 0, 1, 2], [-3, -2, -1, 0, 1], [-4, -3, -2, -1, 0]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).","answer":{"kind":"expression","value":"([5, 5], [[12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12]], [[12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12]], [[12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12]])","explanation":"Check matching shapes before addition, then compute both complete routes independently. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 5], [[12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12]], [[12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12]], [[12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12]])."},"skills":["Same-Shape Matrix Sum and Both Transpose Routes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0051","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba8371558fadf77b759ec8ada53869e48b9c9ef7019984b325d96088b774d49e","family_id":"qf-density16-matrix-three-index-qualification","seed":202610162051,"semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,0],[1,1,0],[0,1,\"untyped\"]],\"shape\":[1,1]}"},"fields":{"title":"Three 1-Based Matrix Queries with Eligibility Guards","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[0]], ordered (i,j) candidates=[[1, 1], [1, 1], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.","answer":{"kind":"expression","value":"([1, 1], [Yes, Yes, No], [0, 0, invalid-index])","explanation":"Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1], [Yes, Yes, No], [0, 0, invalid-index])."},"skills":["Three 1-Based Matrix Queries with Eligibility Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0052","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29f0a594a7f5b5e9c254e2efa913bb9678f3823018049c9e57b4eaf1630a493a","family_id":"qf-density16-matrix-three-index-qualification","seed":202610162052,"semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,-11],[1,2,-12],[0,1,\"untyped\"]],\"shape\":[1,2]}"},"fields":{"title":"Three 1-Based Matrix Queries with Eligibility Guards","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-11, -12]], ordered (i,j) candidates=[[1, 1], [1, 2], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.","answer":{"kind":"expression","value":"([1, 2], [Yes, Yes, No], [-11, -12, invalid-index])","explanation":"Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2], [Yes, Yes, No], [-11, -12, invalid-index])."},"skills":["Three 1-Based Matrix Queries with Eligibility Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0053","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"82ae341d9203b6dfc44c1f24eaf8d1a08cac11acd7d0e090631f0c9a80fae95f","family_id":"qf-density16-matrix-three-index-qualification","seed":202610162053,"semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,-10],[1,3,-12],[0,1,\"untyped\"]],\"shape\":[1,3]}"},"fields":{"title":"Three 1-Based Matrix Queries with Eligibility Guards","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-10, -11, -12]], ordered (i,j) candidates=[[1, 1], [1, 3], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.","answer":{"kind":"expression","value":"([1, 3], [Yes, Yes, No], [-10, -12, invalid-index])","explanation":"Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 3], [Yes, Yes, No], [-10, -12, invalid-index])."},"skills":["Three 1-Based Matrix Queries with Eligibility Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0054","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52f2ecdcd31bd0eb50ce3b305223640bdd1d90bdbfa5c1e701e4ab0f4024d885","family_id":"qf-density16-matrix-three-index-qualification","seed":202610162054,"semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,-9],[1,4,-12],[0,1,\"untyped\"]],\"shape\":[1,4]}"},"fields":{"title":"Three 1-Based Matrix Queries with Eligibility Guards","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-9, -10, -11, -12]], ordered (i,j) candidates=[[1, 1], [1, 4], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.","answer":{"kind":"expression","value":"([1, 4], [Yes, Yes, No], [-9, -12, invalid-index])","explanation":"Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 4], [Yes, Yes, No], [-9, -12, invalid-index])."},"skills":["Three 1-Based Matrix Queries with Eligibility Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0055","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eab1fa05ad433e7c1921ea885eaf6dd64b873419296539a6a23f7f94f53c994a","family_id":"qf-density16-matrix-three-index-qualification","seed":202610162055,"semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,-8],[1,5,-12],[0,1,\"untyped\"]],\"shape\":[1,5]}"},"fields":{"title":"Three 1-Based Matrix Queries with Eligibility Guards","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-8, -9, -10, -11, -12]], ordered (i,j) candidates=[[1, 1], [1, 5], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.","answer":{"kind":"expression","value":"([1, 5], [Yes, Yes, No], [-8, -12, invalid-index])","explanation":"Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 5], [Yes, Yes, No], [-8, -12, invalid-index])."},"skills":["Three 1-Based Matrix Queries with Eligibility Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0056","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5777fb6f1b7fda3f39d2c6d0033a75907222388a4d4a033713b8cc5c7d9405e6","family_id":"qf-density16-matrix-three-index-qualification","seed":202610162056,"semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,-7],[2,1,-6],[0,1,\"untyped\"]],\"shape\":[2,1]}"},"fields":{"title":"Three 1-Based Matrix Queries with Eligibility Guards","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-7], [-6]], ordered (i,j) candidates=[[1, 1], [2, 1], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.","answer":{"kind":"expression","value":"([2, 1], [Yes, Yes, No], [-7, -6, invalid-index])","explanation":"Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1], [Yes, Yes, No], [-7, -6, invalid-index])."},"skills":["Three 1-Based Matrix Queries with Eligibility Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0057","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0d71eddff741569a0d6be00e9a332e931ea759617727724c85b7dab6caccd26","family_id":"qf-density16-matrix-three-index-qualification","seed":202610162057,"semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,-6],[2,2,-6],[0,1,\"untyped\"]],\"shape\":[2,2]}"},"fields":{"title":"Three 1-Based Matrix Queries with Eligibility Guards","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-6, -7], [-5, -6]], ordered (i,j) candidates=[[1, 1], [2, 2], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.","answer":{"kind":"expression","value":"([2, 2], [Yes, Yes, No], [-6, -6, invalid-index])","explanation":"Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2], [Yes, Yes, No], [-6, -6, invalid-index])."},"skills":["Three 1-Based Matrix Queries with Eligibility Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0058","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c52e550c24d5556333c174af17bcd3d73a59c750f52dfe1d876e824c815f36ca","family_id":"qf-density16-matrix-three-index-qualification","seed":202610162058,"semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,-5],[2,3,-6],[0,1,\"untyped\"]],\"shape\":[2,3]}"},"fields":{"title":"Three 1-Based Matrix Queries with Eligibility Guards","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-5, -6, -7], [-4, -5, -6]], ordered (i,j) candidates=[[1, 1], [2, 3], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.","answer":{"kind":"expression","value":"([2, 3], [Yes, Yes, No], [-5, -6, invalid-index])","explanation":"Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 3], [Yes, Yes, No], [-5, -6, invalid-index])."},"skills":["Three 1-Based Matrix Queries with Eligibility Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0059","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7564f607ee5f84fc4776001415286976174269d2fb4362b88f31a41fa32edf5b","family_id":"qf-density16-matrix-three-index-qualification","seed":202610162059,"semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,-4],[2,4,-6],[0,1,\"untyped\"]],\"shape\":[2,4]}"},"fields":{"title":"Three 1-Based Matrix Queries with Eligibility Guards","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-4, -5, -6, -7], [-3, -4, -5, -6]], ordered (i,j) candidates=[[1, 1], [2, 4], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.","answer":{"kind":"expression","value":"([2, 4], [Yes, Yes, No], [-4, -6, invalid-index])","explanation":"Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 4], [Yes, Yes, No], [-4, -6, invalid-index])."},"skills":["Three 1-Based Matrix Queries with Eligibility Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0060","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0bbc0f006c48ce72798de615267347fd2bb424513c8463f653eb1fd3caa2ee3f","family_id":"qf-density16-matrix-three-index-qualification","seed":202610162060,"semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,-3],[2,5,-6],[0,1,\"untyped\"]],\"shape\":[2,5]}"},"fields":{"title":"Three 1-Based Matrix Queries with Eligibility Guards","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-3, -4, -5, -6, -7], [-2, -3, -4, -5, -6]], ordered (i,j) candidates=[[1, 1], [2, 5], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.","answer":{"kind":"expression","value":"([2, 5], [Yes, Yes, No], [-3, -6, invalid-index])","explanation":"Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 5], [Yes, Yes, No], [-3, -6, invalid-index])."},"skills":["Three 1-Based Matrix Queries with Eligibility Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0061","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03ddf08397535e6b4fcd5130fc67a23d8d0891df91074f19d4d129cb5d6a021a","family_id":"qf-density16-matrix-three-index-qualification","seed":202610162061,"semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,-2],[3,1,0],[0,1,\"untyped\"]],\"shape\":[3,1]}"},"fields":{"title":"Three 1-Based Matrix Queries with Eligibility Guards","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-2], [-1], [0]], ordered (i,j) candidates=[[1, 1], [3, 1], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.","answer":{"kind":"expression","value":"([3, 1], [Yes, Yes, No], [-2, 0, invalid-index])","explanation":"Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 1], [Yes, Yes, No], [-2, 0, invalid-index])."},"skills":["Three 1-Based Matrix Queries with Eligibility Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0062","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"82935c2b1190bdc4487d464d29b0c5b4f2cd0e20de875690b857c80c1cc64bc5","family_id":"qf-density16-matrix-three-index-qualification","seed":202610162062,"semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,-1],[3,2,0],[0,1,\"untyped\"]],\"shape\":[3,2]}"},"fields":{"title":"Three 1-Based Matrix Queries with Eligibility Guards","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-1, -2], [0, -1], [1, 0]], ordered (i,j) candidates=[[1, 1], [3, 2], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.","answer":{"kind":"expression","value":"([3, 2], [Yes, Yes, No], [-1, 0, invalid-index])","explanation":"Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 2], [Yes, Yes, No], [-1, 0, invalid-index])."},"skills":["Three 1-Based Matrix Queries with Eligibility Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0063","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"524b66ae6236dc727b200c31644092c672de6c1adf68ae7d0d3aa46e06889281","family_id":"qf-density16-matrix-three-index-qualification","seed":202610162063,"semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,0],[3,3,0],[0,1,\"untyped\"]],\"shape\":[3,3]}"},"fields":{"title":"Three 1-Based Matrix Queries with Eligibility Guards","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[0, -1, -2], [1, 0, -1], [2, 1, 0]], ordered (i,j) candidates=[[1, 1], [3, 3], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.","answer":{"kind":"expression","value":"([3, 3], [Yes, Yes, No], [0, 0, invalid-index])","explanation":"Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [Yes, Yes, No], [0, 0, invalid-index])."},"skills":["Three 1-Based Matrix Queries with Eligibility Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0064","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b22140247950fce8343d8a832dd19b88302d4fd392eba848ab10dba3e9c983e5","family_id":"qf-density16-matrix-three-index-qualification","seed":202610162064,"semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,1],[3,4,0],[0,1,\"untyped\"]],\"shape\":[3,4]}"},"fields":{"title":"Three 1-Based Matrix Queries with Eligibility Guards","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[1, 0, -1, -2], [2, 1, 0, -1], [3, 2, 1, 0]], ordered (i,j) candidates=[[1, 1], [3, 4], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.","answer":{"kind":"expression","value":"([3, 4], [Yes, Yes, No], [1, 0, invalid-index])","explanation":"Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 4], [Yes, Yes, No], [1, 0, invalid-index])."},"skills":["Three 1-Based Matrix Queries with Eligibility Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0065","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"486a3a2656e71a25d2acb8f9c4e49c44c6dd061d7f789fa31ca2b649032fbcd8","family_id":"qf-density16-matrix-three-index-qualification","seed":202610162065,"semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,2],[3,5,0],[0,1,\"untyped\"]],\"shape\":[3,5]}"},"fields":{"title":"Three 1-Based Matrix Queries with Eligibility Guards","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[2, 1, 0, -1, -2], [3, 2, 1, 0, -1], [4, 3, 2, 1, 0]], ordered (i,j) candidates=[[1, 1], [3, 5], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.","answer":{"kind":"expression","value":"([3, 5], [Yes, Yes, No], [2, 0, invalid-index])","explanation":"Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 5], [Yes, Yes, No], [2, 0, invalid-index])."},"skills":["Three 1-Based Matrix Queries with Eligibility Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0066","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88fd80b131be8aba9e29e002bf57988044e7222c81e2f67b6b8957cb915d9895","family_id":"qf-density16-matrix-three-index-qualification","seed":202610162066,"semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,3],[4,1,6],[0,1,\"untyped\"]],\"shape\":[4,1]}"},"fields":{"title":"Three 1-Based Matrix Queries with Eligibility Guards","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[3], [4], [5], [6]], ordered (i,j) candidates=[[1, 1], [4, 1], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.","answer":{"kind":"expression","value":"([4, 1], [Yes, Yes, No], [3, 6, invalid-index])","explanation":"Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 1], [Yes, Yes, No], [3, 6, invalid-index])."},"skills":["Three 1-Based Matrix Queries with Eligibility Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0067","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc4ddce3763696925888733def45d28c175f4b0166b67c5b697abcb40702b40b","family_id":"qf-density16-matrix-three-index-qualification","seed":202610162067,"semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,4],[4,2,6],[0,1,\"untyped\"]],\"shape\":[4,2]}"},"fields":{"title":"Three 1-Based Matrix Queries with Eligibility Guards","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[4, 3], [5, 4], [6, 5], [7, 6]], ordered (i,j) candidates=[[1, 1], [4, 2], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.","answer":{"kind":"expression","value":"([4, 2], [Yes, Yes, No], [4, 6, invalid-index])","explanation":"Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 2], [Yes, Yes, No], [4, 6, invalid-index])."},"skills":["Three 1-Based Matrix Queries with Eligibility Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0068","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a013b7d386aeab7688ae316742bff273d55b6fe691745591c91c98a3908abdf","family_id":"qf-density16-matrix-three-index-qualification","seed":202610162068,"semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,5],[4,3,6],[0,1,\"untyped\"]],\"shape\":[4,3]}"},"fields":{"title":"Three 1-Based Matrix Queries with Eligibility Guards","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[5, 4, 3], [6, 5, 4], [7, 6, 5], [8, 7, 6]], ordered (i,j) candidates=[[1, 1], [4, 3], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.","answer":{"kind":"expression","value":"([4, 3], [Yes, Yes, No], [5, 6, invalid-index])","explanation":"Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 3], [Yes, Yes, No], [5, 6, invalid-index])."},"skills":["Three 1-Based Matrix Queries with Eligibility Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0069","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a41aded076b69b0cb850928e2e3880cf9fe459496d602140e8158af21a2126f","family_id":"qf-density16-matrix-three-index-qualification","seed":202610162069,"semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,6],[4,4,6],[0,1,\"untyped\"]],\"shape\":[4,4]}"},"fields":{"title":"Three 1-Based Matrix Queries with Eligibility Guards","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[6, 5, 4, 3], [7, 6, 5, 4], [8, 7, 6, 5], [9, 8, 7, 6]], ordered (i,j) candidates=[[1, 1], [4, 4], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.","answer":{"kind":"expression","value":"([4, 4], [Yes, Yes, No], [6, 6, invalid-index])","explanation":"Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 4], [Yes, Yes, No], [6, 6, invalid-index])."},"skills":["Three 1-Based Matrix Queries with Eligibility Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0070","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"125f88490997aca9ef5966606b03eac25b2fd5ece10ccecf6b18082c28acd4b5","family_id":"qf-density16-matrix-three-index-qualification","seed":202610162070,"semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,7],[4,5,6],[0,1,\"untyped\"]],\"shape\":[4,5]}"},"fields":{"title":"Three 1-Based Matrix Queries with Eligibility Guards","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[7, 6, 5, 4, 3], [8, 7, 6, 5, 4], [9, 8, 7, 6, 5], [10, 9, 8, 7, 6]], ordered (i,j) candidates=[[1, 1], [4, 5], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.","answer":{"kind":"expression","value":"([4, 5], [Yes, Yes, No], [7, 6, invalid-index])","explanation":"Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 5], [Yes, Yes, No], [7, 6, invalid-index])."},"skills":["Three 1-Based Matrix Queries with Eligibility Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0071","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d60385c5c5575d03b86d724363750b7b55dc724c1031aa533c8daadbd692b815","family_id":"qf-density16-matrix-three-index-qualification","seed":202610162071,"semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,8],[5,1,12],[0,1,\"untyped\"]],\"shape\":[5,1]}"},"fields":{"title":"Three 1-Based Matrix Queries with Eligibility Guards","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[8], [9], [10], [11], [12]], ordered (i,j) candidates=[[1, 1], [5, 1], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.","answer":{"kind":"expression","value":"([5, 1], [Yes, Yes, No], [8, 12, invalid-index])","explanation":"Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 1], [Yes, Yes, No], [8, 12, invalid-index])."},"skills":["Three 1-Based Matrix Queries with Eligibility Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0072","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d68a85c0f00bdc7d26c17b628ca5be06920986385bf0cbee04369cff17779c5","family_id":"qf-density16-matrix-three-index-qualification","seed":202610162072,"semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,9],[5,2,12],[0,1,\"untyped\"]],\"shape\":[5,2]}"},"fields":{"title":"Three 1-Based Matrix Queries with Eligibility Guards","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[9, 8], [10, 9], [11, 10], [12, 11], [13, 12]], ordered (i,j) candidates=[[1, 1], [5, 2], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.","answer":{"kind":"expression","value":"([5, 2], [Yes, Yes, No], [9, 12, invalid-index])","explanation":"Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 2], [Yes, Yes, No], [9, 12, invalid-index])."},"skills":["Three 1-Based Matrix Queries with Eligibility Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0073","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c90e2705cdec6dc2a0b90cd293009557bd1ef4f2a10e1f98dfec84dc23d7c0f4","family_id":"qf-density16-matrix-three-index-qualification","seed":202610162073,"semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,10],[5,3,12],[0,1,\"untyped\"]],\"shape\":[5,3]}"},"fields":{"title":"Three 1-Based Matrix Queries with Eligibility Guards","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[10, 9, 8], [11, 10, 9], [12, 11, 10], [13, 12, 11], [14, 13, 12]], ordered (i,j) candidates=[[1, 1], [5, 3], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.","answer":{"kind":"expression","value":"([5, 3], [Yes, Yes, No], [10, 12, invalid-index])","explanation":"Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 3], [Yes, Yes, No], [10, 12, invalid-index])."},"skills":["Three 1-Based Matrix Queries with Eligibility Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0074","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6194eabd9e9a54d68a635d035e2266f36cdead8ecee4be573db151f3a5999d7d","family_id":"qf-density16-matrix-three-index-qualification","seed":202610162074,"semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,11],[5,4,12],[0,1,\"untyped\"]],\"shape\":[5,4]}"},"fields":{"title":"Three 1-Based Matrix Queries with Eligibility Guards","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[11, 10, 9, 8], [12, 11, 10, 9], [13, 12, 11, 10], [14, 13, 12, 11], [15, 14, 13, 12]], ordered (i,j) candidates=[[1, 1], [5, 4], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.","answer":{"kind":"expression","value":"([5, 4], [Yes, Yes, No], [11, 12, invalid-index])","explanation":"Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 4], [Yes, Yes, No], [11, 12, invalid-index])."},"skills":["Three 1-Based Matrix Queries with Eligibility Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0075","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cef0052a902940d6097da54598e2779a9311c0798a45228b505bebfea2716772","family_id":"qf-density16-matrix-three-index-qualification","seed":202610162075,"semantic_template":"qf-density16-matrix-three-index-qualification/{\"queries\":[[1,1,12],[5,5,12],[0,1,\"untyped\"]],\"shape\":[5,5]}"},"fields":{"title":"Three 1-Based Matrix Queries with Eligibility Guards","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[12, 11, 10, 9, 8], [13, 12, 11, 10, 9], [14, 13, 12, 11, 10], [15, 14, 13, 12, 11], [16, 15, 14, 13, 12]], ordered (i,j) candidates=[[1, 1], [5, 5], [0, 1]]. Give (shape of M, three index-eligibility values, three entries or invalid-index). Do not read an invalid entry.","answer":{"kind":"expression","value":"([5, 5], [Yes, Yes, No], [12, 12, invalid-index])","explanation":"Both indices must lie from 1 through their dimension before lookup; unqueried cells do not change this question’s semantic identity. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 5], [Yes, Yes, No], [12, 12, invalid-index])."},"skills":["Three 1-Based Matrix Queries with Eligibility Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0076","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"084e4425bf6ebf234475c4c2882430097c37f87fd922a0f3a798feb88e0c3110","family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162076,"semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[0]],\"candidate\":[[0]]}"},"fields":{"title":"Candidate Transpose Shape before Entrywise Residuals","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[0]], candidate=[[0]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).","answer":{"kind":"expression","value":"([[0]], [1, 1], Yes, [[0]], Yes)","explanation":"Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0]], [1, 1], Yes, [[0]], Yes)."},"skills":["Candidate Transpose Shape before Entrywise Residuals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0077","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe65f4ef57d3eecbc3e67ce66e93d23bd4561055630a5e94eb5b0467cbb8ad9d","family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162077,"semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[1,2]],\"candidate\":[[1,2]]}"},"fields":{"title":"Candidate Transpose Shape before Entrywise Residuals","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[1, 2]], candidate=[[1, 2]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).","answer":{"kind":"expression","value":"([[1], [2]], [1, 2], No, shape-mismatch, No)","explanation":"Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1], [2]], [1, 2], No, shape-mismatch, No)."},"skills":["Candidate Transpose Shape before Entrywise Residuals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0078","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a0587c34506b26655a5a7482c7bf37b5a6812152fdf9695175c3f188fabbb4a9","family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162078,"semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[1,2],[3,4]],\"candidate\":[[1,3],[2,4]]}"},"fields":{"title":"Candidate Transpose Shape before Entrywise Residuals","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[1, 2], [3, 4]], candidate=[[1, 3], [2, 4]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).","answer":{"kind":"expression","value":"([[1, 3], [2, 4]], [2, 2], Yes, [[0, 0], [0, 0]], Yes)","explanation":"Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 3], [2, 4]], [2, 2], Yes, [[0, 0], [0, 0]], Yes)."},"skills":["Candidate Transpose Shape before Entrywise Residuals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0079","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ac92372300a32914d431e1a23db0e91a0ab4217a21506ba71be60a585521ed9","family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162079,"semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[1,2],[3,4]],\"candidate\":[[1,0],[2,4]]}"},"fields":{"title":"Candidate Transpose Shape before Entrywise Residuals","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[1, 2], [3, 4]], candidate=[[1, 0], [2, 4]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).","answer":{"kind":"expression","value":"([[1, 3], [2, 4]], [2, 2], Yes, [[0, -3], [0, 0]], No)","explanation":"Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 3], [2, 4]], [2, 2], Yes, [[0, -3], [0, 0]], No)."},"skills":["Candidate Transpose Shape before Entrywise Residuals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0080","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cdee9a59f62e1acca7ac587f30465f9c00395d53ea79e90ec819e79552cce7ee","family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162080,"semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[-8,1],[2,-1]],\"candidate\":[[-7,2],[1,-1]]}"},"fields":{"title":"Candidate Transpose Shape before Entrywise Residuals","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-8, 1], [2, -1]], candidate=[[-7, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).","answer":{"kind":"expression","value":"([[-8, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)","explanation":"Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-8, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)."},"skills":["Candidate Transpose Shape before Entrywise Residuals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0081","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"351a6fbd3acebdcf7a6edd8b5566509474271e993728e2a800e6cbab491461d9","family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162081,"semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[-7,1],[2,-1]],\"candidate\":[[-6,2],[1,-1]]}"},"fields":{"title":"Candidate Transpose Shape before Entrywise Residuals","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-7, 1], [2, -1]], candidate=[[-6, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).","answer":{"kind":"expression","value":"([[-7, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)","explanation":"Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-7, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)."},"skills":["Candidate Transpose Shape before Entrywise Residuals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0082","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72addd8286375b76733e76bb33bdae46db23c712d1f7a93f9fda09793752d4a1","family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162082,"semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[-6,1],[2,-1]],\"candidate\":[[-5,2],[1,-1]]}"},"fields":{"title":"Candidate Transpose Shape before Entrywise Residuals","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-6, 1], [2, -1]], candidate=[[-5, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).","answer":{"kind":"expression","value":"([[-6, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)","explanation":"Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-6, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)."},"skills":["Candidate Transpose Shape before Entrywise Residuals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0083","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da02b92ebffeaf2e6b24270aa99e5c1ef4badf9edc69dfcb1dfeaa535da15c1f","family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162083,"semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[-5,1],[2,-1]],\"candidate\":[[-4,2],[1,-1]]}"},"fields":{"title":"Candidate Transpose Shape before Entrywise Residuals","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-5, 1], [2, -1]], candidate=[[-4, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).","answer":{"kind":"expression","value":"([[-5, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)","explanation":"Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-5, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)."},"skills":["Candidate Transpose Shape before Entrywise Residuals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0084","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d4b6f100cc0e5eb8a43d5db268c505b74eb3c12996842d2f630c3e3968dc9062","family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162084,"semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[-4,1],[2,-1]],\"candidate\":[[-3,2],[1,-1]]}"},"fields":{"title":"Candidate Transpose Shape before Entrywise Residuals","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-4, 1], [2, -1]], candidate=[[-3, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).","answer":{"kind":"expression","value":"([[-4, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)","explanation":"Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-4, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)."},"skills":["Candidate Transpose Shape before Entrywise Residuals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0085","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2216c684ebb63f959c528720469d2969048c6e6c0449c7ddd866dc27e9c98f11","family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162085,"semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[-3,1],[2,-1]],\"candidate\":[[-2,2],[1,-1]]}"},"fields":{"title":"Candidate Transpose Shape before Entrywise Residuals","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-3, 1], [2, -1]], candidate=[[-2, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).","answer":{"kind":"expression","value":"([[-3, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)","explanation":"Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-3, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)."},"skills":["Candidate Transpose Shape before Entrywise Residuals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0086","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"693216dafd49cabe54dfff90b5866ff33b4afeda472a559e2892f654efbb83ac","family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162086,"semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[-2,1],[2,-1]],\"candidate\":[[-1,2],[1,-1]]}"},"fields":{"title":"Candidate Transpose Shape before Entrywise Residuals","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-2, 1], [2, -1]], candidate=[[-1, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).","answer":{"kind":"expression","value":"([[-2, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)","explanation":"Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-2, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)."},"skills":["Candidate Transpose Shape before Entrywise Residuals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0087","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"729448c2a8f6233822d78ff67e7a0fd551f2141939dc0a7f3445e8aa13dec166","family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162087,"semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[-1,1],[2,-1]],\"candidate\":[[0,2],[1,-1]]}"},"fields":{"title":"Candidate Transpose Shape before Entrywise Residuals","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[-1, 1], [2, -1]], candidate=[[0, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).","answer":{"kind":"expression","value":"([[-1, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)","explanation":"Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[-1, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)."},"skills":["Candidate Transpose Shape before Entrywise Residuals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0088","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7c757a2add0038b58a3ae698accfca440810bcae95fb20c0751d9aaef2f8270e","family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162088,"semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[0,1],[2,-1]],\"candidate\":[[1,2],[1,-1]]}"},"fields":{"title":"Candidate Transpose Shape before Entrywise Residuals","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[0, 1], [2, -1]], candidate=[[1, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).","answer":{"kind":"expression","value":"([[0, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)","explanation":"Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)."},"skills":["Candidate Transpose Shape before Entrywise Residuals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0089","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6246ad1e50ca1023c542c8e8a07e70c8d6d75c26f9c6c0e7796d9798841735d9","family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162089,"semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[1,1],[2,-1]],\"candidate\":[[2,2],[1,-1]]}"},"fields":{"title":"Candidate Transpose Shape before Entrywise Residuals","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[1, 1], [2, -1]], candidate=[[2, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).","answer":{"kind":"expression","value":"([[1, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)","explanation":"Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[1, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)."},"skills":["Candidate Transpose Shape before Entrywise Residuals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0090","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"156199088e158db795078d09363b8d0c3fe4501ea28704ddabd7148f1b2c351b","family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162090,"semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[2,1],[2,-1]],\"candidate\":[[3,2],[1,-1]]}"},"fields":{"title":"Candidate Transpose Shape before Entrywise Residuals","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[2, 1], [2, -1]], candidate=[[3, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).","answer":{"kind":"expression","value":"([[2, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)","explanation":"Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[2, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)."},"skills":["Candidate Transpose Shape before Entrywise Residuals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0091","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e13ae9c54c1d08074c021c3af476f23ea4e2f05d50e2eedcd75675888aaf13b2","family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162091,"semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[3,1],[2,-1]],\"candidate\":[[4,2],[1,-1]]}"},"fields":{"title":"Candidate Transpose Shape before Entrywise Residuals","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[3, 1], [2, -1]], candidate=[[4, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).","answer":{"kind":"expression","value":"([[3, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)","explanation":"Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[3, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)."},"skills":["Candidate Transpose Shape before Entrywise Residuals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0092","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2cb07c9bc671339e2112ee552edc033a5254c2b4a88240613ad0396b34272ab7","family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162092,"semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[4,1],[2,-1]],\"candidate\":[[5,2],[1,-1]]}"},"fields":{"title":"Candidate Transpose Shape before Entrywise Residuals","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[4, 1], [2, -1]], candidate=[[5, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).","answer":{"kind":"expression","value":"([[4, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)","explanation":"Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[4, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)."},"skills":["Candidate Transpose Shape before Entrywise Residuals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0093","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ed065bf2f915557ddd05bc42d4e9311a78276cb093856e35d490863046a86dd","family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162093,"semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[5,1],[2,-1]],\"candidate\":[[6,2],[1,-1]]}"},"fields":{"title":"Candidate Transpose Shape before Entrywise Residuals","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[5, 1], [2, -1]], candidate=[[6, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).","answer":{"kind":"expression","value":"([[5, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)","explanation":"Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[5, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)."},"skills":["Candidate Transpose Shape before Entrywise Residuals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0094","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"493751215ffadea5b1cfedf26148771e2b1fb4d19bf6c65b874c0cc1bce1fb29","family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162094,"semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[6,1],[2,-1]],\"candidate\":[[7,2],[1,-1]]}"},"fields":{"title":"Candidate Transpose Shape before Entrywise Residuals","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[6, 1], [2, -1]], candidate=[[7, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).","answer":{"kind":"expression","value":"([[6, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)","explanation":"Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[6, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)."},"skills":["Candidate Transpose Shape before Entrywise Residuals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0095","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3d5fde113cec2dca023a9b601bc58b2d03c246b9f69eaddb5c48d80793334e6","family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162095,"semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[7,1],[2,-1]],\"candidate\":[[8,2],[1,-1]]}"},"fields":{"title":"Candidate Transpose Shape before Entrywise Residuals","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[7, 1], [2, -1]], candidate=[[8, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).","answer":{"kind":"expression","value":"([[7, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)","explanation":"Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[7, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)."},"skills":["Candidate Transpose Shape before Entrywise Residuals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0096","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ce8b10853973d2b21fe4e876cf68e3a17933fccfb08318b7e555a94a0fb416f","family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162096,"semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[8,1],[2,-1]],\"candidate\":[[9,2],[1,-1]]}"},"fields":{"title":"Candidate Transpose Shape before Entrywise Residuals","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[8, 1], [2, -1]], candidate=[[9, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).","answer":{"kind":"expression","value":"([[8, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)","explanation":"Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[8, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)."},"skills":["Candidate Transpose Shape before Entrywise Residuals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0097","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c831b4ce417585c6abb9b2eadc47d60a374be77ec167b9da6610830cec0e4dd2","family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162097,"semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[9,1],[2,-1]],\"candidate\":[[10,2],[1,-1]]}"},"fields":{"title":"Candidate Transpose Shape before Entrywise Residuals","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[9, 1], [2, -1]], candidate=[[10, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).","answer":{"kind":"expression","value":"([[9, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)","explanation":"Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[9, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)."},"skills":["Candidate Transpose Shape before Entrywise Residuals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0098","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c3c67e72c5337088b26aa035cac384f10ce78f4af4b6b773b346a2976bf8058","family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162098,"semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[10,1],[2,-1]],\"candidate\":[[11,2],[1,-1]]}"},"fields":{"title":"Candidate Transpose Shape before Entrywise Residuals","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[10, 1], [2, -1]], candidate=[[11, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).","answer":{"kind":"expression","value":"([[10, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)","explanation":"Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[10, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)."},"skills":["Candidate Transpose Shape before Entrywise Residuals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0099","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cac301c0036b3e19c9f620a3e6a967cbb8c678a26e3f361551dbb46ecec6b79b","family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162099,"semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[11,1],[2,-1]],\"candidate\":[[12,2],[1,-1]]}"},"fields":{"title":"Candidate Transpose Shape before Entrywise Residuals","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[11, 1], [2, -1]], candidate=[[12, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).","answer":{"kind":"expression","value":"([[11, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)","explanation":"Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[11, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)."},"skills":["Candidate Transpose Shape before Entrywise Residuals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r02-matrix-0100","batch_id":"batch-20261012-advance-20261016-r02-matrix-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"093f55c7528de32139df42fdccc987fef76aa9157edd60dbca7f98f35306ad9a","family_id":"qf-density16-matrix-candidate-transpose-audit","seed":202610162100,"semantic_template":"qf-density16-matrix-candidate-transpose-audit/{\"M\":[[12,1],[2,-1]],\"candidate\":[[13,2],[1,-1]]}"},"fields":{"title":"Candidate Transpose Shape before Entrywise Residuals","stem":"Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. M=[[12, 1], [2, -1]], candidate=[[13, 2], [1, -1]]. Give (Mᵀ, candidate shape, shape eligibility, candidate−Mᵀ residual matrix or shape-mismatch, candidate equality).","answer":{"kind":"expression","value":"([[12, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)","explanation":"Compute the transpose shape first and subtract entries only for a matching shape; do not manufacture residuals for a mismatch. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[12, 2], [1, -1]], [2, 2], Yes, [[1, 0], [0, 0]], No)."},"skills":["Candidate Transpose Shape before Entrywise Residuals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}