{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r01-vector-0001","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc8c10a7bb5f627b4e420da73929ba96ef5457289460f789c3280febd4568708","family_id":"qf-density16-vector-weighted-combination","seed":202610161001,"semantic_template":"qf-density16-vector-weighted-combination/{\"a\":0,\"b\":0,\"u\":[0,0],\"v\":[0,0]}"},"fields":{"title":"Ordered Weighted Vector Combination","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[0, 0], v=[0, 0], a=0, b=0. Give (a*u, b*v, a*u+b*v) in that order.","answer":{"kind":"expression","value":"([0, 0], [0, 0], [0, 0])","explanation":"Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0], [0, 0], [0, 0])."},"skills":["Ordered Weighted Vector Combination"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0002","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f297a50c69a3033ef0142455cc29ca6854b97def1eb881561c8820ddd598383","family_id":"qf-density16-vector-weighted-combination","seed":202610161002,"semantic_template":"qf-density16-vector-weighted-combination/{\"a\":0,\"b\":1,\"u\":[99,1],\"v\":[1,2]}"},"fields":{"title":"Ordered Weighted Vector Combination","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[99, 1], v=[1, 2], a=0, b=1. Give (a*u, b*v, a*u+b*v) in that order.","answer":{"kind":"expression","value":"([0, 0], [1, 2], [1, 2])","explanation":"Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0], [1, 2], [1, 2])."},"skills":["Ordered Weighted Vector Combination"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0003","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f42c2053a814b533ae43e38ee75ce54a349a9c76d211a79b52f2bf2d60dacad","family_id":"qf-density16-vector-weighted-combination","seed":202610161003,"semantic_template":"qf-density16-vector-weighted-combination/{\"a\":2,\"b\":-2,\"u\":[1,2],\"v\":[1,2]}"},"fields":{"title":"Ordered Weighted Vector Combination","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[1, 2], v=[1, 2], a=2, b=-2. Give (a*u, b*v, a*u+b*v) in that order.","answer":{"kind":"expression","value":"([2, 4], [-2, -4], [0, 0])","explanation":"Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 4], [-2, -4], [0, 0])."},"skills":["Ordered Weighted Vector Combination"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0004","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88cba0451bca2db9779f4332225b9ece62cc8688d0502e8f2e78c259faab795f","family_id":"qf-density16-vector-weighted-combination","seed":202610161004,"semantic_template":"qf-density16-vector-weighted-combination/{\"a\":9007199254740991,\"b\":-9007199254740991,\"u\":[2,-3],\"v\":[1,4]}"},"fields":{"title":"Ordered Weighted Vector Combination","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[2, -3], v=[1, 4], a=9007199254740991, b=-9007199254740991. Give (a*u, b*v, a*u+b*v) in that order.","answer":{"kind":"expression","value":"([18014398509481982, -27021597764222973], [-9007199254740991, -36028797018963964], [9007199254740991, -63050394783186937])","explanation":"Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([18014398509481982, -27021597764222973], [-9007199254740991, -36028797018963964], [9007199254740991, -63050394783186937])."},"skills":["Ordered Weighted Vector Combination"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0005","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5eceff3ccd69af9dc7be681857db7c404b2ee5dd9efc8c4d6de9e043f523962","family_id":"qf-density16-vector-weighted-combination","seed":202610161005,"semantic_template":"qf-density16-vector-weighted-combination/{\"a\":2,\"b\":-1,\"u\":[-8,1],\"v\":[1,2]}"},"fields":{"title":"Ordered Weighted Vector Combination","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-8, 1], v=[1, 2], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.","answer":{"kind":"expression","value":"([-16, 2], [-1, -2], [-17, 0])","explanation":"Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-16, 2], [-1, -2], [-17, 0])."},"skills":["Ordered Weighted Vector Combination"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0006","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91003d43755508c897957b3d0bf00fe74a9e60217fb1e66521bc40435b621009","family_id":"qf-density16-vector-weighted-combination","seed":202610161006,"semantic_template":"qf-density16-vector-weighted-combination/{\"a\":2,\"b\":-1,\"u\":[-7,1],\"v\":[1,3]}"},"fields":{"title":"Ordered Weighted Vector Combination","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-7, 1], v=[1, 3], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.","answer":{"kind":"expression","value":"([-14, 2], [-1, -3], [-15, -1])","explanation":"Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-14, 2], [-1, -3], [-15, -1])."},"skills":["Ordered Weighted Vector Combination"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0007","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"780e2e24df789ba00cb9a4f64b089eb943b7d727fc551b47f6dd71ac8272a66b","family_id":"qf-density16-vector-weighted-combination","seed":202610161007,"semantic_template":"qf-density16-vector-weighted-combination/{\"a\":2,\"b\":-1,\"u\":[-6,1],\"v\":[1,4]}"},"fields":{"title":"Ordered Weighted Vector Combination","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-6, 1], v=[1, 4], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.","answer":{"kind":"expression","value":"([-12, 2], [-1, -4], [-13, -2])","explanation":"Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-12, 2], [-1, -4], [-13, -2])."},"skills":["Ordered Weighted Vector Combination"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0008","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65fb92f7dd5dee680d3457bf9024e1d74260053f248876779bffbf65a5e2ea63","family_id":"qf-density16-vector-weighted-combination","seed":202610161008,"semantic_template":"qf-density16-vector-weighted-combination/{\"a\":2,\"b\":-1,\"u\":[-5,1],\"v\":[1,5]}"},"fields":{"title":"Ordered Weighted Vector Combination","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-5, 1], v=[1, 5], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.","answer":{"kind":"expression","value":"([-10, 2], [-1, -5], [-11, -3])","explanation":"Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-10, 2], [-1, -5], [-11, -3])."},"skills":["Ordered Weighted Vector Combination"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0009","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55d550cbac0c02ce1999bc314e9c99fe3ef61fcccc9a4b90eab1dcb97809fe1f","family_id":"qf-density16-vector-weighted-combination","seed":202610161009,"semantic_template":"qf-density16-vector-weighted-combination/{\"a\":2,\"b\":-1,\"u\":[-4,1],\"v\":[1,6]}"},"fields":{"title":"Ordered Weighted Vector Combination","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-4, 1], v=[1, 6], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.","answer":{"kind":"expression","value":"([-8, 2], [-1, -6], [-9, -4])","explanation":"Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-8, 2], [-1, -6], [-9, -4])."},"skills":["Ordered Weighted Vector Combination"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0010","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16d96a9b8884e323de10789f45bf3977cdc5ece21764740885d18941d702afc9","family_id":"qf-density16-vector-weighted-combination","seed":202610161010,"semantic_template":"qf-density16-vector-weighted-combination/{\"a\":2,\"b\":-1,\"u\":[-3,1],\"v\":[1,7]}"},"fields":{"title":"Ordered Weighted Vector Combination","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-3, 1], v=[1, 7], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.","answer":{"kind":"expression","value":"([-6, 2], [-1, -7], [-7, -5])","explanation":"Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-6, 2], [-1, -7], [-7, -5])."},"skills":["Ordered Weighted Vector Combination"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0011","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ac93c16b9962fbf8b9f8ac198004aa7754044c2a78a761a81675a02b9cf87b8","family_id":"qf-density16-vector-weighted-combination","seed":202610161011,"semantic_template":"qf-density16-vector-weighted-combination/{\"a\":2,\"b\":-1,\"u\":[-2,1],\"v\":[1,8]}"},"fields":{"title":"Ordered Weighted Vector Combination","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-2, 1], v=[1, 8], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.","answer":{"kind":"expression","value":"([-4, 2], [-1, -8], [-5, -6])","explanation":"Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-4, 2], [-1, -8], [-5, -6])."},"skills":["Ordered Weighted Vector Combination"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0012","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f40f37710ca4d7a5f6e207b9090f2b752c61cd927e36dc53c01b5de62339acb","family_id":"qf-density16-vector-weighted-combination","seed":202610161012,"semantic_template":"qf-density16-vector-weighted-combination/{\"a\":2,\"b\":-1,\"u\":[-1,1],\"v\":[1,9]}"},"fields":{"title":"Ordered Weighted Vector Combination","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-1, 1], v=[1, 9], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.","answer":{"kind":"expression","value":"([-2, 2], [-1, -9], [-3, -7])","explanation":"Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-2, 2], [-1, -9], [-3, -7])."},"skills":["Ordered Weighted Vector Combination"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0013","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2c6b2513b7aacb1466d3e64b5fed6b1c63667587b9159c0f66c4bf6415a7382","family_id":"qf-density16-vector-weighted-combination","seed":202610161013,"semantic_template":"qf-density16-vector-weighted-combination/{\"a\":2,\"b\":-1,\"u\":[0,1],\"v\":[1,10]}"},"fields":{"title":"Ordered Weighted Vector Combination","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[0, 1], v=[1, 10], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.","answer":{"kind":"expression","value":"([0, 2], [-1, -10], [-1, -8])","explanation":"Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2], [-1, -10], [-1, -8])."},"skills":["Ordered Weighted Vector Combination"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0014","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3de2c2c0edba05ddfc48caccc0ca2ebbb442658b6d1cb86ae9d1771ed7543678","family_id":"qf-density16-vector-weighted-combination","seed":202610161014,"semantic_template":"qf-density16-vector-weighted-combination/{\"a\":2,\"b\":-1,\"u\":[1,1],\"v\":[1,11]}"},"fields":{"title":"Ordered Weighted Vector Combination","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[1, 1], v=[1, 11], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.","answer":{"kind":"expression","value":"([2, 2], [-1, -11], [1, -9])","explanation":"Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2], [-1, -11], [1, -9])."},"skills":["Ordered Weighted Vector Combination"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0015","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d58837b56d4a6ba737760905b1549deee6cb46c4a35ebbcd70b42b4f458dd108","family_id":"qf-density16-vector-weighted-combination","seed":202610161015,"semantic_template":"qf-density16-vector-weighted-combination/{\"a\":2,\"b\":-1,\"u\":[2,1],\"v\":[1,12]}"},"fields":{"title":"Ordered Weighted Vector Combination","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[2, 1], v=[1, 12], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.","answer":{"kind":"expression","value":"([4, 2], [-1, -12], [3, -10])","explanation":"Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 2], [-1, -12], [3, -10])."},"skills":["Ordered Weighted Vector Combination"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0016","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"09a07b6969316f0fc5bf3b92fdfef4fc8188a91872b09b8c0c389f0d88e5f8ea","family_id":"qf-density16-vector-weighted-combination","seed":202610161016,"semantic_template":"qf-density16-vector-weighted-combination/{\"a\":2,\"b\":-1,\"u\":[3,1],\"v\":[1,13]}"},"fields":{"title":"Ordered Weighted Vector Combination","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[3, 1], v=[1, 13], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.","answer":{"kind":"expression","value":"([6, 2], [-1, -13], [5, -11])","explanation":"Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([6, 2], [-1, -13], [5, -11])."},"skills":["Ordered Weighted Vector Combination"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0017","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a55d55b8fbc7b4353832b880f8f8877c3dd6884efdc43568781f2f6fd1b1256","family_id":"qf-density16-vector-weighted-combination","seed":202610161017,"semantic_template":"qf-density16-vector-weighted-combination/{\"a\":2,\"b\":-1,\"u\":[4,1],\"v\":[1,14]}"},"fields":{"title":"Ordered Weighted Vector Combination","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[4, 1], v=[1, 14], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.","answer":{"kind":"expression","value":"([8, 2], [-1, -14], [7, -12])","explanation":"Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([8, 2], [-1, -14], [7, -12])."},"skills":["Ordered Weighted Vector Combination"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0018","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2337997cb4e53cf7dd2383a232ae84e12ffd9896b9fd2f09d7727efab11a98e","family_id":"qf-density16-vector-weighted-combination","seed":202610161018,"semantic_template":"qf-density16-vector-weighted-combination/{\"a\":2,\"b\":-1,\"u\":[5,1],\"v\":[1,15]}"},"fields":{"title":"Ordered Weighted Vector Combination","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[5, 1], v=[1, 15], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.","answer":{"kind":"expression","value":"([10, 2], [-1, -15], [9, -13])","explanation":"Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([10, 2], [-1, -15], [9, -13])."},"skills":["Ordered Weighted Vector Combination"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0019","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c480466bb4b253787c58edfa8ba329ca523556aaa3ac790d5937c78ee8a35ac6","family_id":"qf-density16-vector-weighted-combination","seed":202610161019,"semantic_template":"qf-density16-vector-weighted-combination/{\"a\":2,\"b\":-1,\"u\":[6,1],\"v\":[1,16]}"},"fields":{"title":"Ordered Weighted Vector Combination","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[6, 1], v=[1, 16], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.","answer":{"kind":"expression","value":"([12, 2], [-1, -16], [11, -14])","explanation":"Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([12, 2], [-1, -16], [11, -14])."},"skills":["Ordered Weighted Vector Combination"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0020","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"803ea7feaf5364578d2c0c6ecd78a57943e9dd0bb9b80fec5e4b1ca3d70de19e","family_id":"qf-density16-vector-weighted-combination","seed":202610161020,"semantic_template":"qf-density16-vector-weighted-combination/{\"a\":2,\"b\":-1,\"u\":[7,1],\"v\":[1,17]}"},"fields":{"title":"Ordered Weighted Vector Combination","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[7, 1], v=[1, 17], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.","answer":{"kind":"expression","value":"([14, 2], [-1, -17], [13, -15])","explanation":"Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([14, 2], [-1, -17], [13, -15])."},"skills":["Ordered Weighted Vector Combination"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0021","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6dc0255369f19eacf34f5ad4632536467c1aa928d226d9fd80dfc13f5820c4d6","family_id":"qf-density16-vector-weighted-combination","seed":202610161021,"semantic_template":"qf-density16-vector-weighted-combination/{\"a\":2,\"b\":-1,\"u\":[8,1],\"v\":[1,18]}"},"fields":{"title":"Ordered Weighted Vector Combination","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[8, 1], v=[1, 18], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.","answer":{"kind":"expression","value":"([16, 2], [-1, -18], [15, -16])","explanation":"Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([16, 2], [-1, -18], [15, -16])."},"skills":["Ordered Weighted Vector Combination"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0022","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a623e185eecf3ca46e42e90e44ff70fc7d8638735b5bbc2d5eaf6585d6bff6a1","family_id":"qf-density16-vector-weighted-combination","seed":202610161022,"semantic_template":"qf-density16-vector-weighted-combination/{\"a\":2,\"b\":-1,\"u\":[9,1],\"v\":[1,19]}"},"fields":{"title":"Ordered Weighted Vector Combination","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[9, 1], v=[1, 19], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.","answer":{"kind":"expression","value":"([18, 2], [-1, -19], [17, -17])","explanation":"Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([18, 2], [-1, -19], [17, -17])."},"skills":["Ordered Weighted Vector Combination"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0023","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9844d415ea473946252df5b159bd7c5054449715cb9727bbc5117ec0e010a308","family_id":"qf-density16-vector-weighted-combination","seed":202610161023,"semantic_template":"qf-density16-vector-weighted-combination/{\"a\":2,\"b\":-1,\"u\":[10,1],\"v\":[1,20]}"},"fields":{"title":"Ordered Weighted Vector Combination","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[10, 1], v=[1, 20], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.","answer":{"kind":"expression","value":"([20, 2], [-1, -20], [19, -18])","explanation":"Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([20, 2], [-1, -20], [19, -18])."},"skills":["Ordered Weighted Vector Combination"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0024","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a91b49672af7e7c330124968715c81596989d82d3c11862035737a821e1c494","family_id":"qf-density16-vector-weighted-combination","seed":202610161024,"semantic_template":"qf-density16-vector-weighted-combination/{\"a\":2,\"b\":-1,\"u\":[11,1],\"v\":[1,21]}"},"fields":{"title":"Ordered Weighted Vector Combination","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[11, 1], v=[1, 21], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.","answer":{"kind":"expression","value":"([22, 2], [-1, -21], [21, -19])","explanation":"Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([22, 2], [-1, -21], [21, -19])."},"skills":["Ordered Weighted Vector Combination"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0025","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eada5a88c6c2b6c89200fa15d4009f474331e64e9f9e4271600bd47bc4ebaf96","family_id":"qf-density16-vector-weighted-combination","seed":202610161025,"semantic_template":"qf-density16-vector-weighted-combination/{\"a\":2,\"b\":-1,\"u\":[12,1],\"v\":[1,22]}"},"fields":{"title":"Ordered Weighted Vector Combination","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[12, 1], v=[1, 22], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order.","answer":{"kind":"expression","value":"([24, 2], [-1, -22], [23, -20])","explanation":"Scale both components separately, then add componentwise. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([24, 2], [-1, -22], [23, -20])."},"skills":["Ordered Weighted Vector Combination"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0026","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0362f941c77ba71038f6e36c95933d52240e2938f65366db44e3af6ef4ce2c26","family_id":"qf-density16-vector-difference-reversal","seed":202610161026,"semantic_template":"qf-density16-vector-difference-reversal/{\"u\":[0,0],\"v\":[0,0]}"},"fields":{"title":"Both Directed Vector Differences and Their Sum","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[0, 0], v=[0, 0]. Give (u, v, u−v, v−u, sum of the two differences).","answer":{"kind":"expression","value":"([0, 0], [0, 0], [0, 0], [0, 0], [0, 0])","explanation":"Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0], [0, 0], [0, 0], [0, 0], [0, 0])."},"skills":["Both Directed Vector Differences and Their Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0027","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b37a9e9a044219054ec82804221300e960dfe049a0bfab8287ef5e214226a26","family_id":"qf-density16-vector-difference-reversal","seed":202610161027,"semantic_template":"qf-density16-vector-difference-reversal/{\"u\":[1,-1],\"v\":[1,-1]}"},"fields":{"title":"Both Directed Vector Differences and Their Sum","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[1, -1], v=[1, -1]. Give (u, v, u−v, v−u, sum of the two differences).","answer":{"kind":"expression","value":"([1, -1], [1, -1], [0, 0], [0, 0], [0, 0])","explanation":"Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, -1], [1, -1], [0, 0], [0, 0], [0, 0])."},"skills":["Both Directed Vector Differences and Their Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0028","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06105e367356e8e37a7cc50904a80e91165a9fa159ed217e8cdfc34406f7b1f3","family_id":"qf-density16-vector-difference-reversal","seed":202610161028,"semantic_template":"qf-density16-vector-difference-reversal/{\"u\":[-10,1],\"v\":[0,-1]}"},"fields":{"title":"Both Directed Vector Differences and Their Sum","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-10, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).","answer":{"kind":"expression","value":"([-10, 1], [0, -1], [-10, 2], [10, -2], [0, 0])","explanation":"Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-10, 1], [0, -1], [-10, 2], [10, -2], [0, 0])."},"skills":["Both Directed Vector Differences and Their Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0029","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"365b01c2ef0e976558c3809aa6a4b0bb02431f7684cbfba5fe51dbde2b772807","family_id":"qf-density16-vector-difference-reversal","seed":202610161029,"semantic_template":"qf-density16-vector-difference-reversal/{\"u\":[-9,1],\"v\":[0,-1]}"},"fields":{"title":"Both Directed Vector Differences and Their Sum","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-9, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).","answer":{"kind":"expression","value":"([-9, 1], [0, -1], [-9, 2], [9, -2], [0, 0])","explanation":"Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-9, 1], [0, -1], [-9, 2], [9, -2], [0, 0])."},"skills":["Both Directed Vector Differences and Their Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0030","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c7ba02337ebe6eb6d5ef1e39e69a9cd3f4a86258883bcf99882796b60b04bcb","family_id":"qf-density16-vector-difference-reversal","seed":202610161030,"semantic_template":"qf-density16-vector-difference-reversal/{\"u\":[-8,1],\"v\":[0,-1]}"},"fields":{"title":"Both Directed Vector Differences and Their Sum","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-8, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).","answer":{"kind":"expression","value":"([-8, 1], [0, -1], [-8, 2], [8, -2], [0, 0])","explanation":"Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-8, 1], [0, -1], [-8, 2], [8, -2], [0, 0])."},"skills":["Both Directed Vector Differences and Their Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0031","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ea4cf435a77c8c60f877fda3af77b92a7f4489be2ea5011ecd349a186894c5e","family_id":"qf-density16-vector-difference-reversal","seed":202610161031,"semantic_template":"qf-density16-vector-difference-reversal/{\"u\":[-7,1],\"v\":[0,-1]}"},"fields":{"title":"Both Directed Vector Differences and Their Sum","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-7, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).","answer":{"kind":"expression","value":"([-7, 1], [0, -1], [-7, 2], [7, -2], [0, 0])","explanation":"Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-7, 1], [0, -1], [-7, 2], [7, -2], [0, 0])."},"skills":["Both Directed Vector Differences and Their Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0032","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c3c7ea196f9078c2875878d35fc07ce0a610974658b3b6a72b3485621bbf334","family_id":"qf-density16-vector-difference-reversal","seed":202610161032,"semantic_template":"qf-density16-vector-difference-reversal/{\"u\":[-6,1],\"v\":[0,-1]}"},"fields":{"title":"Both Directed Vector Differences and Their Sum","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-6, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).","answer":{"kind":"expression","value":"([-6, 1], [0, -1], [-6, 2], [6, -2], [0, 0])","explanation":"Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-6, 1], [0, -1], [-6, 2], [6, -2], [0, 0])."},"skills":["Both Directed Vector Differences and Their Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0033","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea88934c6b9e3b009d2e7424d628a9a68c23b9aa52a51da851d30ab11b7a3c32","family_id":"qf-density16-vector-difference-reversal","seed":202610161033,"semantic_template":"qf-density16-vector-difference-reversal/{\"u\":[-5,1],\"v\":[0,-1]}"},"fields":{"title":"Both Directed Vector Differences and Their Sum","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-5, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).","answer":{"kind":"expression","value":"([-5, 1], [0, -1], [-5, 2], [5, -2], [0, 0])","explanation":"Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-5, 1], [0, -1], [-5, 2], [5, -2], [0, 0])."},"skills":["Both Directed Vector Differences and Their Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0034","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e32b73dc887e46ee48f478ab834e03a21bf7f53fea059c52827620d9a4727932","family_id":"qf-density16-vector-difference-reversal","seed":202610161034,"semantic_template":"qf-density16-vector-difference-reversal/{\"u\":[-4,1],\"v\":[0,-1]}"},"fields":{"title":"Both Directed Vector Differences and Their Sum","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-4, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).","answer":{"kind":"expression","value":"([-4, 1], [0, -1], [-4, 2], [4, -2], [0, 0])","explanation":"Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-4, 1], [0, -1], [-4, 2], [4, -2], [0, 0])."},"skills":["Both Directed Vector Differences and Their Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0035","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7a4241b9001c49a42271cc82cd3caea8451faad1322c01b27f96f7eb86b1ed3","family_id":"qf-density16-vector-difference-reversal","seed":202610161035,"semantic_template":"qf-density16-vector-difference-reversal/{\"u\":[-3,1],\"v\":[0,-1]}"},"fields":{"title":"Both Directed Vector Differences and Their Sum","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-3, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).","answer":{"kind":"expression","value":"([-3, 1], [0, -1], [-3, 2], [3, -2], [0, 0])","explanation":"Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-3, 1], [0, -1], [-3, 2], [3, -2], [0, 0])."},"skills":["Both Directed Vector Differences and Their Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0036","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"498496c18deefeecc6b66592a1f41f298e69a9381134182eff51cd33277d072d","family_id":"qf-density16-vector-difference-reversal","seed":202610161036,"semantic_template":"qf-density16-vector-difference-reversal/{\"u\":[-2,1],\"v\":[0,-1]}"},"fields":{"title":"Both Directed Vector Differences and Their Sum","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-2, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).","answer":{"kind":"expression","value":"([-2, 1], [0, -1], [-2, 2], [2, -2], [0, 0])","explanation":"Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-2, 1], [0, -1], [-2, 2], [2, -2], [0, 0])."},"skills":["Both Directed Vector Differences and Their Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0037","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d78dd18d11cb9111bac397e9ec82f97e9d6e6914718f6d6d08076536baa5fc63","family_id":"qf-density16-vector-difference-reversal","seed":202610161037,"semantic_template":"qf-density16-vector-difference-reversal/{\"u\":[-1,1],\"v\":[0,-1]}"},"fields":{"title":"Both Directed Vector Differences and Their Sum","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-1, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).","answer":{"kind":"expression","value":"([-1, 1], [0, -1], [-1, 2], [1, -2], [0, 0])","explanation":"Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1, 1], [0, -1], [-1, 2], [1, -2], [0, 0])."},"skills":["Both Directed Vector Differences and Their Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0038","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2240651185b80770b516cbd96188e716f4f498b23178749af56736d91833cc8","family_id":"qf-density16-vector-difference-reversal","seed":202610161038,"semantic_template":"qf-density16-vector-difference-reversal/{\"u\":[0,1],\"v\":[0,-1]}"},"fields":{"title":"Both Directed Vector Differences and Their Sum","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[0, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).","answer":{"kind":"expression","value":"([0, 1], [0, -1], [0, 2], [0, -2], [0, 0])","explanation":"Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1], [0, -1], [0, 2], [0, -2], [0, 0])."},"skills":["Both Directed Vector Differences and Their Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0039","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"560b3dac99b06d309ffb32a704c8cc5b4f405aae590a4597595cd55fe663832c","family_id":"qf-density16-vector-difference-reversal","seed":202610161039,"semantic_template":"qf-density16-vector-difference-reversal/{\"u\":[1,1],\"v\":[0,-1]}"},"fields":{"title":"Both Directed Vector Differences and Their Sum","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[1, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).","answer":{"kind":"expression","value":"([1, 1], [0, -1], [1, 2], [-1, -2], [0, 0])","explanation":"Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1], [0, -1], [1, 2], [-1, -2], [0, 0])."},"skills":["Both Directed Vector Differences and Their Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0040","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef69252438849a9658243bfaeb00cfef8f944a5460687b1d30ff76b3f55945e6","family_id":"qf-density16-vector-difference-reversal","seed":202610161040,"semantic_template":"qf-density16-vector-difference-reversal/{\"u\":[2,1],\"v\":[0,-1]}"},"fields":{"title":"Both Directed Vector Differences and Their Sum","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[2, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).","answer":{"kind":"expression","value":"([2, 1], [0, -1], [2, 2], [-2, -2], [0, 0])","explanation":"Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1], [0, -1], [2, 2], [-2, -2], [0, 0])."},"skills":["Both Directed Vector Differences and Their Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0041","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b157e20b323ddfed976920db88b5e172467f248d525585e39ef2c754e6e49715","family_id":"qf-density16-vector-difference-reversal","seed":202610161041,"semantic_template":"qf-density16-vector-difference-reversal/{\"u\":[3,1],\"v\":[0,-1]}"},"fields":{"title":"Both Directed Vector Differences and Their Sum","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[3, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).","answer":{"kind":"expression","value":"([3, 1], [0, -1], [3, 2], [-3, -2], [0, 0])","explanation":"Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 1], [0, -1], [3, 2], [-3, -2], [0, 0])."},"skills":["Both Directed Vector Differences and Their Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0042","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f5e70db0a24b0c52d1091003576fd1160ae64fe77a1c0bbc7d6fc9ad719bf2d","family_id":"qf-density16-vector-difference-reversal","seed":202610161042,"semantic_template":"qf-density16-vector-difference-reversal/{\"u\":[4,1],\"v\":[0,-1]}"},"fields":{"title":"Both Directed Vector Differences and Their Sum","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[4, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).","answer":{"kind":"expression","value":"([4, 1], [0, -1], [4, 2], [-4, -2], [0, 0])","explanation":"Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 1], [0, -1], [4, 2], [-4, -2], [0, 0])."},"skills":["Both Directed Vector Differences and Their Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0043","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2885db11a7fa464fd827d6d5e02de135847b8fcea93d37e645786024c92b1f71","family_id":"qf-density16-vector-difference-reversal","seed":202610161043,"semantic_template":"qf-density16-vector-difference-reversal/{\"u\":[5,1],\"v\":[0,-1]}"},"fields":{"title":"Both Directed Vector Differences and Their Sum","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[5, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).","answer":{"kind":"expression","value":"([5, 1], [0, -1], [5, 2], [-5, -2], [0, 0])","explanation":"Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 1], [0, -1], [5, 2], [-5, -2], [0, 0])."},"skills":["Both Directed Vector Differences and Their Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0044","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1b8bf10b175119301ceb022295da8698a4344c0066e7023b4757d3f3adc241b","family_id":"qf-density16-vector-difference-reversal","seed":202610161044,"semantic_template":"qf-density16-vector-difference-reversal/{\"u\":[6,1],\"v\":[0,-1]}"},"fields":{"title":"Both Directed Vector Differences and Their Sum","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[6, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).","answer":{"kind":"expression","value":"([6, 1], [0, -1], [6, 2], [-6, -2], [0, 0])","explanation":"Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([6, 1], [0, -1], [6, 2], [-6, -2], [0, 0])."},"skills":["Both Directed Vector Differences and Their Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0045","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"784922a537db9d46d44ed893c325ac2ac828d5d310f6f4d73f0ecd8c6f083ed6","family_id":"qf-density16-vector-difference-reversal","seed":202610161045,"semantic_template":"qf-density16-vector-difference-reversal/{\"u\":[7,1],\"v\":[0,-1]}"},"fields":{"title":"Both Directed Vector Differences and Their Sum","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[7, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).","answer":{"kind":"expression","value":"([7, 1], [0, -1], [7, 2], [-7, -2], [0, 0])","explanation":"Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([7, 1], [0, -1], [7, 2], [-7, -2], [0, 0])."},"skills":["Both Directed Vector Differences and Their Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0046","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8268a51363497e0f59aa4a64400c961932be3df2414ba62cf832871f862fda33","family_id":"qf-density16-vector-difference-reversal","seed":202610161046,"semantic_template":"qf-density16-vector-difference-reversal/{\"u\":[8,1],\"v\":[0,-1]}"},"fields":{"title":"Both Directed Vector Differences and Their Sum","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[8, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).","answer":{"kind":"expression","value":"([8, 1], [0, -1], [8, 2], [-8, -2], [0, 0])","explanation":"Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([8, 1], [0, -1], [8, 2], [-8, -2], [0, 0])."},"skills":["Both Directed Vector Differences and Their Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0047","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b13abcf2443f76de3700cf8d1b56d7d569f7fb7bdc19ca87c18021f429ffb5a","family_id":"qf-density16-vector-difference-reversal","seed":202610161047,"semantic_template":"qf-density16-vector-difference-reversal/{\"u\":[9,1],\"v\":[0,-1]}"},"fields":{"title":"Both Directed Vector Differences and Their Sum","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[9, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).","answer":{"kind":"expression","value":"([9, 1], [0, -1], [9, 2], [-9, -2], [0, 0])","explanation":"Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([9, 1], [0, -1], [9, 2], [-9, -2], [0, 0])."},"skills":["Both Directed Vector Differences and Their Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0048","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4137bb7aac69d97f8d4e1259426c47dc8ec74b2e539c0b03ff03f8e7a6481a0c","family_id":"qf-density16-vector-difference-reversal","seed":202610161048,"semantic_template":"qf-density16-vector-difference-reversal/{\"u\":[10,1],\"v\":[0,-1]}"},"fields":{"title":"Both Directed Vector Differences and Their Sum","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[10, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).","answer":{"kind":"expression","value":"([10, 1], [0, -1], [10, 2], [-10, -2], [0, 0])","explanation":"Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([10, 1], [0, -1], [10, 2], [-10, -2], [0, 0])."},"skills":["Both Directed Vector Differences and Their Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0049","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62657ef30087985827ce73160209d5671b941f139d02170437810c047859799d","family_id":"qf-density16-vector-difference-reversal","seed":202610161049,"semantic_template":"qf-density16-vector-difference-reversal/{\"u\":[11,1],\"v\":[0,-1]}"},"fields":{"title":"Both Directed Vector Differences and Their Sum","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[11, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).","answer":{"kind":"expression","value":"([11, 1], [0, -1], [11, 2], [-11, -2], [0, 0])","explanation":"Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([11, 1], [0, -1], [11, 2], [-11, -2], [0, 0])."},"skills":["Both Directed Vector Differences and Their Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0050","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b041bb7ddc0a37a494370381247476774f9a41df1c55ebb50b374d9c0d2ae97c","family_id":"qf-density16-vector-difference-reversal","seed":202610161050,"semantic_template":"qf-density16-vector-difference-reversal/{\"u\":[12,1],\"v\":[0,-1]}"},"fields":{"title":"Both Directed Vector Differences and Their Sum","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[12, 1], v=[0, -1]. Give (u, v, u−v, v−u, sum of the two differences).","answer":{"kind":"expression","value":"([12, 1], [0, -1], [12, 2], [-12, -2], [0, 0])","explanation":"Subtract in each specified direction independently, preserving component order before adding. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([12, 1], [0, -1], [12, 2], [-12, -2], [0, 0])."},"skills":["Both Directed Vector Differences and Their Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0051","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"25f292fa8eb3f1c542e222c13447af09da378f1021775cb730752acae576a608","family_id":"qf-density16-vector-two-scalar-additivity","seed":202610161051,"semantic_template":"qf-density16-vector-two-scalar-additivity/{\"a\":0,\"b\":0,\"u\":[0,0]}"},"fields":{"title":"Two Scalar-Addition Routes and Component Defect","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[0, 0], a=0, b=0. Give (u, (a+b)u, au+bu, difference of the last two routes).","answer":{"kind":"expression","value":"([0, 0], [0, 0], [0, 0], [0, 0])","explanation":"One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0], [0, 0], [0, 0], [0, 0])."},"skills":["Two Scalar-Addition Routes and Component Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0052","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aaee8972acb232d99328c08bc6040da8e6da2b1acb0ea9e59e76e3566e368e08","family_id":"qf-density16-vector-two-scalar-additivity","seed":202610161052,"semantic_template":"qf-density16-vector-two-scalar-additivity/{\"a\":2,\"b\":-2,\"u\":[1,2]}"},"fields":{"title":"Two Scalar-Addition Routes and Component Defect","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[1, 2], a=2, b=-2. Give (u, (a+b)u, au+bu, difference of the last two routes).","answer":{"kind":"expression","value":"([1, 2], [0, 0], [0, 0], [0, 0])","explanation":"One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2], [0, 0], [0, 0], [0, 0])."},"skills":["Two Scalar-Addition Routes and Component Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0053","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a66c8feb1b0a12d46fecf0a98764e74914eabd6d5fa96c35889319e48fa8257","family_id":"qf-density16-vector-two-scalar-additivity","seed":202610161053,"semantic_template":"qf-density16-vector-two-scalar-additivity/{\"a\":-1,\"b\":2,\"u\":[1,-1]}"},"fields":{"title":"Two Scalar-Addition Routes and Component Defect","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[1, -1], a=-1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).","answer":{"kind":"expression","value":"([1, -1], [1, -1], [1, -1], [0, 0])","explanation":"One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, -1], [1, -1], [1, -1], [0, 0])."},"skills":["Two Scalar-Addition Routes and Component Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0054","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cda36dc3d2f9975f242402fd0a08491b26ac0b9332d08ff0ee5c5a7759b37b1f","family_id":"qf-density16-vector-two-scalar-additivity","seed":202610161054,"semantic_template":"qf-density16-vector-two-scalar-additivity/{\"a\":1,\"b\":2,\"u\":[-9,2]}"},"fields":{"title":"Two Scalar-Addition Routes and Component Defect","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-9, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).","answer":{"kind":"expression","value":"([-9, 2], [-27, 6], [-27, 6], [0, 0])","explanation":"One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-9, 2], [-27, 6], [-27, 6], [0, 0])."},"skills":["Two Scalar-Addition Routes and Component Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0055","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9188a1e5adab4652b8edc183028c3ad88a6d3a65ef5549ed1986f90a7ee304b7","family_id":"qf-density16-vector-two-scalar-additivity","seed":202610161055,"semantic_template":"qf-density16-vector-two-scalar-additivity/{\"a\":1,\"b\":2,\"u\":[-8,2]}"},"fields":{"title":"Two Scalar-Addition Routes and Component Defect","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-8, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).","answer":{"kind":"expression","value":"([-8, 2], [-24, 6], [-24, 6], [0, 0])","explanation":"One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-8, 2], [-24, 6], [-24, 6], [0, 0])."},"skills":["Two Scalar-Addition Routes and Component Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0056","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d694125aef75412728627cbdfad9fa06025c74e7d66e94b32f6a2ce40a6a0782","family_id":"qf-density16-vector-two-scalar-additivity","seed":202610161056,"semantic_template":"qf-density16-vector-two-scalar-additivity/{\"a\":1,\"b\":2,\"u\":[-7,2]}"},"fields":{"title":"Two Scalar-Addition Routes and Component Defect","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-7, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).","answer":{"kind":"expression","value":"([-7, 2], [-21, 6], [-21, 6], [0, 0])","explanation":"One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-7, 2], [-21, 6], [-21, 6], [0, 0])."},"skills":["Two Scalar-Addition Routes and Component Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0057","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0a84df6ef4e756f226a021e6d30b8b9f2bce7002223276a932f87d782bf0872","family_id":"qf-density16-vector-two-scalar-additivity","seed":202610161057,"semantic_template":"qf-density16-vector-two-scalar-additivity/{\"a\":1,\"b\":2,\"u\":[-6,2]}"},"fields":{"title":"Two Scalar-Addition Routes and Component Defect","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-6, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).","answer":{"kind":"expression","value":"([-6, 2], [-18, 6], [-18, 6], [0, 0])","explanation":"One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-6, 2], [-18, 6], [-18, 6], [0, 0])."},"skills":["Two Scalar-Addition Routes and Component Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0058","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b7bebe4178eea6bbdfae89a41f24f92e3e5b06c2e08ccd7f2df8b09f08223ea","family_id":"qf-density16-vector-two-scalar-additivity","seed":202610161058,"semantic_template":"qf-density16-vector-two-scalar-additivity/{\"a\":1,\"b\":2,\"u\":[-5,2]}"},"fields":{"title":"Two Scalar-Addition Routes and Component Defect","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-5, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).","answer":{"kind":"expression","value":"([-5, 2], [-15, 6], [-15, 6], [0, 0])","explanation":"One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-5, 2], [-15, 6], [-15, 6], [0, 0])."},"skills":["Two Scalar-Addition Routes and Component Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0059","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"661e1c2ba6da5345a73570609ad2d02265acae2f7a711e77e84bd63c38c04d49","family_id":"qf-density16-vector-two-scalar-additivity","seed":202610161059,"semantic_template":"qf-density16-vector-two-scalar-additivity/{\"a\":1,\"b\":2,\"u\":[-4,2]}"},"fields":{"title":"Two Scalar-Addition Routes and Component Defect","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-4, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).","answer":{"kind":"expression","value":"([-4, 2], [-12, 6], [-12, 6], [0, 0])","explanation":"One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-4, 2], [-12, 6], [-12, 6], [0, 0])."},"skills":["Two Scalar-Addition Routes and Component Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0060","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0571429f35c7b472ad9a6a84746280095aaee88b4113b9414b9920da1521632","family_id":"qf-density16-vector-two-scalar-additivity","seed":202610161060,"semantic_template":"qf-density16-vector-two-scalar-additivity/{\"a\":1,\"b\":2,\"u\":[-3,2]}"},"fields":{"title":"Two Scalar-Addition Routes and Component Defect","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-3, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).","answer":{"kind":"expression","value":"([-3, 2], [-9, 6], [-9, 6], [0, 0])","explanation":"One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-3, 2], [-9, 6], [-9, 6], [0, 0])."},"skills":["Two Scalar-Addition Routes and Component Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0061","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27e35e572dd3e51308effb641d184771486d78ed852d35a7c07f163a630cb8f1","family_id":"qf-density16-vector-two-scalar-additivity","seed":202610161061,"semantic_template":"qf-density16-vector-two-scalar-additivity/{\"a\":1,\"b\":2,\"u\":[-2,2]}"},"fields":{"title":"Two Scalar-Addition Routes and Component Defect","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-2, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).","answer":{"kind":"expression","value":"([-2, 2], [-6, 6], [-6, 6], [0, 0])","explanation":"One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-2, 2], [-6, 6], [-6, 6], [0, 0])."},"skills":["Two Scalar-Addition Routes and Component Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0062","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f505f7e47f7bf3e77409c89cd6a023bb6e72e23c41056f7fe6ccb392012791dc","family_id":"qf-density16-vector-two-scalar-additivity","seed":202610161062,"semantic_template":"qf-density16-vector-two-scalar-additivity/{\"a\":1,\"b\":2,\"u\":[-1,2]}"},"fields":{"title":"Two Scalar-Addition Routes and Component Defect","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-1, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).","answer":{"kind":"expression","value":"([-1, 2], [-3, 6], [-3, 6], [0, 0])","explanation":"One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1, 2], [-3, 6], [-3, 6], [0, 0])."},"skills":["Two Scalar-Addition Routes and Component Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0063","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8c22fd86ee3129d93921d60f79a395f2b8a03804dca6775a856128746ab33cf","family_id":"qf-density16-vector-two-scalar-additivity","seed":202610161063,"semantic_template":"qf-density16-vector-two-scalar-additivity/{\"a\":1,\"b\":2,\"u\":[0,2]}"},"fields":{"title":"Two Scalar-Addition Routes and Component Defect","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[0, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).","answer":{"kind":"expression","value":"([0, 2], [0, 6], [0, 6], [0, 0])","explanation":"One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2], [0, 6], [0, 6], [0, 0])."},"skills":["Two Scalar-Addition Routes and Component Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0064","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aefea379b239fe7e04d15a4ac4be6b3f31535ccf8e05e92b343df16c8212d2e1","family_id":"qf-density16-vector-two-scalar-additivity","seed":202610161064,"semantic_template":"qf-density16-vector-two-scalar-additivity/{\"a\":1,\"b\":2,\"u\":[1,2]}"},"fields":{"title":"Two Scalar-Addition Routes and Component Defect","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[1, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).","answer":{"kind":"expression","value":"([1, 2], [3, 6], [3, 6], [0, 0])","explanation":"One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2], [3, 6], [3, 6], [0, 0])."},"skills":["Two Scalar-Addition Routes and Component Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0065","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b283c139d2e36bdf7f618c3706aa444950182f682ea96853bebc0e81847c4aae","family_id":"qf-density16-vector-two-scalar-additivity","seed":202610161065,"semantic_template":"qf-density16-vector-two-scalar-additivity/{\"a\":1,\"b\":2,\"u\":[2,2]}"},"fields":{"title":"Two Scalar-Addition Routes and Component Defect","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[2, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).","answer":{"kind":"expression","value":"([2, 2], [6, 6], [6, 6], [0, 0])","explanation":"One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2], [6, 6], [6, 6], [0, 0])."},"skills":["Two Scalar-Addition Routes and Component Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0066","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0efa95f67516b8fd78f26a82077f0e95ba246811eef6f7497685f957b6e0079c","family_id":"qf-density16-vector-two-scalar-additivity","seed":202610161066,"semantic_template":"qf-density16-vector-two-scalar-additivity/{\"a\":1,\"b\":2,\"u\":[3,2]}"},"fields":{"title":"Two Scalar-Addition Routes and Component Defect","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[3, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).","answer":{"kind":"expression","value":"([3, 2], [9, 6], [9, 6], [0, 0])","explanation":"One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 2], [9, 6], [9, 6], [0, 0])."},"skills":["Two Scalar-Addition Routes and Component Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0067","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f84fe645f5ca3d72d92c831de083b871bf191cc4c3b4f1051cf7cfca31f5b2dd","family_id":"qf-density16-vector-two-scalar-additivity","seed":202610161067,"semantic_template":"qf-density16-vector-two-scalar-additivity/{\"a\":1,\"b\":2,\"u\":[4,2]}"},"fields":{"title":"Two Scalar-Addition Routes and Component Defect","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[4, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).","answer":{"kind":"expression","value":"([4, 2], [12, 6], [12, 6], [0, 0])","explanation":"One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4, 2], [12, 6], [12, 6], [0, 0])."},"skills":["Two Scalar-Addition Routes and Component Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0068","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8fbecae6f81bb16db7be5f2929416cacd7504f390a3eba11b9b72fdd6be0282d","family_id":"qf-density16-vector-two-scalar-additivity","seed":202610161068,"semantic_template":"qf-density16-vector-two-scalar-additivity/{\"a\":1,\"b\":2,\"u\":[5,2]}"},"fields":{"title":"Two Scalar-Addition Routes and Component Defect","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[5, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).","answer":{"kind":"expression","value":"([5, 2], [15, 6], [15, 6], [0, 0])","explanation":"One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 2], [15, 6], [15, 6], [0, 0])."},"skills":["Two Scalar-Addition Routes and Component Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0069","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"230f9a59433dd78509692c9fddc4e2b5297bf792b3b4f92c81e705c3013f2a17","family_id":"qf-density16-vector-two-scalar-additivity","seed":202610161069,"semantic_template":"qf-density16-vector-two-scalar-additivity/{\"a\":1,\"b\":2,\"u\":[6,2]}"},"fields":{"title":"Two Scalar-Addition Routes and Component Defect","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[6, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).","answer":{"kind":"expression","value":"([6, 2], [18, 6], [18, 6], [0, 0])","explanation":"One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([6, 2], [18, 6], [18, 6], [0, 0])."},"skills":["Two Scalar-Addition Routes and Component Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0070","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81a35874e3ff3a579c249ad446a4065094decd266d6244fbbf7dcc77e440be09","family_id":"qf-density16-vector-two-scalar-additivity","seed":202610161070,"semantic_template":"qf-density16-vector-two-scalar-additivity/{\"a\":1,\"b\":2,\"u\":[7,2]}"},"fields":{"title":"Two Scalar-Addition Routes and Component Defect","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[7, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).","answer":{"kind":"expression","value":"([7, 2], [21, 6], [21, 6], [0, 0])","explanation":"One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([7, 2], [21, 6], [21, 6], [0, 0])."},"skills":["Two Scalar-Addition Routes and Component Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0071","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ebafd3796b2b670517d6f2e6c93ea41e4ae7206b9785606571a00c0f91209f7","family_id":"qf-density16-vector-two-scalar-additivity","seed":202610161071,"semantic_template":"qf-density16-vector-two-scalar-additivity/{\"a\":1,\"b\":2,\"u\":[8,2]}"},"fields":{"title":"Two Scalar-Addition Routes and Component Defect","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[8, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).","answer":{"kind":"expression","value":"([8, 2], [24, 6], [24, 6], [0, 0])","explanation":"One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([8, 2], [24, 6], [24, 6], [0, 0])."},"skills":["Two Scalar-Addition Routes and Component Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0072","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a460c74a94fecf8d445b154e88fcfdafc9b4a9eeee5298ce2cb91ab409de33a","family_id":"qf-density16-vector-two-scalar-additivity","seed":202610161072,"semantic_template":"qf-density16-vector-two-scalar-additivity/{\"a\":1,\"b\":2,\"u\":[9,2]}"},"fields":{"title":"Two Scalar-Addition Routes and Component Defect","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[9, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).","answer":{"kind":"expression","value":"([9, 2], [27, 6], [27, 6], [0, 0])","explanation":"One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([9, 2], [27, 6], [27, 6], [0, 0])."},"skills":["Two Scalar-Addition Routes and Component Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0073","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d35631b7f2767437182afae2b4db2d46065bf7c1cdbe58322bb246577dabdb0c","family_id":"qf-density16-vector-two-scalar-additivity","seed":202610161073,"semantic_template":"qf-density16-vector-two-scalar-additivity/{\"a\":1,\"b\":2,\"u\":[10,2]}"},"fields":{"title":"Two Scalar-Addition Routes and Component Defect","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[10, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).","answer":{"kind":"expression","value":"([10, 2], [30, 6], [30, 6], [0, 0])","explanation":"One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([10, 2], [30, 6], [30, 6], [0, 0])."},"skills":["Two Scalar-Addition Routes and Component Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0074","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"350f94dbcc60fe6e8962dabbc22ce59c57dae8474b0f7207f45a9e8a4a3fb69e","family_id":"qf-density16-vector-two-scalar-additivity","seed":202610161074,"semantic_template":"qf-density16-vector-two-scalar-additivity/{\"a\":1,\"b\":2,\"u\":[11,2]}"},"fields":{"title":"Two Scalar-Addition Routes and Component Defect","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[11, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).","answer":{"kind":"expression","value":"([11, 2], [33, 6], [33, 6], [0, 0])","explanation":"One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([11, 2], [33, 6], [33, 6], [0, 0])."},"skills":["Two Scalar-Addition Routes and Component Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0075","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f9ce2c1fc09e2f5cacda71386b7d8d892fb0ccce2d2c5122305a40d88fb0ca4","family_id":"qf-density16-vector-two-scalar-additivity","seed":202610161075,"semantic_template":"qf-density16-vector-two-scalar-additivity/{\"a\":1,\"b\":2,\"u\":[12,2]}"},"fields":{"title":"Two Scalar-Addition Routes and Component Defect","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[12, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes).","answer":{"kind":"expression","value":"([12, 2], [36, 6], [36, 6], [0, 0])","explanation":"One route adds scalars before scaling; the other scales before adding. Compare the actual vectors, not norms. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([12, 2], [36, 6], [36, 6], [0, 0])."},"skills":["Two Scalar-Addition Routes and Component Defect"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0076","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06c0dd2816c3ee701adade29ba6a0d8aaea258c52670b7af6f8738ff9bfaa4d0","family_id":"qf-density16-vector-candidate-residual","seed":202610161076,"semantic_template":"qf-density16-vector-candidate-residual/{\"a\":0,\"b\":0,\"candidate\":[0,0],\"u\":[0,0],\"v\":[0,0]}"},"fields":{"title":"Weighted Vector Candidate and Ordered Residual","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[0, 0], v=[0, 0], a=0, b=0, candidate=[0, 0]. Give (au+bv, candidate−actual value, is the candidate equal?).","answer":{"kind":"expression","value":"([0, 0], [0, 0], Yes)","explanation":"Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0], [0, 0], Yes)."},"skills":["Weighted Vector Candidate and Ordered Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0077","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55fd937d50cab65bf71fae94523f34ced0e1556a3319f6027ce213c4743c5594","family_id":"qf-density16-vector-candidate-residual","seed":202610161077,"semantic_template":"qf-density16-vector-candidate-residual/{\"a\":1,\"b\":1,\"candidate\":[1,0],\"u\":[1,0],\"v\":[0,1]}"},"fields":{"title":"Weighted Vector Candidate and Ordered Residual","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[1, 0], v=[0, 1], a=1, b=1, candidate=[1, 0]. Give (au+bv, candidate−actual value, is the candidate equal?).","answer":{"kind":"expression","value":"([1, 1], [0, -1], No)","explanation":"Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1], [0, -1], No)."},"skills":["Weighted Vector Candidate and Ordered Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0078","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4da9738cb984cdc946a588351ab51b26669768363de1e4e12d9b182cca1cfeac","family_id":"qf-density16-vector-candidate-residual","seed":202610161078,"semantic_template":"qf-density16-vector-candidate-residual/{\"a\":2,\"b\":-1,\"candidate\":[1,2],\"u\":[1,2],\"v\":[1,2]}"},"fields":{"title":"Weighted Vector Candidate and Ordered Residual","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[1, 2], v=[1, 2], a=2, b=-1, candidate=[1, 2]. Give (au+bv, candidate−actual value, is the candidate equal?).","answer":{"kind":"expression","value":"([1, 2], [0, 0], Yes)","explanation":"Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2], [0, 0], Yes)."},"skills":["Weighted Vector Candidate and Ordered Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0079","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"caa13d9ec40573ec07f57a8a001484029915809ca961bc6bad2469d5eedf0bc5","family_id":"qf-density16-vector-candidate-residual","seed":202610161079,"semantic_template":"qf-density16-vector-candidate-residual/{\"a\":2,\"b\":1,\"candidate\":[-17,4],\"u\":[-9,1],\"v\":[1,2]}"},"fields":{"title":"Weighted Vector Candidate and Ordered Residual","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-9, 1], v=[1, 2], a=2, b=1, candidate=[-17, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).","answer":{"kind":"expression","value":"([-17, 4], [0, 0], Yes)","explanation":"Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-17, 4], [0, 0], Yes)."},"skills":["Weighted Vector Candidate and Ordered Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0080","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2cb94bdc0958f9e052ea25a2a5fb87d5f184a83c6b4b289b91414f19a8557734","family_id":"qf-density16-vector-candidate-residual","seed":202610161080,"semantic_template":"qf-density16-vector-candidate-residual/{\"a\":2,\"b\":1,\"candidate\":[-14,4],\"u\":[-8,1],\"v\":[1,2]}"},"fields":{"title":"Weighted Vector Candidate and Ordered Residual","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-8, 1], v=[1, 2], a=2, b=1, candidate=[-14, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).","answer":{"kind":"expression","value":"([-15, 4], [1, 0], No)","explanation":"Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-15, 4], [1, 0], No)."},"skills":["Weighted Vector Candidate and Ordered Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0081","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"847e414f9c46c2b27312d9aff70cffd75cdcd3bf5dc359f8549326f30dbce2d2","family_id":"qf-density16-vector-candidate-residual","seed":202610161081,"semantic_template":"qf-density16-vector-candidate-residual/{\"a\":2,\"b\":1,\"candidate\":[-12,4],\"u\":[-7,1],\"v\":[1,2]}"},"fields":{"title":"Weighted Vector Candidate and Ordered Residual","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-7, 1], v=[1, 2], a=2, b=1, candidate=[-12, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).","answer":{"kind":"expression","value":"([-13, 4], [1, 0], No)","explanation":"Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-13, 4], [1, 0], No)."},"skills":["Weighted Vector Candidate and Ordered Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0082","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b46a19a4d9fd19085af781469108d0a1850ff68205a02a4cdb4f23aad5c2d2e","family_id":"qf-density16-vector-candidate-residual","seed":202610161082,"semantic_template":"qf-density16-vector-candidate-residual/{\"a\":2,\"b\":1,\"candidate\":[-11,4],\"u\":[-6,1],\"v\":[1,2]}"},"fields":{"title":"Weighted Vector Candidate and Ordered Residual","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-6, 1], v=[1, 2], a=2, b=1, candidate=[-11, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).","answer":{"kind":"expression","value":"([-11, 4], [0, 0], Yes)","explanation":"Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-11, 4], [0, 0], Yes)."},"skills":["Weighted Vector Candidate and Ordered Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0083","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e0474c356cacdaca4b9b3341530a1a3dfbc84107ac089f2625ef0b3f18a2071","family_id":"qf-density16-vector-candidate-residual","seed":202610161083,"semantic_template":"qf-density16-vector-candidate-residual/{\"a\":2,\"b\":1,\"candidate\":[-8,4],\"u\":[-5,1],\"v\":[1,2]}"},"fields":{"title":"Weighted Vector Candidate and Ordered Residual","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-5, 1], v=[1, 2], a=2, b=1, candidate=[-8, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).","answer":{"kind":"expression","value":"([-9, 4], [1, 0], No)","explanation":"Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-9, 4], [1, 0], No)."},"skills":["Weighted Vector Candidate and Ordered Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0084","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81528ddfb597eac7a67b3677df93b5567645490ebe155798b335748d13b32f11","family_id":"qf-density16-vector-candidate-residual","seed":202610161084,"semantic_template":"qf-density16-vector-candidate-residual/{\"a\":2,\"b\":1,\"candidate\":[-6,4],\"u\":[-4,1],\"v\":[1,2]}"},"fields":{"title":"Weighted Vector Candidate and Ordered Residual","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-4, 1], v=[1, 2], a=2, b=1, candidate=[-6, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).","answer":{"kind":"expression","value":"([-7, 4], [1, 0], No)","explanation":"Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-7, 4], [1, 0], No)."},"skills":["Weighted Vector Candidate and Ordered Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0085","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"248c146e0c8e1d3cc92072c27e70363b7516a14e7c1158668c9566d77bce6529","family_id":"qf-density16-vector-candidate-residual","seed":202610161085,"semantic_template":"qf-density16-vector-candidate-residual/{\"a\":2,\"b\":1,\"candidate\":[-5,4],\"u\":[-3,1],\"v\":[1,2]}"},"fields":{"title":"Weighted Vector Candidate and Ordered Residual","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-3, 1], v=[1, 2], a=2, b=1, candidate=[-5, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).","answer":{"kind":"expression","value":"([-5, 4], [0, 0], Yes)","explanation":"Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-5, 4], [0, 0], Yes)."},"skills":["Weighted Vector Candidate and Ordered Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0086","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad40423e6a8f8896d5f283aa8de1f2e8bc6bfc3a9f8fc2e7fa7c730d1a1937fd","family_id":"qf-density16-vector-candidate-residual","seed":202610161086,"semantic_template":"qf-density16-vector-candidate-residual/{\"a\":2,\"b\":1,\"candidate\":[-2,4],\"u\":[-2,1],\"v\":[1,2]}"},"fields":{"title":"Weighted Vector Candidate and Ordered Residual","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-2, 1], v=[1, 2], a=2, b=1, candidate=[-2, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).","answer":{"kind":"expression","value":"([-3, 4], [1, 0], No)","explanation":"Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-3, 4], [1, 0], No)."},"skills":["Weighted Vector Candidate and Ordered Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0087","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2a9a24209556f7c353b26f2f819bf5a29ba48bbec6e4c334365222fe149cd62","family_id":"qf-density16-vector-candidate-residual","seed":202610161087,"semantic_template":"qf-density16-vector-candidate-residual/{\"a\":2,\"b\":1,\"candidate\":[0,4],\"u\":[-1,1],\"v\":[1,2]}"},"fields":{"title":"Weighted Vector Candidate and Ordered Residual","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-1, 1], v=[1, 2], a=2, b=1, candidate=[0, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).","answer":{"kind":"expression","value":"([-1, 4], [1, 0], No)","explanation":"Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1, 4], [1, 0], No)."},"skills":["Weighted Vector Candidate and Ordered Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0088","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14a8fa2c34e74515e58651e69e6d84f657329633e6c80eb6ca7aa50470504f62","family_id":"qf-density16-vector-candidate-residual","seed":202610161088,"semantic_template":"qf-density16-vector-candidate-residual/{\"a\":2,\"b\":1,\"candidate\":[1,4],\"u\":[0,1],\"v\":[1,2]}"},"fields":{"title":"Weighted Vector Candidate and Ordered Residual","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[0, 1], v=[1, 2], a=2, b=1, candidate=[1, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).","answer":{"kind":"expression","value":"([1, 4], [0, 0], Yes)","explanation":"Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 4], [0, 0], Yes)."},"skills":["Weighted Vector Candidate and Ordered Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0089","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"abeffb19624dc518c45dc404d25a4fb9c7f457206ea31c16797ce96de3c710cf","family_id":"qf-density16-vector-candidate-residual","seed":202610161089,"semantic_template":"qf-density16-vector-candidate-residual/{\"a\":2,\"b\":1,\"candidate\":[4,4],\"u\":[1,1],\"v\":[1,2]}"},"fields":{"title":"Weighted Vector Candidate and Ordered Residual","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[1, 1], v=[1, 2], a=2, b=1, candidate=[4, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).","answer":{"kind":"expression","value":"([3, 4], [1, 0], No)","explanation":"Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 4], [1, 0], No)."},"skills":["Weighted Vector Candidate and Ordered Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0090","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d8bfe977bd58e8bd8ec8eb015ac1c42c0b731c9bfd8e70433cbc764a5cb5cf8","family_id":"qf-density16-vector-candidate-residual","seed":202610161090,"semantic_template":"qf-density16-vector-candidate-residual/{\"a\":2,\"b\":1,\"candidate\":[6,4],\"u\":[2,1],\"v\":[1,2]}"},"fields":{"title":"Weighted Vector Candidate and Ordered Residual","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[2, 1], v=[1, 2], a=2, b=1, candidate=[6, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).","answer":{"kind":"expression","value":"([5, 4], [1, 0], No)","explanation":"Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5, 4], [1, 0], No)."},"skills":["Weighted Vector Candidate and Ordered Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0091","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ac518c2e76470eb5df45bad6cebc78be2cfb952350092fd51b8cc9e55a504c6","family_id":"qf-density16-vector-candidate-residual","seed":202610161091,"semantic_template":"qf-density16-vector-candidate-residual/{\"a\":2,\"b\":1,\"candidate\":[7,4],\"u\":[3,1],\"v\":[1,2]}"},"fields":{"title":"Weighted Vector Candidate and Ordered Residual","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[3, 1], v=[1, 2], a=2, b=1, candidate=[7, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).","answer":{"kind":"expression","value":"([7, 4], [0, 0], Yes)","explanation":"Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([7, 4], [0, 0], Yes)."},"skills":["Weighted Vector Candidate and Ordered Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0092","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16a9cee036ad7c98f61932fbb296d32f024134c41323e6ce5fe93d13ccc565eb","family_id":"qf-density16-vector-candidate-residual","seed":202610161092,"semantic_template":"qf-density16-vector-candidate-residual/{\"a\":2,\"b\":1,\"candidate\":[10,4],\"u\":[4,1],\"v\":[1,2]}"},"fields":{"title":"Weighted Vector Candidate and Ordered Residual","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[4, 1], v=[1, 2], a=2, b=1, candidate=[10, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).","answer":{"kind":"expression","value":"([9, 4], [1, 0], No)","explanation":"Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([9, 4], [1, 0], No)."},"skills":["Weighted Vector Candidate and Ordered Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0093","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2636c29c382308b9e18d129d7c70707a7e840d924f6165f8de34978772fdffa6","family_id":"qf-density16-vector-candidate-residual","seed":202610161093,"semantic_template":"qf-density16-vector-candidate-residual/{\"a\":2,\"b\":1,\"candidate\":[12,4],\"u\":[5,1],\"v\":[1,2]}"},"fields":{"title":"Weighted Vector Candidate and Ordered Residual","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[5, 1], v=[1, 2], a=2, b=1, candidate=[12, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).","answer":{"kind":"expression","value":"([11, 4], [1, 0], No)","explanation":"Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([11, 4], [1, 0], No)."},"skills":["Weighted Vector Candidate and Ordered Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0094","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59f4fbc5ff7037f62b138ea893b0549548177b8fc304b33a22332f3389b0e900","family_id":"qf-density16-vector-candidate-residual","seed":202610161094,"semantic_template":"qf-density16-vector-candidate-residual/{\"a\":2,\"b\":1,\"candidate\":[13,4],\"u\":[6,1],\"v\":[1,2]}"},"fields":{"title":"Weighted Vector Candidate and Ordered Residual","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[6, 1], v=[1, 2], a=2, b=1, candidate=[13, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).","answer":{"kind":"expression","value":"([13, 4], [0, 0], Yes)","explanation":"Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([13, 4], [0, 0], Yes)."},"skills":["Weighted Vector Candidate and Ordered Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0095","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae070d8e4ccbabb00fb0cc891e1ac05c94a1d6df2bf2073bff1e55f3e948930e","family_id":"qf-density16-vector-candidate-residual","seed":202610161095,"semantic_template":"qf-density16-vector-candidate-residual/{\"a\":2,\"b\":1,\"candidate\":[16,4],\"u\":[7,1],\"v\":[1,2]}"},"fields":{"title":"Weighted Vector Candidate and Ordered Residual","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[7, 1], v=[1, 2], a=2, b=1, candidate=[16, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).","answer":{"kind":"expression","value":"([15, 4], [1, 0], No)","explanation":"Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([15, 4], [1, 0], No)."},"skills":["Weighted Vector Candidate and Ordered Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0096","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12fc79b08cadedb87b4ab0a57148abdd85d6d30c4a2c1b5606f03810fc0d0e33","family_id":"qf-density16-vector-candidate-residual","seed":202610161096,"semantic_template":"qf-density16-vector-candidate-residual/{\"a\":2,\"b\":1,\"candidate\":[18,4],\"u\":[8,1],\"v\":[1,2]}"},"fields":{"title":"Weighted Vector Candidate and Ordered Residual","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[8, 1], v=[1, 2], a=2, b=1, candidate=[18, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).","answer":{"kind":"expression","value":"([17, 4], [1, 0], No)","explanation":"Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([17, 4], [1, 0], No)."},"skills":["Weighted Vector Candidate and Ordered Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0097","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eaac6659092cac9fa0d93e947991db530bd30974ee835c1cba8fe7e287dbf495","family_id":"qf-density16-vector-candidate-residual","seed":202610161097,"semantic_template":"qf-density16-vector-candidate-residual/{\"a\":2,\"b\":1,\"candidate\":[19,4],\"u\":[9,1],\"v\":[1,2]}"},"fields":{"title":"Weighted Vector Candidate and Ordered Residual","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[9, 1], v=[1, 2], a=2, b=1, candidate=[19, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).","answer":{"kind":"expression","value":"([19, 4], [0, 0], Yes)","explanation":"Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([19, 4], [0, 0], Yes)."},"skills":["Weighted Vector Candidate and Ordered Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0098","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64fcabaea849cd8b416bcc4cd1640f2331381b771f8df4f735fe621e7eab7476","family_id":"qf-density16-vector-candidate-residual","seed":202610161098,"semantic_template":"qf-density16-vector-candidate-residual/{\"a\":2,\"b\":1,\"candidate\":[22,4],\"u\":[10,1],\"v\":[1,2]}"},"fields":{"title":"Weighted Vector Candidate and Ordered Residual","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[10, 1], v=[1, 2], a=2, b=1, candidate=[22, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).","answer":{"kind":"expression","value":"([21, 4], [1, 0], No)","explanation":"Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([21, 4], [1, 0], No)."},"skills":["Weighted Vector Candidate and Ordered Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0099","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92db309ce3a8804b0a90ae39a8011ec9d2a806775e9225af963f29bac6ebb93a","family_id":"qf-density16-vector-candidate-residual","seed":202610161099,"semantic_template":"qf-density16-vector-candidate-residual/{\"a\":2,\"b\":1,\"candidate\":[24,4],\"u\":[11,1],\"v\":[1,2]}"},"fields":{"title":"Weighted Vector Candidate and Ordered Residual","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[11, 1], v=[1, 2], a=2, b=1, candidate=[24, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).","answer":{"kind":"expression","value":"([23, 4], [1, 0], No)","explanation":"Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([23, 4], [1, 0], No)."},"skills":["Weighted Vector Candidate and Ordered Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r01-vector-0100","batch_id":"batch-20261012-advance-20261016-r01-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c412e491a2278a54a98240b05bb249cb25e0371f615a54858e2d485f369c6c46","family_id":"qf-density16-vector-candidate-residual","seed":202610161100,"semantic_template":"qf-density16-vector-candidate-residual/{\"a\":2,\"b\":1,\"candidate\":[25,4],\"u\":[12,1],\"v\":[1,2]}"},"fields":{"title":"Weighted Vector Candidate and Ordered Residual","stem":"Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[12, 1], v=[1, 2], a=2, b=1, candidate=[25, 4]. Give (au+bv, candidate−actual value, is the candidate equal?).","answer":{"kind":"expression","value":"([25, 4], [0, 0], Yes)","explanation":"Compute both actual components first; equality requires both signed residuals to be zero. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([25, 4], [0, 0], Yes)."},"skills":["Weighted Vector Candidate and Ordered Residual"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}