{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0001","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b6bb6eeb86941fd4830feef1aa40e4f3bb22eca1c9d7bb4cbc40893c2036b06c","family_id":"qf-mod23-span-weighted-sum","seed":202610232001,"semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[-12,-3,-2],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"An Ordered Generator-Coefficient Weighted Sum","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-12, -3, -2]. Give (original vectors,original coefficients,the coordinate sum).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-12, -3, -2], [-14, -5])","explanation":"Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-12, -3, -2], [-14, -5])."},"skills":["An Ordered Generator-Coefficient Weighted 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-20261013-advance-20261023-r02-linear-combination-span-0002","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"083f1313ba0e497556ecf45ffcb33e6cbc5936672cc1bcef12ef9bd54ace6e9e","family_id":"qf-mod23-span-weighted-sum","seed":202610232002,"semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[-11,-2,-1],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"An Ordered Generator-Coefficient Weighted Sum","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-11, -2, -1]. Give (original vectors,original coefficients,the coordinate sum).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-11, -2, -1], [-12, -3])","explanation":"Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-11, -2, -1], [-12, -3])."},"skills":["An Ordered Generator-Coefficient Weighted 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-20261013-advance-20261023-r02-linear-combination-span-0003","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13301257b3c141b5bd5e6f2b5a51f7422cac2786436e7adc2907fa06339348dc","family_id":"qf-mod23-span-weighted-sum","seed":202610232003,"semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[-10,-1,0],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"An Ordered Generator-Coefficient Weighted Sum","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-10, -1, 0]. Give (original vectors,original coefficients,the coordinate sum).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-10, -1, 0], [-10, -1])","explanation":"Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-10, -1, 0], [-10, -1])."},"skills":["An Ordered Generator-Coefficient Weighted 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-20261013-advance-20261023-r02-linear-combination-span-0004","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3fef89b8d5b206bfd4f15d88ef64589305e5a35757050222caee97309f481df4","family_id":"qf-mod23-span-weighted-sum","seed":202610232004,"semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[-9,0,1],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"An Ordered Generator-Coefficient Weighted Sum","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-9, 0, 1]. Give (original vectors,original coefficients,the coordinate sum).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-9, 0, 1], [-8, 1])","explanation":"Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-9, 0, 1], [-8, 1])."},"skills":["An Ordered Generator-Coefficient Weighted 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-20261013-advance-20261023-r02-linear-combination-span-0005","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a930533cd2b60f673cc2ec6a34544fc3461eb066de90a33046a2840ad4ad2ef","family_id":"qf-mod23-span-weighted-sum","seed":202610232005,"semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[-8,1,2],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"An Ordered Generator-Coefficient Weighted Sum","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-8, 1, 2]. Give (original vectors,original coefficients,the coordinate sum).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-8, 1, 2], [-6, 3])","explanation":"Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-8, 1, 2], [-6, 3])."},"skills":["An Ordered Generator-Coefficient Weighted 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-20261013-advance-20261023-r02-linear-combination-span-0006","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7cec179c2e3b595344f74198ae2b8bc4f286b9cc01f63995a1bf7bb72cc6dc2a","family_id":"qf-mod23-span-weighted-sum","seed":202610232006,"semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[-7,2,-2],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"An Ordered Generator-Coefficient Weighted Sum","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-7, 2, -2]. Give (original vectors,original coefficients,the coordinate sum).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-7, 2, -2], [-9, 0])","explanation":"Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-7, 2, -2], [-9, 0])."},"skills":["An Ordered Generator-Coefficient Weighted 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-20261013-advance-20261023-r02-linear-combination-span-0007","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4214bbf05d5b7ffee2039a23b267ab88af0fa40031af388d1460e094bcfa8b0","family_id":"qf-mod23-span-weighted-sum","seed":202610232007,"semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[-6,3,-1],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"An Ordered Generator-Coefficient Weighted Sum","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-6, 3, -1]. Give (original vectors,original coefficients,the coordinate sum).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-6, 3, -1], [-7, 2])","explanation":"Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-6, 3, -1], [-7, 2])."},"skills":["An Ordered Generator-Coefficient Weighted 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-20261013-advance-20261023-r02-linear-combination-span-0008","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7d5a7cd8b8e2ade7087e288e1210ef0cd0e1163ed5b79da602098a818d81004","family_id":"qf-mod23-span-weighted-sum","seed":202610232008,"semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[-5,-3,0],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"An Ordered Generator-Coefficient Weighted Sum","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-5, -3, 0]. Give (original vectors,original coefficients,the coordinate sum).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-5, -3, 0], [-5, -3])","explanation":"Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-5, -3, 0], [-5, -3])."},"skills":["An Ordered Generator-Coefficient Weighted 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-20261013-advance-20261023-r02-linear-combination-span-0009","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d42b5cc93415bfeea2bd5ab67e22a1a2b87534c839d923adbc4ff3ac3cb6c281","family_id":"qf-mod23-span-weighted-sum","seed":202610232009,"semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[-4,-2,1],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"An Ordered Generator-Coefficient Weighted Sum","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-4, -2, 1]. Give (original vectors,original coefficients,the coordinate sum).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-4, -2, 1], [-3, -1])","explanation":"Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-4, -2, 1], [-3, -1])."},"skills":["An Ordered Generator-Coefficient Weighted 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-20261013-advance-20261023-r02-linear-combination-span-0010","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4601b845286399e1a5bafbb663f04d2d9a8de7003c646041f3ca943b3190d07","family_id":"qf-mod23-span-weighted-sum","seed":202610232010,"semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[-3,-1,2],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"An Ordered Generator-Coefficient Weighted Sum","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-3, -1, 2]. Give (original vectors,original coefficients,the coordinate sum).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-3, -1, 2], [-1, 1])","explanation":"Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-3, -1, 2], [-1, 1])."},"skills":["An Ordered Generator-Coefficient Weighted 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-20261013-advance-20261023-r02-linear-combination-span-0011","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3943543b39eacc149a2915e8c2dc6668e555b8e5e4a24f023e5780c2bc3565d","family_id":"qf-mod23-span-weighted-sum","seed":202610232011,"semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[-2,0,-2],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"An Ordered Generator-Coefficient Weighted Sum","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-2, 0, -2]. Give (original vectors,original coefficients,the coordinate sum).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-2, 0, -2], [-4, -2])","explanation":"Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-2, 0, -2], [-4, -2])."},"skills":["An Ordered Generator-Coefficient Weighted 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-20261013-advance-20261023-r02-linear-combination-span-0012","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72d1d3f14f04e873d83d2fae36b2b60e740c529fc56f364be5de8f2090b16bab","family_id":"qf-mod23-span-weighted-sum","seed":202610232012,"semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[-1,1,-1],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"An Ordered Generator-Coefficient Weighted Sum","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-1, 1, -1]. Give (original vectors,original coefficients,the coordinate sum).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-1, 1, -1], [-2, 0])","explanation":"Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-1, 1, -1], [-2, 0])."},"skills":["An Ordered Generator-Coefficient Weighted 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-20261013-advance-20261023-r02-linear-combination-span-0013","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79578b14abb12135817561dcda9b2e19f058e42d84ede82819bdc7845e0632b0","family_id":"qf-mod23-span-weighted-sum","seed":202610232013,"semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[0,2,0],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"An Ordered Generator-Coefficient Weighted Sum","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[0, 2, 0]. Give (original vectors,original coefficients,the coordinate sum).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [0, 2, 0], [0, 2])","explanation":"Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [0, 2, 0], [0, 2])."},"skills":["An Ordered Generator-Coefficient Weighted 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-20261013-advance-20261023-r02-linear-combination-span-0014","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f336c2ed1a6e4f25b5a656e7c4f0984a3a6d53eceadfc82470f5a2127d6ed06f","family_id":"qf-mod23-span-weighted-sum","seed":202610232014,"semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[1,3,1],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"An Ordered Generator-Coefficient Weighted Sum","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[1, 3, 1]. Give (original vectors,original coefficients,the coordinate sum).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [1, 3, 1], [2, 4])","explanation":"Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [1, 3, 1], [2, 4])."},"skills":["An Ordered Generator-Coefficient Weighted 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-20261013-advance-20261023-r02-linear-combination-span-0015","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"193564a4515d3f009cf62dc8db58bafc4682217b42e8117422ac2bf8192878ca","family_id":"qf-mod23-span-weighted-sum","seed":202610232015,"semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[2,-3,2],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"An Ordered Generator-Coefficient Weighted Sum","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[2, -3, 2]. Give (original vectors,original coefficients,the coordinate sum).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [2, -3, 2], [4, -1])","explanation":"Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [2, -3, 2], [4, -1])."},"skills":["An Ordered Generator-Coefficient Weighted 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-20261013-advance-20261023-r02-linear-combination-span-0016","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aebee5a23a3f9363b11e9ba33775590f065b4c9a27fb7122e27f165c011d22ca","family_id":"qf-mod23-span-weighted-sum","seed":202610232016,"semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[3,-2,-2],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"An Ordered Generator-Coefficient Weighted Sum","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[3, -2, -2]. Give (original vectors,original coefficients,the coordinate sum).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [3, -2, -2], [1, -4])","explanation":"Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [3, -2, -2], [1, -4])."},"skills":["An Ordered Generator-Coefficient Weighted 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-20261013-advance-20261023-r02-linear-combination-span-0017","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f3b551047e6825b1e4cd6d443f5fbc005b330d426de5a64e7c325c63740e9d6","family_id":"qf-mod23-span-weighted-sum","seed":202610232017,"semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[4,-1,-1],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"An Ordered Generator-Coefficient Weighted Sum","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[4, -1, -1]. Give (original vectors,original coefficients,the coordinate sum).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [4, -1, -1], [3, -2])","explanation":"Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [4, -1, -1], [3, -2])."},"skills":["An Ordered Generator-Coefficient Weighted 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-20261013-advance-20261023-r02-linear-combination-span-0018","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eca5819ff9039668e96c28212cbe5b2188f0e6a9a3019e0c9868ae3d5e8242d8","family_id":"qf-mod23-span-weighted-sum","seed":202610232018,"semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[5,0,0],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"An Ordered Generator-Coefficient Weighted Sum","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[5, 0, 0]. Give (original vectors,original coefficients,the coordinate sum).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [5, 0, 0], [5, 0])","explanation":"Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [5, 0, 0], [5, 0])."},"skills":["An Ordered Generator-Coefficient Weighted 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-20261013-advance-20261023-r02-linear-combination-span-0019","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8536b4cca62f2a47e68f66cae5e414f4c5ce09d3edfaf84913b5f783f422c6f","family_id":"qf-mod23-span-weighted-sum","seed":202610232019,"semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[6,1,1],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"An Ordered Generator-Coefficient Weighted Sum","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[6, 1, 1]. Give (original vectors,original coefficients,the coordinate sum).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [6, 1, 1], [7, 2])","explanation":"Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [6, 1, 1], [7, 2])."},"skills":["An Ordered Generator-Coefficient Weighted 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-20261013-advance-20261023-r02-linear-combination-span-0020","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86c3442ed7ae4dfa22c9faaae65149b88cf62ac3e94593a1dea7174d4ce77ed6","family_id":"qf-mod23-span-weighted-sum","seed":202610232020,"semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[7,2,2],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"An Ordered Generator-Coefficient Weighted Sum","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[7, 2, 2]. Give (original vectors,original coefficients,the coordinate sum).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [7, 2, 2], [9, 4])","explanation":"Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [7, 2, 2], [9, 4])."},"skills":["An Ordered Generator-Coefficient Weighted 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-20261013-advance-20261023-r02-linear-combination-span-0021","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"725477c9e55e020ccc54cc1a52022f9dab47337592fbef157c2ad9728a52b500","family_id":"qf-mod23-span-weighted-sum","seed":202610232021,"semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[8,3,-2],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"An Ordered Generator-Coefficient Weighted Sum","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[8, 3, -2]. Give (original vectors,original coefficients,the coordinate sum).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [8, 3, -2], [6, 1])","explanation":"Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [8, 3, -2], [6, 1])."},"skills":["An Ordered Generator-Coefficient Weighted 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-20261013-advance-20261023-r02-linear-combination-span-0022","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb4ff9ba65813cc3d2002eb70a034aa9c0723a2ca5e0e9d29e9257ffc269d694","family_id":"qf-mod23-span-weighted-sum","seed":202610232022,"semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[9,-3,-1],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"An Ordered Generator-Coefficient Weighted Sum","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[9, -3, -1]. Give (original vectors,original coefficients,the coordinate sum).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [9, -3, -1], [8, -4])","explanation":"Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [9, -3, -1], [8, -4])."},"skills":["An Ordered Generator-Coefficient Weighted 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-20261013-advance-20261023-r02-linear-combination-span-0023","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8af7848371f5d1edee748db116621493a9ec083b1d00e375d5e9b9385b59008","family_id":"qf-mod23-span-weighted-sum","seed":202610232023,"semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[10,-2,0],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"An Ordered Generator-Coefficient Weighted Sum","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[10, -2, 0]. Give (original vectors,original coefficients,the coordinate sum).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [10, -2, 0], [10, -2])","explanation":"Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [10, -2, 0], [10, -2])."},"skills":["An Ordered Generator-Coefficient Weighted 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-20261013-advance-20261023-r02-linear-combination-span-0024","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c372aab80a290809dc1eea79d6af5c22e6ddc9ff32b8eea8f1f9947e36858744","family_id":"qf-mod23-span-weighted-sum","seed":202610232024,"semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[11,-1,1],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"An Ordered Generator-Coefficient Weighted Sum","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[11, -1, 1]. Give (original vectors,original coefficients,the coordinate sum).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [11, -1, 1], [12, 0])","explanation":"Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [11, -1, 1], [12, 0])."},"skills":["An Ordered Generator-Coefficient Weighted 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-20261013-advance-20261023-r02-linear-combination-span-0025","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"779ef2b68ad2e4f48db621f417a537bf9000574339d4beaa44609ab4a816b0ba","family_id":"qf-mod23-span-weighted-sum","seed":202610232025,"semantic_template":"qf-mod23-span-weighted-sum/{\"coefficients\":[12,0,2],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"An Ordered Generator-Coefficient Weighted Sum","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[12, 0, 2]. Give (original vectors,original coefficients,the coordinate sum).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [12, 0, 2], [14, 2])","explanation":"Scale each supplied vector by its aligned coefficient, then add componentwise. Do not sort or deduplicate generators before weighting. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [12, 0, 2], [14, 2])."},"skills":["An Ordered Generator-Coefficient Weighted 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-20261013-advance-20261023-r02-linear-combination-span-0026","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fdd57ce8d0781142d593b046d162da2446c19ed0f078eac8aaabc603516f6e7e","family_id":"qf-mod23-span-witness-audit","seed":202610232026,"semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[-12,-3,-2],\"target\":[-13,-5],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"A Given Coefficient Witness against a Target Vector","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-12, -3, -2], target=[-13, -5]. Give (vectors,coefficients,target,reconstructed sum,matches?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-12, -3, -2], [-13, -5], [-14, -5], No)","explanation":"Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-12, -3, -2], [-13, -5], [-14, -5], No)."},"skills":["A Given Coefficient Witness against a Target Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0027","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c04ef5ef23da72bf4ceb61284c81cbc259bb7b9f5a4c06f91c29ec398121ca6","family_id":"qf-mod23-span-witness-audit","seed":202610232027,"semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[-11,-2,-1],\"target\":[-12,-3],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"A Given Coefficient Witness against a Target Vector","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-11, -2, -1], target=[-12, -3]. Give (vectors,coefficients,target,reconstructed sum,matches?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-11, -2, -1], [-12, -3], [-12, -3], Yes)","explanation":"Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-11, -2, -1], [-12, -3], [-12, -3], Yes)."},"skills":["A Given Coefficient Witness against a Target Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0028","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0bd2ff519a83329dcc158c38cca462c9fa580d5048469d9812ced28dfde8f03e","family_id":"qf-mod23-span-witness-audit","seed":202610232028,"semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[-10,-1,0],\"target\":[-10,-1],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"A Given Coefficient Witness against a Target Vector","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-10, -1, 0], target=[-10, -1]. Give (vectors,coefficients,target,reconstructed sum,matches?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-10, -1, 0], [-10, -1], [-10, -1], Yes)","explanation":"Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-10, -1, 0], [-10, -1], [-10, -1], Yes)."},"skills":["A Given Coefficient Witness against a Target Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0029","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53f270ccfc2e2bd2c9dfb472f1ce51c7e756153d83f06ec849f96945077a04cb","family_id":"qf-mod23-span-witness-audit","seed":202610232029,"semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[-9,0,1],\"target\":[-8,1],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"A Given Coefficient Witness against a Target Vector","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-9, 0, 1], target=[-8, 1]. Give (vectors,coefficients,target,reconstructed sum,matches?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-9, 0, 1], [-8, 1], [-8, 1], Yes)","explanation":"Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-9, 0, 1], [-8, 1], [-8, 1], Yes)."},"skills":["A Given Coefficient Witness against a Target Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0030","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49b00197885389110c394baa04aa2979b6b4848c3d162f361ba00bdb544b8598","family_id":"qf-mod23-span-witness-audit","seed":202610232030,"semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[-8,1,2],\"target\":[-5,3],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"A Given Coefficient Witness against a Target Vector","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-8, 1, 2], target=[-5, 3]. Give (vectors,coefficients,target,reconstructed sum,matches?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-8, 1, 2], [-5, 3], [-6, 3], No)","explanation":"Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-8, 1, 2], [-5, 3], [-6, 3], No)."},"skills":["A Given Coefficient Witness against a Target Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0031","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"980e5691eba5e828501b1087609ab4a0f1bcaaacdfacd4d58c6f23c1a7937ae4","family_id":"qf-mod23-span-witness-audit","seed":202610232031,"semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[-7,2,-2],\"target\":[-9,0],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"A Given Coefficient Witness against a Target Vector","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-7, 2, -2], target=[-9, 0]. Give (vectors,coefficients,target,reconstructed sum,matches?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-7, 2, -2], [-9, 0], [-9, 0], Yes)","explanation":"Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-7, 2, -2], [-9, 0], [-9, 0], Yes)."},"skills":["A Given Coefficient Witness against a Target Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0032","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4ba81df5d23ec3f09d2a18c68936d318e391f92202300b9c695a7480e120b94","family_id":"qf-mod23-span-witness-audit","seed":202610232032,"semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[-6,3,-1],\"target\":[-7,2],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"A Given Coefficient Witness against a Target Vector","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-6, 3, -1], target=[-7, 2]. Give (vectors,coefficients,target,reconstructed sum,matches?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-6, 3, -1], [-7, 2], [-7, 2], Yes)","explanation":"Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-6, 3, -1], [-7, 2], [-7, 2], Yes)."},"skills":["A Given Coefficient Witness against a Target Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0033","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cbf398f25c8c3440e0b8434bbb8911505f5a86b2450ebb85690dcb77753d3008","family_id":"qf-mod23-span-witness-audit","seed":202610232033,"semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[-5,-3,0],\"target\":[-5,-3],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"A Given Coefficient Witness against a Target Vector","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-5, -3, 0], target=[-5, -3]. Give (vectors,coefficients,target,reconstructed sum,matches?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-5, -3, 0], [-5, -3], [-5, -3], Yes)","explanation":"Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-5, -3, 0], [-5, -3], [-5, -3], Yes)."},"skills":["A Given Coefficient Witness against a Target Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0034","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a700c0a43622bb3a69ddcbadeec7b3d5530703ceb1734d98f48aade88eb186e5","family_id":"qf-mod23-span-witness-audit","seed":202610232034,"semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[-4,-2,1],\"target\":[-2,-1],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"A Given Coefficient Witness against a Target Vector","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-4, -2, 1], target=[-2, -1]. Give (vectors,coefficients,target,reconstructed sum,matches?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-4, -2, 1], [-2, -1], [-3, -1], No)","explanation":"Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-4, -2, 1], [-2, -1], [-3, -1], No)."},"skills":["A Given Coefficient Witness against a Target Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0035","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01978d1291ba36745562bd3206ced40c34f6b730970c4da11bac6143e61f825d","family_id":"qf-mod23-span-witness-audit","seed":202610232035,"semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[-3,-1,2],\"target\":[-1,1],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"A Given Coefficient Witness against a Target Vector","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-3, -1, 2], target=[-1, 1]. Give (vectors,coefficients,target,reconstructed sum,matches?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-3, -1, 2], [-1, 1], [-1, 1], Yes)","explanation":"Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-3, -1, 2], [-1, 1], [-1, 1], Yes)."},"skills":["A Given Coefficient Witness against a Target Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0036","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c374be716b49ca7acd60637211e75ff7427f843b10c56b9a3690f117f7ba14a","family_id":"qf-mod23-span-witness-audit","seed":202610232036,"semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[-2,0,-2],\"target\":[-4,-2],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"A Given Coefficient Witness against a Target Vector","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-2, 0, -2], target=[-4, -2]. Give (vectors,coefficients,target,reconstructed sum,matches?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-2, 0, -2], [-4, -2], [-4, -2], Yes)","explanation":"Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-2, 0, -2], [-4, -2], [-4, -2], Yes)."},"skills":["A Given Coefficient Witness against a Target Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0037","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b6958fa9638e9504e0fec7015e9b8caffd32c66c835a401d416232a9bc6e88d","family_id":"qf-mod23-span-witness-audit","seed":202610232037,"semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[-1,1,-1],\"target\":[-2,0],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"A Given Coefficient Witness against a Target Vector","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[-1, 1, -1], target=[-2, 0]. Give (vectors,coefficients,target,reconstructed sum,matches?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [-1, 1, -1], [-2, 0], [-2, 0], Yes)","explanation":"Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [-1, 1, -1], [-2, 0], [-2, 0], Yes)."},"skills":["A Given Coefficient Witness against a Target Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0038","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f29763f97d8240826910eebec5a55607e7c5fed289a8ad52a4ba656615c70d47","family_id":"qf-mod23-span-witness-audit","seed":202610232038,"semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[0,2,0],\"target\":[1,2],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"A Given Coefficient Witness against a Target Vector","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[0, 2, 0], target=[1, 2]. Give (vectors,coefficients,target,reconstructed sum,matches?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [0, 2, 0], [1, 2], [0, 2], No)","explanation":"Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [0, 2, 0], [1, 2], [0, 2], No)."},"skills":["A Given Coefficient Witness against a Target Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0039","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef0c41a45d6366e20ded1ab5076e886ecad32d177e5f3e3bd97d39a7f33558d3","family_id":"qf-mod23-span-witness-audit","seed":202610232039,"semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[1,3,1],\"target\":[2,4],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"A Given Coefficient Witness against a Target Vector","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[1, 3, 1], target=[2, 4]. Give (vectors,coefficients,target,reconstructed sum,matches?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [1, 3, 1], [2, 4], [2, 4], Yes)","explanation":"Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [1, 3, 1], [2, 4], [2, 4], Yes)."},"skills":["A Given Coefficient Witness against a Target Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0040","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b18d179b2a19a50f5afc6ab73358a2de4d2206737c6c26980a3fb3fc63f9f5c1","family_id":"qf-mod23-span-witness-audit","seed":202610232040,"semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[2,-3,2],\"target\":[4,-1],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"A Given Coefficient Witness against a Target Vector","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[2, -3, 2], target=[4, -1]. Give (vectors,coefficients,target,reconstructed sum,matches?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [2, -3, 2], [4, -1], [4, -1], Yes)","explanation":"Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [2, -3, 2], [4, -1], [4, -1], Yes)."},"skills":["A Given Coefficient Witness against a Target Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0041","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f436e0b2eb57abae93cf2a5fc63b398fa60816c31295412d45b2f92b21493cb3","family_id":"qf-mod23-span-witness-audit","seed":202610232041,"semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[3,-2,-2],\"target\":[1,-4],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"A Given Coefficient Witness against a Target Vector","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[3, -2, -2], target=[1, -4]. Give (vectors,coefficients,target,reconstructed sum,matches?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [3, -2, -2], [1, -4], [1, -4], Yes)","explanation":"Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [3, -2, -2], [1, -4], [1, -4], Yes)."},"skills":["A Given Coefficient Witness against a Target Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0042","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"564030dd94c097e430163d7f9ec14b9262c1c6dcc8b280b2d2aa61ddfc6da550","family_id":"qf-mod23-span-witness-audit","seed":202610232042,"semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[4,-1,-1],\"target\":[4,-2],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"A Given Coefficient Witness against a Target Vector","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[4, -1, -1], target=[4, -2]. Give (vectors,coefficients,target,reconstructed sum,matches?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [4, -1, -1], [4, -2], [3, -2], No)","explanation":"Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [4, -1, -1], [4, -2], [3, -2], No)."},"skills":["A Given Coefficient Witness against a Target Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0043","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30a4bdb91f6edc7f67424473e8abe36d45bda58ac7dff5733d9bc3f06cdb416f","family_id":"qf-mod23-span-witness-audit","seed":202610232043,"semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[5,0,0],\"target\":[5,0],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"A Given Coefficient Witness against a Target Vector","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[5, 0, 0], target=[5, 0]. Give (vectors,coefficients,target,reconstructed sum,matches?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [5, 0, 0], [5, 0], [5, 0], Yes)","explanation":"Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [5, 0, 0], [5, 0], [5, 0], Yes)."},"skills":["A Given Coefficient Witness against a Target Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0044","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1f0463c2e373e3b23cfd8735abe8e682a2cfb737a592185e11448d1df01ef4e","family_id":"qf-mod23-span-witness-audit","seed":202610232044,"semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[6,1,1],\"target\":[7,2],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"A Given Coefficient Witness against a Target Vector","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[6, 1, 1], target=[7, 2]. Give (vectors,coefficients,target,reconstructed sum,matches?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [6, 1, 1], [7, 2], [7, 2], Yes)","explanation":"Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [6, 1, 1], [7, 2], [7, 2], Yes)."},"skills":["A Given Coefficient Witness against a Target Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0045","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"082ab795cac6f8499034dc83b0bfd227edbf792642a7f4e8fbb5afdacd2c3da4","family_id":"qf-mod23-span-witness-audit","seed":202610232045,"semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[7,2,2],\"target\":[9,4],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"A Given Coefficient Witness against a Target Vector","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[7, 2, 2], target=[9, 4]. Give (vectors,coefficients,target,reconstructed sum,matches?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [7, 2, 2], [9, 4], [9, 4], Yes)","explanation":"Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [7, 2, 2], [9, 4], [9, 4], Yes)."},"skills":["A Given Coefficient Witness against a Target Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0046","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7379dc4d46563841aa2450e16a465b619ef444910684b27283a34980e47bb59","family_id":"qf-mod23-span-witness-audit","seed":202610232046,"semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[8,3,-2],\"target\":[7,1],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"A Given Coefficient Witness against a Target Vector","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[8, 3, -2], target=[7, 1]. Give (vectors,coefficients,target,reconstructed sum,matches?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [8, 3, -2], [7, 1], [6, 1], No)","explanation":"Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [8, 3, -2], [7, 1], [6, 1], No)."},"skills":["A Given Coefficient Witness against a Target Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0047","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2783b81c2173f99fada6e69947f1a9521085f2ddc312ae25aff9408b46f6ef19","family_id":"qf-mod23-span-witness-audit","seed":202610232047,"semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[9,-3,-1],\"target\":[8,-4],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"A Given Coefficient Witness against a Target Vector","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[9, -3, -1], target=[8, -4]. Give (vectors,coefficients,target,reconstructed sum,matches?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [9, -3, -1], [8, -4], [8, -4], Yes)","explanation":"Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [9, -3, -1], [8, -4], [8, -4], Yes)."},"skills":["A Given Coefficient Witness against a Target Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0048","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93cf155a278031dc790fd935cc7a1648f110d7253904d19e7649ced15ec340a3","family_id":"qf-mod23-span-witness-audit","seed":202610232048,"semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[10,-2,0],\"target\":[10,-2],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"A Given Coefficient Witness against a Target Vector","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[10, -2, 0], target=[10, -2]. Give (vectors,coefficients,target,reconstructed sum,matches?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [10, -2, 0], [10, -2], [10, -2], Yes)","explanation":"Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [10, -2, 0], [10, -2], [10, -2], Yes)."},"skills":["A Given Coefficient Witness against a Target Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0049","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c17b147a86c844d5d62862cc4db460f9a0ed422ed2b5e3f3f04b298a0a194e5c","family_id":"qf-mod23-span-witness-audit","seed":202610232049,"semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[11,-1,1],\"target\":[12,0],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"A Given Coefficient Witness against a Target Vector","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[11, -1, 1], target=[12, 0]. Give (vectors,coefficients,target,reconstructed sum,matches?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [11, -1, 1], [12, 0], [12, 0], Yes)","explanation":"Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [11, -1, 1], [12, 0], [12, 0], Yes)."},"skills":["A Given Coefficient Witness against a Target Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0050","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5c42cca4815640a30f0c06b93e6a08ab479b4e1412609a56db784f42c094c88","family_id":"qf-mod23-span-witness-audit","seed":202610232050,"semantic_template":"qf-mod23-span-witness-audit/{\"coefficients\":[12,0,2],\"target\":[15,2],\"vectors\":[[0,1],[1,0],[1,1]]}"},"fields":{"title":"A Given Coefficient Witness against a Target Vector","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. vectors=[[1, 0], [0, 1], [1, 1]], coefficients=[12, 0, 2], target=[15, 2]. Give (vectors,coefficients,target,reconstructed sum,matches?).","answer":{"kind":"expression","value":"([[1, 0], [0, 1], [1, 1]], [12, 0, 2], [15, 2], [14, 2], No)","explanation":"Reconstruct the target using this particular coefficient witness. A failed witness alone does not establish that every possible real combination fails. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [0, 1], [1, 1]], [12, 0, 2], [15, 2], [14, 2], No)."},"skills":["A Given Coefficient Witness against a Target Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0051","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"33ca08b34d1ee264b0e023fe5c737e33e2c8bd340db48f3e15fdb3b1dacdd5b7","family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232051,"semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[-12,-3],\"v1\":[1,1],\"v2\":[1,-1]}"},"fields":{"title":"Unique Real Coordinates in Two Independent Plane Generators","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[-12, -3]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([1, 1], [1, -1], [-12, -3], -2, -15/2, -9/2, [-12, -3])","explanation":"Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [-12, -3], -2, -15/2, -9/2, [-12, -3])."},"skills":["Unique Real Coordinates in Two Independent Plane Generators"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0052","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd88d76dc00b3c6ae515611c5eaf0bc11cfcb3c4f87d3ae9380c30f5a648c8d1","family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232052,"semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[-11,-2],\"v1\":[1,1],\"v2\":[1,-1]}"},"fields":{"title":"Unique Real Coordinates in Two Independent Plane Generators","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[-11, -2]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([1, 1], [1, -1], [-11, -2], -2, -13/2, -9/2, [-11, -2])","explanation":"Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [-11, -2], -2, -13/2, -9/2, [-11, -2])."},"skills":["Unique Real Coordinates in Two Independent Plane Generators"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0053","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ecd1f4965f3a4ff35d9832d68eb28b694ca9411aec0f2ff5ff03ec65610c7bbf","family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232053,"semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[-10,-1],\"v1\":[1,1],\"v2\":[1,-1]}"},"fields":{"title":"Unique Real Coordinates in Two Independent Plane Generators","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[-10, -1]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([1, 1], [1, -1], [-10, -1], -2, -11/2, -9/2, [-10, -1])","explanation":"Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [-10, -1], -2, -11/2, -9/2, [-10, -1])."},"skills":["Unique Real Coordinates in Two Independent Plane Generators"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0054","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22fbcdad50bc485af813bd952ac9eaa7bfce3a777585bb75f501002414de928b","family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232054,"semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[-9,0],\"v1\":[1,1],\"v2\":[1,-1]}"},"fields":{"title":"Unique Real Coordinates in Two Independent Plane Generators","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[-9, 0]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([1, 1], [1, -1], [-9, 0], -2, -9/2, -9/2, [-9, 0])","explanation":"Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [-9, 0], -2, -9/2, -9/2, [-9, 0])."},"skills":["Unique Real Coordinates in Two Independent Plane Generators"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0055","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2afc12f9a687e28ded124324ccc20d03d873081e84137e0114ce4a1dca380ca3","family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232055,"semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[-8,1],\"v1\":[1,1],\"v2\":[1,-1]}"},"fields":{"title":"Unique Real Coordinates in Two Independent Plane Generators","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[-8, 1]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([1, 1], [1, -1], [-8, 1], -2, -7/2, -9/2, [-8, 1])","explanation":"Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [-8, 1], -2, -7/2, -9/2, [-8, 1])."},"skills":["Unique Real Coordinates in Two Independent Plane Generators"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0056","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f2c0f0fb621e850c87204c9aa16b4e3e113297cb62e218199bc0445f2fd69cd","family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232056,"semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[-7,2],\"v1\":[1,1],\"v2\":[1,-1]}"},"fields":{"title":"Unique Real Coordinates in Two Independent Plane Generators","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[-7, 2]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([1, 1], [1, -1], [-7, 2], -2, -5/2, -9/2, [-7, 2])","explanation":"Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [-7, 2], -2, -5/2, -9/2, [-7, 2])."},"skills":["Unique Real Coordinates in Two Independent Plane Generators"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0057","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0fb468004cc9fcd3ecfe4421facef149cb8e1a7d48ce0a58d2c75607c1dbdbaa","family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232057,"semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[-6,3],\"v1\":[1,1],\"v2\":[1,-1]}"},"fields":{"title":"Unique Real Coordinates in Two Independent Plane Generators","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[-6, 3]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([1, 1], [1, -1], [-6, 3], -2, -3/2, -9/2, [-6, 3])","explanation":"Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [-6, 3], -2, -3/2, -9/2, [-6, 3])."},"skills":["Unique Real Coordinates in Two Independent Plane Generators"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0058","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1cd3231ca34dc441bf08c12665ef0d24921c26be2277d667a1c4401cc34d076c","family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232058,"semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[-5,-3],\"v1\":[1,1],\"v2\":[1,-1]}"},"fields":{"title":"Unique Real Coordinates in Two Independent Plane Generators","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[-5, -3]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([1, 1], [1, -1], [-5, -3], -2, -4, -1, [-5, -3])","explanation":"Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [-5, -3], -2, -4, -1, [-5, -3])."},"skills":["Unique Real Coordinates in Two Independent Plane Generators"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0059","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc00fd255780ab0ade216a4d3d77c54b883eade0b9f199c4e248df25e38cfc27","family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232059,"semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[-4,-2],\"v1\":[1,1],\"v2\":[1,-1]}"},"fields":{"title":"Unique Real Coordinates in Two Independent Plane Generators","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[-4, -2]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([1, 1], [1, -1], [-4, -2], -2, -3, -1, [-4, -2])","explanation":"Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [-4, -2], -2, -3, -1, [-4, -2])."},"skills":["Unique Real Coordinates in Two Independent Plane Generators"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0060","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"572d57a37bb30a72e8bc24aa3535120b5dcae088f0670c06de35bc5ce0a99f87","family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232060,"semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[-3,-1],\"v1\":[1,1],\"v2\":[1,-1]}"},"fields":{"title":"Unique Real Coordinates in Two Independent Plane Generators","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[-3, -1]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([1, 1], [1, -1], [-3, -1], -2, -2, -1, [-3, -1])","explanation":"Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [-3, -1], -2, -2, -1, [-3, -1])."},"skills":["Unique Real Coordinates in Two Independent Plane Generators"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0061","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e55c16786942d93d4e1c054a279b1849e3b49fd6150ebdcb81f593e116d6c9a","family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232061,"semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[-2,0],\"v1\":[1,1],\"v2\":[1,-1]}"},"fields":{"title":"Unique Real Coordinates in Two Independent Plane Generators","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[-2, 0]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([1, 1], [1, -1], [-2, 0], -2, -1, -1, [-2, 0])","explanation":"Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [-2, 0], -2, -1, -1, [-2, 0])."},"skills":["Unique Real Coordinates in Two Independent Plane Generators"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0062","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd06d1ddc681728c393bd1099b9af45f2952b215cadc8cd326e469a321218c1c","family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232062,"semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[-1,1],\"v1\":[1,1],\"v2\":[1,-1]}"},"fields":{"title":"Unique Real Coordinates in Two Independent Plane Generators","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[-1, 1]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([1, 1], [1, -1], [-1, 1], -2, 0, -1, [-1, 1])","explanation":"Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [-1, 1], -2, 0, -1, [-1, 1])."},"skills":["Unique Real Coordinates in Two Independent Plane Generators"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0063","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9dd803ac916f42872266089f35ab9e3b817ca8eb70b16dbeb3803a588f4c272","family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232063,"semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[0,2],\"v1\":[1,1],\"v2\":[1,-1]}"},"fields":{"title":"Unique Real Coordinates in Two Independent Plane Generators","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[0, 2]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([1, 1], [1, -1], [0, 2], -2, 1, -1, [0, 2])","explanation":"Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [0, 2], -2, 1, -1, [0, 2])."},"skills":["Unique Real Coordinates in Two Independent Plane Generators"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0064","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e3c2878a517039d50dea3b176c855851174b81754b145551c73e9ac6e0e56659","family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232064,"semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[1,3],\"v1\":[1,1],\"v2\":[1,-1]}"},"fields":{"title":"Unique Real Coordinates in Two Independent Plane Generators","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[1, 3]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([1, 1], [1, -1], [1, 3], -2, 2, -1, [1, 3])","explanation":"Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [1, 3], -2, 2, -1, [1, 3])."},"skills":["Unique Real Coordinates in Two Independent Plane Generators"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0065","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6edbbff72eaef7b403236b1aaf78b69f6242a9d69b27ffc60e791ade5d2222cc","family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232065,"semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[2,-3],\"v1\":[1,1],\"v2\":[1,-1]}"},"fields":{"title":"Unique Real Coordinates in Two Independent Plane Generators","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[2, -3]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([1, 1], [1, -1], [2, -3], -2, -1/2, 5/2, [2, -3])","explanation":"Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [2, -3], -2, -1/2, 5/2, [2, -3])."},"skills":["Unique Real Coordinates in Two Independent Plane Generators"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0066","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6347953243e4555dec9fe57edcfeef179b139f8b9b6ab1f82d778ff62719327a","family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232066,"semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[3,-2],\"v1\":[1,1],\"v2\":[1,-1]}"},"fields":{"title":"Unique Real Coordinates in Two Independent Plane Generators","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[3, -2]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([1, 1], [1, -1], [3, -2], -2, 1/2, 5/2, [3, -2])","explanation":"Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [3, -2], -2, 1/2, 5/2, [3, -2])."},"skills":["Unique Real Coordinates in Two Independent Plane Generators"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0067","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b823cca1f8a5c27d451661286e1ae883c531b6ec0193424194f8168842689e92","family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232067,"semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[4,-1],\"v1\":[1,1],\"v2\":[1,-1]}"},"fields":{"title":"Unique Real Coordinates in Two Independent Plane Generators","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[4, -1]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([1, 1], [1, -1], [4, -1], -2, 3/2, 5/2, [4, -1])","explanation":"Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [4, -1], -2, 3/2, 5/2, [4, -1])."},"skills":["Unique Real Coordinates in Two Independent Plane Generators"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0068","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd65b867e34e1d67dddc97bfdd18e9afe07868f0ec0d5f41bc0072f6ac44d6db","family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232068,"semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[5,0],\"v1\":[1,1],\"v2\":[1,-1]}"},"fields":{"title":"Unique Real Coordinates in Two Independent Plane Generators","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[5, 0]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([1, 1], [1, -1], [5, 0], -2, 5/2, 5/2, [5, 0])","explanation":"Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [5, 0], -2, 5/2, 5/2, [5, 0])."},"skills":["Unique Real Coordinates in Two Independent Plane Generators"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0069","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c72a712bbc889f58b7460880b96b73ed28d38321bd19686076a4c36be479f53e","family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232069,"semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[6,1],\"v1\":[1,1],\"v2\":[1,-1]}"},"fields":{"title":"Unique Real Coordinates in Two Independent Plane Generators","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[6, 1]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([1, 1], [1, -1], [6, 1], -2, 7/2, 5/2, [6, 1])","explanation":"Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [6, 1], -2, 7/2, 5/2, [6, 1])."},"skills":["Unique Real Coordinates in Two Independent Plane Generators"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0070","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"491926efe5132ff9833545254d12bf0bbbf417920b298b08c1a62d96292a334f","family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232070,"semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[7,2],\"v1\":[1,1],\"v2\":[1,-1]}"},"fields":{"title":"Unique Real Coordinates in Two Independent Plane Generators","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[7, 2]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([1, 1], [1, -1], [7, 2], -2, 9/2, 5/2, [7, 2])","explanation":"Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [7, 2], -2, 9/2, 5/2, [7, 2])."},"skills":["Unique Real Coordinates in Two Independent Plane Generators"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0071","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e409c70870d0d77b054277d348a78ad0383c823296d80f1a88c7428c7b55731","family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232071,"semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[8,3],\"v1\":[1,1],\"v2\":[1,-1]}"},"fields":{"title":"Unique Real Coordinates in Two Independent Plane Generators","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[8, 3]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([1, 1], [1, -1], [8, 3], -2, 11/2, 5/2, [8, 3])","explanation":"Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [8, 3], -2, 11/2, 5/2, [8, 3])."},"skills":["Unique Real Coordinates in Two Independent Plane Generators"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0072","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"075087757a3425f6a005b99e648630e873861081b1c8710e32533292819471b2","family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232072,"semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[9,-3],\"v1\":[1,1],\"v2\":[1,-1]}"},"fields":{"title":"Unique Real Coordinates in Two Independent Plane Generators","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[9, -3]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([1, 1], [1, -1], [9, -3], -2, 3, 6, [9, -3])","explanation":"Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [9, -3], -2, 3, 6, [9, -3])."},"skills":["Unique Real Coordinates in Two Independent Plane Generators"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0073","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0223a7896f0626b284235e428047f19fd5a71a603402c83022bcbaa316c6a4b","family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232073,"semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[10,-2],\"v1\":[1,1],\"v2\":[1,-1]}"},"fields":{"title":"Unique Real Coordinates in Two Independent Plane Generators","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[10, -2]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([1, 1], [1, -1], [10, -2], -2, 4, 6, [10, -2])","explanation":"Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [10, -2], -2, 4, 6, [10, -2])."},"skills":["Unique Real Coordinates in Two Independent Plane Generators"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0074","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76d41dfb5eeac58d258068294a6385cd6f5911b8680495903ac1d92d4b831f8a","family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232074,"semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[11,-1],\"v1\":[1,1],\"v2\":[1,-1]}"},"fields":{"title":"Unique Real Coordinates in Two Independent Plane Generators","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[11, -1]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([1, 1], [1, -1], [11, -1], -2, 5, 6, [11, -1])","explanation":"Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [11, -1], -2, 5, 6, [11, -1])."},"skills":["Unique Real Coordinates in Two Independent Plane Generators"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0075","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74ec2ca91160a7e58128e6dc73a1f0fea5f0dc2e6e7961e8dc609af47fc3a692","family_id":"qf-mod23-span-r2-coordinate-solve","seed":202610232075,"semantic_template":"qf-mod23-span-r2-coordinate-solve/{\"target\":[12,0],\"v1\":[1,1],\"v2\":[1,-1]}"},"fields":{"title":"Unique Real Coordinates in Two Independent Plane Generators","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. v1=[1, 1], v2=[1, -1], target=[12, 0]. Treat the two generators as columns, require determinant≠0, and give (v1,v2,target,signed determinant,coefficient of v1,coefficient of v2,reconstructed target). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([1, 1], [1, -1], [12, 0], -2, 6, 6, [12, 0])","explanation":"Solve the exact two-variable system using the declared column order. A negative determinant changes numerator/denominator signs, not the coordinate roles. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [1, -1], [12, 0], -2, 6, 6, [12, 0])."},"skills":["Unique Real Coordinates in Two Independent Plane Generators"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0076","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24a7240a8849c83ac490477d7ecda39d566c0f96941d33b5e5b65ee9f4f0164c","family_id":"qf-mod23-span-single-generator-membership","seed":202610232076,"semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[-20,-39]}"},"fields":{"title":"A Single Generator, Including the Zero-Vector Case","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[-20, -39]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([2, 4], [-20, -39], -10, No)","explanation":"Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [-20, -39], -10, No)."},"skills":["A Single Generator, Including the Zero-Vector Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0077","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0aa2c8d1d10d28ee866eac38d6f00e9e2d50e83d7d1305b7e19e59097f33fa12","family_id":"qf-mod23-span-single-generator-membership","seed":202610232077,"semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[-18,-36]}"},"fields":{"title":"A Single Generator, Including the Zero-Vector Case","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[-18, -36]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([2, 4], [-18, -36], -9, Yes)","explanation":"Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [-18, -36], -9, Yes)."},"skills":["A Single Generator, Including the Zero-Vector Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0078","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13f90fb5f2da8d54b2a5d75526d8d190bb58cf6c61ee18dafd70cf4da6f12085","family_id":"qf-mod23-span-single-generator-membership","seed":202610232078,"semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[-16,-32]}"},"fields":{"title":"A Single Generator, Including the Zero-Vector Case","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[-16, -32]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([2, 4], [-16, -32], -8, Yes)","explanation":"Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [-16, -32], -8, Yes)."},"skills":["A Single Generator, Including the Zero-Vector Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0079","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7783382195ee24071a7216f13f8ed94670f05db454dd1f588f9e9d7b25ea1426","family_id":"qf-mod23-span-single-generator-membership","seed":202610232079,"semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[-14,-28]}"},"fields":{"title":"A Single Generator, Including the Zero-Vector Case","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[-14, -28]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([2, 4], [-14, -28], -7, Yes)","explanation":"Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [-14, -28], -7, Yes)."},"skills":["A Single Generator, Including the Zero-Vector Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0080","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35284768c0e7c51053f863d175ad6fb5046c1d29cd813031129322849764929a","family_id":"qf-mod23-span-single-generator-membership","seed":202610232080,"semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[-12,-23]}"},"fields":{"title":"A Single Generator, Including the Zero-Vector Case","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[-12, -23]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([2, 4], [-12, -23], -6, No)","explanation":"Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [-12, -23], -6, No)."},"skills":["A Single Generator, Including the Zero-Vector Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0081","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3019d8028e93b5339604e5dc052cffcc18de38af2e02ce031d9f63faaaa77ef5","family_id":"qf-mod23-span-single-generator-membership","seed":202610232081,"semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[-10,-20]}"},"fields":{"title":"A Single Generator, Including the Zero-Vector Case","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[-10, -20]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([2, 4], [-10, -20], -5, Yes)","explanation":"Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [-10, -20], -5, Yes)."},"skills":["A Single Generator, Including the Zero-Vector Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0082","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7246e6c039753f5d0033ce141b100026654e6559d8ee0b69fb3c6e2c5afcb34c","family_id":"qf-mod23-span-single-generator-membership","seed":202610232082,"semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[-8,-16]}"},"fields":{"title":"A Single Generator, Including the Zero-Vector Case","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[-8, -16]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([2, 4], [-8, -16], -4, Yes)","explanation":"Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [-8, -16], -4, Yes)."},"skills":["A Single Generator, Including the Zero-Vector Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0083","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1a36e634bbd736c7b1ddfeb55f2645736d5a788eef03e100668f32793945dec","family_id":"qf-mod23-span-single-generator-membership","seed":202610232083,"semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[-6,-12]}"},"fields":{"title":"A Single Generator, Including the Zero-Vector Case","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[-6, -12]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([2, 4], [-6, -12], -3, Yes)","explanation":"Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [-6, -12], -3, Yes)."},"skills":["A Single Generator, Including the Zero-Vector Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0084","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55f82bc1058682faab4bb36c0e8f0dda40013ee1d91342bdef3f8b93bcaeea63","family_id":"qf-mod23-span-single-generator-membership","seed":202610232084,"semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[-4,-7]}"},"fields":{"title":"A Single Generator, Including the Zero-Vector Case","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[-4, -7]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([2, 4], [-4, -7], -2, No)","explanation":"Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [-4, -7], -2, No)."},"skills":["A Single Generator, Including the Zero-Vector Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0085","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c804c2da8a7179014f3f154f8b15afe404679c99562fb18da827902c51159b58","family_id":"qf-mod23-span-single-generator-membership","seed":202610232085,"semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[-2,-4]}"},"fields":{"title":"A Single Generator, Including the Zero-Vector Case","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[-2, -4]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([2, 4], [-2, -4], -1, Yes)","explanation":"Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [-2, -4], -1, Yes)."},"skills":["A Single Generator, Including the Zero-Vector Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0086","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08d649721cf492a3cf13826d5eb053860b4a6a131e8908ec91c94dc227fc599e","family_id":"qf-mod23-span-single-generator-membership","seed":202610232086,"semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[0,0]}"},"fields":{"title":"A Single Generator, Including the Zero-Vector Case","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[0, 0]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([2, 4], [0, 0], 0, Yes)","explanation":"Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [0, 0], 0, Yes)."},"skills":["A Single Generator, Including the Zero-Vector Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0087","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45f9674d31a25a74d434adcfc0b26dbfe352f2974465ad2f94ec52a428c86003","family_id":"qf-mod23-span-single-generator-membership","seed":202610232087,"semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[2,4]}"},"fields":{"title":"A Single Generator, Including the Zero-Vector Case","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[2, 4]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([2, 4], [2, 4], 1, Yes)","explanation":"Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [2, 4], 1, Yes)."},"skills":["A Single Generator, Including the Zero-Vector Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0088","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f1279191be94627859503a46f85be0a4c8172b882b4df1a48a8048f9413e795","family_id":"qf-mod23-span-single-generator-membership","seed":202610232088,"semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[4,9]}"},"fields":{"title":"A Single Generator, Including the Zero-Vector Case","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[4, 9]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([2, 4], [4, 9], 2, No)","explanation":"Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [4, 9], 2, No)."},"skills":["A Single Generator, Including the Zero-Vector Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0089","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"850d08b80033a581c6c482f34e4f3e4443b7f0ff4afce415b75effe2f6423ef7","family_id":"qf-mod23-span-single-generator-membership","seed":202610232089,"semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[6,12]}"},"fields":{"title":"A Single Generator, Including the Zero-Vector Case","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[6, 12]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([2, 4], [6, 12], 3, Yes)","explanation":"Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [6, 12], 3, Yes)."},"skills":["A Single Generator, Including the Zero-Vector Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0090","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb8e303efdcaa1cc27c7c71b5be0998fa5f23113b0f51d6159adfde2f66fd73f","family_id":"qf-mod23-span-single-generator-membership","seed":202610232090,"semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[8,16]}"},"fields":{"title":"A Single Generator, Including the Zero-Vector Case","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[8, 16]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([2, 4], [8, 16], 4, Yes)","explanation":"Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [8, 16], 4, Yes)."},"skills":["A Single Generator, Including the Zero-Vector Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0091","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bcd0b366d32f9d8d52f160282365b937e3b9b85d6854f86a0d372fbf6ea1cba7","family_id":"qf-mod23-span-single-generator-membership","seed":202610232091,"semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[10,20]}"},"fields":{"title":"A Single Generator, Including the Zero-Vector Case","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[10, 20]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([2, 4], [10, 20], 5, Yes)","explanation":"Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [10, 20], 5, Yes)."},"skills":["A Single Generator, Including the Zero-Vector Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0092","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49a74885d350aa558587d85fe5e19a541586413da81af1a38dd02b0d38b6a103","family_id":"qf-mod23-span-single-generator-membership","seed":202610232092,"semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[12,25]}"},"fields":{"title":"A Single Generator, Including the Zero-Vector Case","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[12, 25]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([2, 4], [12, 25], 6, No)","explanation":"Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [12, 25], 6, No)."},"skills":["A Single Generator, Including the Zero-Vector Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0093","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50b0e459bfa3eac0acb3c414a5985c40e7420b0d504f6fb9ebafdfe7a34f42ac","family_id":"qf-mod23-span-single-generator-membership","seed":202610232093,"semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[14,28]}"},"fields":{"title":"A Single Generator, Including the Zero-Vector Case","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[14, 28]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([2, 4], [14, 28], 7, Yes)","explanation":"Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [14, 28], 7, Yes)."},"skills":["A Single Generator, Including the Zero-Vector Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0094","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d05a837114351a7eac07833e02a5e14813859629a457dfa87962675accf82c39","family_id":"qf-mod23-span-single-generator-membership","seed":202610232094,"semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[16,32]}"},"fields":{"title":"A Single Generator, Including the Zero-Vector Case","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[16, 32]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([2, 4], [16, 32], 8, Yes)","explanation":"Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [16, 32], 8, Yes)."},"skills":["A Single Generator, Including the Zero-Vector Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0095","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2218c4d6a920c3bc93718b21cbcf3ba8839adf6c9174c92ec694c3315eef7008","family_id":"qf-mod23-span-single-generator-membership","seed":202610232095,"semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[18,36]}"},"fields":{"title":"A Single Generator, Including the Zero-Vector Case","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[18, 36]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([2, 4], [18, 36], 9, Yes)","explanation":"Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [18, 36], 9, Yes)."},"skills":["A Single Generator, Including the Zero-Vector Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0096","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"622c1a8e8d7d2a9d06f43ff04f9b291f32a46a8606fd8abb6369a86891ef66cd","family_id":"qf-mod23-span-single-generator-membership","seed":202610232096,"semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[20,41]}"},"fields":{"title":"A Single Generator, Including the Zero-Vector Case","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[20, 41]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([2, 4], [20, 41], 10, No)","explanation":"Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [20, 41], 10, No)."},"skills":["A Single Generator, Including the Zero-Vector Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0097","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2eb473ee0c7d5c1f87c470d63db19a9fd96ca01aeb20262f81e9888dbba50a93","family_id":"qf-mod23-span-single-generator-membership","seed":202610232097,"semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[22,44]}"},"fields":{"title":"A Single Generator, Including the Zero-Vector Case","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[22, 44]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([2, 4], [22, 44], 11, Yes)","explanation":"Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [22, 44], 11, Yes)."},"skills":["A Single Generator, Including the Zero-Vector Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0098","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b4c90cc00b365d6cd2adcb1ec4ff5d4982981f8003cf533522e542d1db1b3ab","family_id":"qf-mod23-span-single-generator-membership","seed":202610232098,"semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[24,48]}"},"fields":{"title":"A Single Generator, Including the Zero-Vector Case","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[24, 48]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([2, 4], [24, 48], 12, Yes)","explanation":"Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [24, 48], 12, Yes)."},"skills":["A Single Generator, Including the Zero-Vector Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0099","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96b00b4690b92edf3daa37d062796d6ba02b6de0b52fc27b88b125ae00c92591","family_id":"qf-mod23-span-single-generator-membership","seed":202610232099,"semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[26,52]}"},"fields":{"title":"A Single Generator, Including the Zero-Vector Case","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[26, 52]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([2, 4], [26, 52], 13, Yes)","explanation":"Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [26, 52], 13, Yes)."},"skills":["A Single Generator, Including the Zero-Vector Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r02-linear-combination-span-0100","batch_id":"batch-20261013-advance-20261023-r02-linear-combination-span-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0d0daee14854c8e86d95ae487edd98b2a4684a379b6018de3648160646e1a49","family_id":"qf-mod23-span-single-generator-membership","seed":202610232100,"semantic_template":"qf-mod23-span-single-generator-membership/{\"generator\":[2,4],\"target\":[28,57]}"},"fields":{"title":"A Single Generator, Including the Zero-Vector Case","stem":"Use a declared finite generator family in a fixed real coordinate space. Preserve generator order, duplicates and its coefficient pairing; all vector dimensions and coefficient counts must match. A span quantifies over all finite linear combinations, not a fixed-arity sample or an undefined infinite series. generator=[2, 4], target=[28, 57]. Give (generator,target,scalar or status,in span?). For a nonzero generator, obtain the candidate scalar from its first nonzero coordinate and retain it even if another coordinate fails. For a zero generator, output any-scalar for a zero target, otherwise does-not-exist. Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"([2, 4], [28, 57], 14, No)","explanation":"Check all coordinates after choosing the pivot ratio. A zero generator spans only the zero vector, for which every real scalar is a witness. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 4], [28, 57], 14, No)."},"skills":["A Single Generator, Including the Zero-Vector Case"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}