{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r05-finite-convex-combination-0001","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"747bf58e177a713298df5627e410c4fa504dd21fb335fcba4a0e9ef1a24b2033","family_id":"qf-mod24-convcomb-qualification","seed":202610245001,"semantic_template":"qf-mod24-convcomb-qualification/{\"vectors\":[[-12,0],[-10,2]],\"weights\":[[2,3],[2,3]]}"},"fields":{"title":"Nonnegativity and Exact Total One as Separate Weight Tests","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[-12, 0], [-10, 2]], weights=[2/3, 2/3]. This family admits unqualified candidates for inspection. Give (vectors,normalized weights,all nonnegative?,exact sum,sum equals 1?,both conditions?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[-12, 0], [-10, 2]], [2/3, 2/3], Yes, 4/3, No, No)","explanation":"Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-12, 0], [-10, 2]], [2/3, 2/3], Yes, 4/3, No, No)."},"skills":["Nonnegativity and Exact Total One as Separate Weight Tests"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0002","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54c7d361ae9e8d2639a9a432a774a29ba9931152d30dd9f97a9bed6a5e509cdb","family_id":"qf-mod24-convcomb-qualification","seed":202610245002,"semantic_template":"qf-mod24-convcomb-qualification/{\"vectors\":[[-11,0],[-9,2]],\"weights\":[[2,4],[2,4]]}"},"fields":{"title":"Nonnegativity and Exact Total One as Separate Weight Tests","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[-11, 0], [-9, 2]], weights=[2/4, 2/4]. This family admits unqualified candidates for inspection. Give (vectors,normalized weights,all nonnegative?,exact sum,sum equals 1?,both conditions?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[-11, 0], [-9, 2]], [1/2, 1/2], Yes, 1, Yes, Yes)","explanation":"Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-11, 0], [-9, 2]], [1/2, 1/2], Yes, 1, Yes, Yes)."},"skills":["Nonnegativity and Exact Total One as Separate Weight Tests"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0003","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"793a445ab987de597361774f13d784cd77ade1c1366b8183b8121ad342cd890f","family_id":"qf-mod24-convcomb-qualification","seed":202610245003,"semantic_template":"qf-mod24-convcomb-qualification/{\"vectors\":[[-10,0],[-8,2]],\"weights\":[[3,5],[2,5]]}"},"fields":{"title":"Nonnegativity and Exact Total One as Separate Weight Tests","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[-10, 0], [-8, 2]], weights=[3/5, 2/5]. This family admits unqualified candidates for inspection. Give (vectors,normalized weights,all nonnegative?,exact sum,sum equals 1?,both conditions?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[-10, 0], [-8, 2]], [3/5, 2/5], Yes, 1, Yes, Yes)","explanation":"Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-10, 0], [-8, 2]], [3/5, 2/5], Yes, 1, Yes, Yes)."},"skills":["Nonnegativity and Exact Total One as Separate Weight Tests"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0004","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9ab555d51e5d90624def87cf8fd952cf428b2df7f8a9952e5168edf85d7569c","family_id":"qf-mod24-convcomb-qualification","seed":202610245004,"semantic_template":"qf-mod24-convcomb-qualification/{\"vectors\":[[-9,0],[-7,2]],\"weights\":[[4,6],[2,6]]}"},"fields":{"title":"Nonnegativity and Exact Total One as Separate Weight Tests","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[-9, 0], [-7, 2]], weights=[4/6, 2/6]. This family admits unqualified candidates for inspection. Give (vectors,normalized weights,all nonnegative?,exact sum,sum equals 1?,both conditions?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[-9, 0], [-7, 2]], [2/3, 1/3], Yes, 1, Yes, Yes)","explanation":"Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-9, 0], [-7, 2]], [2/3, 1/3], Yes, 1, Yes, Yes)."},"skills":["Nonnegativity and Exact Total One as Separate Weight Tests"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0005","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0bef4ef649754451eaaec74c8c856dbf6385f58423e0dfac91969c6db34681a","family_id":"qf-mod24-convcomb-qualification","seed":202610245005,"semantic_template":"qf-mod24-convcomb-qualification/{\"vectors\":[[-8,0],[-6,2]],\"weights\":[[5,7],[2,7]]}"},"fields":{"title":"Nonnegativity and Exact Total One as Separate Weight Tests","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[-8, 0], [-6, 2]], weights=[5/7, 2/7]. This family admits unqualified candidates for inspection. Give (vectors,normalized weights,all nonnegative?,exact sum,sum equals 1?,both conditions?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[-8, 0], [-6, 2]], [5/7, 2/7], Yes, 1, Yes, Yes)","explanation":"Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-8, 0], [-6, 2]], [5/7, 2/7], Yes, 1, Yes, Yes)."},"skills":["Nonnegativity and Exact Total One as Separate Weight Tests"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0006","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a6c6977634d0fabda29937cd4942ee364e5ecce1df708c1b366812d6bbe33dd","family_id":"qf-mod24-convcomb-qualification","seed":202610245006,"semantic_template":"qf-mod24-convcomb-qualification/{\"vectors\":[[-7,0],[-5,2]],\"weights\":[[2,3],[2,3]]}"},"fields":{"title":"Nonnegativity and Exact Total One as Separate Weight Tests","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[-7, 0], [-5, 2]], weights=[2/3, 2/3]. This family admits unqualified candidates for inspection. Give (vectors,normalized weights,all nonnegative?,exact sum,sum equals 1?,both conditions?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[-7, 0], [-5, 2]], [2/3, 2/3], Yes, 4/3, No, No)","explanation":"Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-7, 0], [-5, 2]], [2/3, 2/3], Yes, 4/3, No, No)."},"skills":["Nonnegativity and Exact Total One as Separate Weight Tests"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0007","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8757468d7a88bbb6e2a559fc50ef7813516ddfb01b39101f5e385256ab7030a","family_id":"qf-mod24-convcomb-qualification","seed":202610245007,"semantic_template":"qf-mod24-convcomb-qualification/{\"vectors\":[[-6,0],[-4,2]],\"weights\":[[7,9],[2,9]]}"},"fields":{"title":"Nonnegativity and Exact Total One as Separate Weight Tests","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[-6, 0], [-4, 2]], weights=[7/9, 2/9]. This family admits unqualified candidates for inspection. Give (vectors,normalized weights,all nonnegative?,exact sum,sum equals 1?,both conditions?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[-6, 0], [-4, 2]], [7/9, 2/9], Yes, 1, Yes, Yes)","explanation":"Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-6, 0], [-4, 2]], [7/9, 2/9], Yes, 1, Yes, Yes)."},"skills":["Nonnegativity and Exact Total One as Separate Weight Tests"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0008","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d7b52d56177fef33cb09c9f84921b06fd3184402f74f04b8826ac00c727c017","family_id":"qf-mod24-convcomb-qualification","seed":202610245008,"semantic_template":"qf-mod24-convcomb-qualification/{\"vectors\":[[-5,0],[-3,2]],\"weights\":[[8,10],[2,10]]}"},"fields":{"title":"Nonnegativity and Exact Total One as Separate Weight Tests","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[-5, 0], [-3, 2]], weights=[8/10, 2/10]. This family admits unqualified candidates for inspection. Give (vectors,normalized weights,all nonnegative?,exact sum,sum equals 1?,both conditions?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[-5, 0], [-3, 2]], [4/5, 1/5], Yes, 1, Yes, Yes)","explanation":"Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-5, 0], [-3, 2]], [4/5, 1/5], Yes, 1, Yes, Yes)."},"skills":["Nonnegativity and Exact Total One as Separate Weight Tests"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0009","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76bdf9a2d3897354a3348a34f372b2a5b0f2acea6452ba33d8156bb1c837f431","family_id":"qf-mod24-convcomb-qualification","seed":202610245009,"semantic_template":"qf-mod24-convcomb-qualification/{\"vectors\":[[-4,0],[-2,2]],\"weights\":[[9,11],[2,11]]}"},"fields":{"title":"Nonnegativity and Exact Total One as Separate Weight Tests","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[-4, 0], [-2, 2]], weights=[9/11, 2/11]. This family admits unqualified candidates for inspection. Give (vectors,normalized weights,all nonnegative?,exact sum,sum equals 1?,both conditions?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[-4, 0], [-2, 2]], [9/11, 2/11], Yes, 1, Yes, Yes)","explanation":"Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-4, 0], [-2, 2]], [9/11, 2/11], Yes, 1, Yes, Yes)."},"skills":["Nonnegativity and Exact Total One as Separate Weight Tests"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0010","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60a0f8b5af687c8507274105a94f81c16090b0871f471800e7ef496ec69b1256","family_id":"qf-mod24-convcomb-qualification","seed":202610245010,"semantic_template":"qf-mod24-convcomb-qualification/{\"vectors\":[[-3,0],[-1,2]],\"weights\":[[10,12],[2,12]]}"},"fields":{"title":"Nonnegativity and Exact Total One as Separate Weight Tests","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[-3, 0], [-1, 2]], weights=[10/12, 2/12]. This family admits unqualified candidates for inspection. Give (vectors,normalized weights,all nonnegative?,exact sum,sum equals 1?,both conditions?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[-3, 0], [-1, 2]], [5/6, 1/6], Yes, 1, Yes, Yes)","explanation":"Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-3, 0], [-1, 2]], [5/6, 1/6], Yes, 1, Yes, Yes)."},"skills":["Nonnegativity and Exact Total One as Separate Weight Tests"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0011","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52270b9be3cc134796cf93f5c3619b07bebe488a8d5731a2fe91f06f95b95a50","family_id":"qf-mod24-convcomb-qualification","seed":202610245011,"semantic_template":"qf-mod24-convcomb-qualification/{\"vectors\":[[-2,0],[0,2]],\"weights\":[[2,3],[2,3]]}"},"fields":{"title":"Nonnegativity and Exact Total One as Separate Weight Tests","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[-2, 0], [0, 2]], weights=[2/3, 2/3]. This family admits unqualified candidates for inspection. Give (vectors,normalized weights,all nonnegative?,exact sum,sum equals 1?,both conditions?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[-2, 0], [0, 2]], [2/3, 2/3], Yes, 4/3, No, No)","explanation":"Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-2, 0], [0, 2]], [2/3, 2/3], Yes, 4/3, No, No)."},"skills":["Nonnegativity and Exact Total One as Separate Weight Tests"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0012","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d23496b698a69d2fd096c609737a63fe27d78d4bc25c1ea3696968b764fba63","family_id":"qf-mod24-convcomb-qualification","seed":202610245012,"semantic_template":"qf-mod24-convcomb-qualification/{\"vectors\":[[-1,0],[1,2]],\"weights\":[[12,14],[2,14]]}"},"fields":{"title":"Nonnegativity and Exact Total One as Separate Weight Tests","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[-1, 0], [1, 2]], weights=[12/14, 2/14]. This family admits unqualified candidates for inspection. Give (vectors,normalized weights,all nonnegative?,exact sum,sum equals 1?,both conditions?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[-1, 0], [1, 2]], [6/7, 1/7], Yes, 1, Yes, Yes)","explanation":"Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-1, 0], [1, 2]], [6/7, 1/7], Yes, 1, Yes, Yes)."},"skills":["Nonnegativity and Exact Total One as Separate Weight Tests"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0013","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58e0fa61bd45e48823003c02e58300095405ec224cbf3b8bf5155cae4c2fceee","family_id":"qf-mod24-convcomb-qualification","seed":202610245013,"semantic_template":"qf-mod24-convcomb-qualification/{\"vectors\":[[0,0],[2,2]],\"weights\":[[13,15],[2,15]]}"},"fields":{"title":"Nonnegativity and Exact Total One as Separate Weight Tests","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[0, 0], [2, 2]], weights=[13/15, 2/15]. This family admits unqualified candidates for inspection. Give (vectors,normalized weights,all nonnegative?,exact sum,sum equals 1?,both conditions?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [2, 2]], [13/15, 2/15], Yes, 1, Yes, Yes)","explanation":"Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [2, 2]], [13/15, 2/15], Yes, 1, Yes, Yes)."},"skills":["Nonnegativity and Exact Total One as Separate Weight Tests"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0014","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b31fc4b7a5828b08608253b71e1c15795b75da63fa14c79f6eab233c3d0e8691","family_id":"qf-mod24-convcomb-qualification","seed":202610245014,"semantic_template":"qf-mod24-convcomb-qualification/{\"vectors\":[[1,0],[3,2]],\"weights\":[[14,16],[2,16]]}"},"fields":{"title":"Nonnegativity and Exact Total One as Separate Weight Tests","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[1, 0], [3, 2]], weights=[14/16, 2/16]. This family admits unqualified candidates for inspection. Give (vectors,normalized weights,all nonnegative?,exact sum,sum equals 1?,both conditions?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[1, 0], [3, 2]], [7/8, 1/8], Yes, 1, Yes, Yes)","explanation":"Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [3, 2]], [7/8, 1/8], Yes, 1, Yes, Yes)."},"skills":["Nonnegativity and Exact Total One as Separate Weight Tests"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0015","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f6cea0d6cc278e77072e7659fc2aea7e28e77e109b8b8db0e03e0167df401cb","family_id":"qf-mod24-convcomb-qualification","seed":202610245015,"semantic_template":"qf-mod24-convcomb-qualification/{\"vectors\":[[2,0],[4,2]],\"weights\":[[15,17],[2,17]]}"},"fields":{"title":"Nonnegativity and Exact Total One as Separate Weight Tests","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[2, 0], [4, 2]], weights=[15/17, 2/17]. This family admits unqualified candidates for inspection. Give (vectors,normalized weights,all nonnegative?,exact sum,sum equals 1?,both conditions?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[2, 0], [4, 2]], [15/17, 2/17], Yes, 1, Yes, Yes)","explanation":"Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[2, 0], [4, 2]], [15/17, 2/17], Yes, 1, Yes, Yes)."},"skills":["Nonnegativity and Exact Total One as Separate Weight Tests"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0016","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f9d58125a45c858f8e41732ee21a290bae1b39677ff0b445f5752b171c2a334","family_id":"qf-mod24-convcomb-qualification","seed":202610245016,"semantic_template":"qf-mod24-convcomb-qualification/{\"vectors\":[[3,0],[5,2]],\"weights\":[[2,3],[2,3]]}"},"fields":{"title":"Nonnegativity and Exact Total One as Separate Weight Tests","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[3, 0], [5, 2]], weights=[2/3, 2/3]. This family admits unqualified candidates for inspection. Give (vectors,normalized weights,all nonnegative?,exact sum,sum equals 1?,both conditions?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[3, 0], [5, 2]], [2/3, 2/3], Yes, 4/3, No, No)","explanation":"Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[3, 0], [5, 2]], [2/3, 2/3], Yes, 4/3, No, No)."},"skills":["Nonnegativity and Exact Total One as Separate Weight Tests"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0017","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a90d5fc95af9a179391f1f8aaed2a90905d92d5a27e075a0028c5afd80a95394","family_id":"qf-mod24-convcomb-qualification","seed":202610245017,"semantic_template":"qf-mod24-convcomb-qualification/{\"vectors\":[[4,0],[6,2]],\"weights\":[[17,19],[2,19]]}"},"fields":{"title":"Nonnegativity and Exact Total One as Separate Weight Tests","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[4, 0], [6, 2]], weights=[17/19, 2/19]. This family admits unqualified candidates for inspection. Give (vectors,normalized weights,all nonnegative?,exact sum,sum equals 1?,both conditions?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[4, 0], [6, 2]], [17/19, 2/19], Yes, 1, Yes, Yes)","explanation":"Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[4, 0], [6, 2]], [17/19, 2/19], Yes, 1, Yes, Yes)."},"skills":["Nonnegativity and Exact Total One as Separate Weight Tests"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0018","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92ce19d6a503ea56d37adb78db24bcd9536e3c84349a049bd5c7ffd49b617e00","family_id":"qf-mod24-convcomb-qualification","seed":202610245018,"semantic_template":"qf-mod24-convcomb-qualification/{\"vectors\":[[5,0],[7,2]],\"weights\":[[18,20],[2,20]]}"},"fields":{"title":"Nonnegativity and Exact Total One as Separate Weight Tests","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[5, 0], [7, 2]], weights=[18/20, 2/20]. This family admits unqualified candidates for inspection. Give (vectors,normalized weights,all nonnegative?,exact sum,sum equals 1?,both conditions?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[5, 0], [7, 2]], [9/10, 1/10], Yes, 1, Yes, Yes)","explanation":"Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[5, 0], [7, 2]], [9/10, 1/10], Yes, 1, Yes, Yes)."},"skills":["Nonnegativity and Exact Total One as Separate Weight Tests"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0019","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ee1f7a0bc88281a8d2b6296bcd07ca721b315d0e0a37d2103358f74f73c8aea","family_id":"qf-mod24-convcomb-qualification","seed":202610245019,"semantic_template":"qf-mod24-convcomb-qualification/{\"vectors\":[[6,0],[8,2]],\"weights\":[[19,21],[2,21]]}"},"fields":{"title":"Nonnegativity and Exact Total One as Separate Weight Tests","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[6, 0], [8, 2]], weights=[19/21, 2/21]. This family admits unqualified candidates for inspection. Give (vectors,normalized weights,all nonnegative?,exact sum,sum equals 1?,both conditions?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[6, 0], [8, 2]], [19/21, 2/21], Yes, 1, Yes, Yes)","explanation":"Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[6, 0], [8, 2]], [19/21, 2/21], Yes, 1, Yes, Yes)."},"skills":["Nonnegativity and Exact Total One as Separate Weight Tests"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0020","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7b6b18c2c5f69e5d6301302ae4cde054868088f5f86d9510ef262fdeb0dbe3a","family_id":"qf-mod24-convcomb-qualification","seed":202610245020,"semantic_template":"qf-mod24-convcomb-qualification/{\"vectors\":[[7,0],[9,2]],\"weights\":[[20,22],[2,22]]}"},"fields":{"title":"Nonnegativity and Exact Total One as Separate Weight Tests","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[7, 0], [9, 2]], weights=[20/22, 2/22]. This family admits unqualified candidates for inspection. Give (vectors,normalized weights,all nonnegative?,exact sum,sum equals 1?,both conditions?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[7, 0], [9, 2]], [10/11, 1/11], Yes, 1, Yes, Yes)","explanation":"Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[7, 0], [9, 2]], [10/11, 1/11], Yes, 1, Yes, Yes)."},"skills":["Nonnegativity and Exact Total One as Separate Weight Tests"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0021","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6503253051d63768d173ba246eec6a33072b8e29504dd1fd99ae63ee4a6b55a","family_id":"qf-mod24-convcomb-qualification","seed":202610245021,"semantic_template":"qf-mod24-convcomb-qualification/{\"vectors\":[[8,0],[10,2]],\"weights\":[[2,3],[2,3]]}"},"fields":{"title":"Nonnegativity and Exact Total One as Separate Weight Tests","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[8, 0], [10, 2]], weights=[2/3, 2/3]. This family admits unqualified candidates for inspection. Give (vectors,normalized weights,all nonnegative?,exact sum,sum equals 1?,both conditions?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[8, 0], [10, 2]], [2/3, 2/3], Yes, 4/3, No, No)","explanation":"Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[8, 0], [10, 2]], [2/3, 2/3], Yes, 4/3, No, No)."},"skills":["Nonnegativity and Exact Total One as Separate Weight Tests"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0022","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0cfcaf8d77a21966761ee397ba1d8a99cbc20a23912b8a0e7346c612317cf2fa","family_id":"qf-mod24-convcomb-qualification","seed":202610245022,"semantic_template":"qf-mod24-convcomb-qualification/{\"vectors\":[[9,0],[11,2]],\"weights\":[[22,24],[2,24]]}"},"fields":{"title":"Nonnegativity and Exact Total One as Separate Weight Tests","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[9, 0], [11, 2]], weights=[22/24, 2/24]. This family admits unqualified candidates for inspection. Give (vectors,normalized weights,all nonnegative?,exact sum,sum equals 1?,both conditions?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[9, 0], [11, 2]], [11/12, 1/12], Yes, 1, Yes, Yes)","explanation":"Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[9, 0], [11, 2]], [11/12, 1/12], Yes, 1, Yes, Yes)."},"skills":["Nonnegativity and Exact Total One as Separate Weight Tests"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0023","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9d30ed3b0caeb086606b49450b32759922b324dd55294a024d9b6ea6e0298fd","family_id":"qf-mod24-convcomb-qualification","seed":202610245023,"semantic_template":"qf-mod24-convcomb-qualification/{\"vectors\":[[10,0],[12,2]],\"weights\":[[23,25],[2,25]]}"},"fields":{"title":"Nonnegativity and Exact Total One as Separate Weight Tests","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[10, 0], [12, 2]], weights=[23/25, 2/25]. This family admits unqualified candidates for inspection. Give (vectors,normalized weights,all nonnegative?,exact sum,sum equals 1?,both conditions?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[10, 0], [12, 2]], [23/25, 2/25], Yes, 1, Yes, Yes)","explanation":"Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[10, 0], [12, 2]], [23/25, 2/25], Yes, 1, Yes, Yes)."},"skills":["Nonnegativity and Exact Total One as Separate Weight Tests"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0024","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54984b7a8f8ebabfbc1d6b93dafe398c0be79d6db35dfcde61ca6730427240bb","family_id":"qf-mod24-convcomb-qualification","seed":202610245024,"semantic_template":"qf-mod24-convcomb-qualification/{\"vectors\":[[11,0],[13,2]],\"weights\":[[24,26],[2,26]]}"},"fields":{"title":"Nonnegativity and Exact Total One as Separate Weight Tests","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[11, 0], [13, 2]], weights=[24/26, 2/26]. This family admits unqualified candidates for inspection. Give (vectors,normalized weights,all nonnegative?,exact sum,sum equals 1?,both conditions?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[11, 0], [13, 2]], [12/13, 1/13], Yes, 1, Yes, Yes)","explanation":"Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[11, 0], [13, 2]], [12/13, 1/13], Yes, 1, Yes, Yes)."},"skills":["Nonnegativity and Exact Total One as Separate Weight Tests"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0025","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57ad4b40219044f5fb4adcc3ff9ab320c798f43f625197a56d36e82eb1cff4b0","family_id":"qf-mod24-convcomb-qualification","seed":202610245025,"semantic_template":"qf-mod24-convcomb-qualification/{\"vectors\":[[12,0],[14,2]],\"weights\":[[25,27],[2,27]]}"},"fields":{"title":"Nonnegativity and Exact Total One as Separate Weight Tests","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[12, 0], [14, 2]], weights=[25/27, 2/27]. This family admits unqualified candidates for inspection. Give (vectors,normalized weights,all nonnegative?,exact sum,sum equals 1?,both conditions?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[12, 0], [14, 2]], [25/27, 2/27], Yes, 1, Yes, Yes)","explanation":"Test both weight conditions independently without clipping negatives or rounding a near-one total to one. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[12, 0], [14, 2]], [25/27, 2/27], Yes, 1, Yes, Yes)."},"skills":["Nonnegativity and Exact Total One as Separate Weight Tests"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0026","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a85a4df74510edc79273e65468c555783c9bd37c679df380551eaeb0e78a4f90","family_id":"qf-mod24-convcomb-weighted-point","seed":202610245026,"semantic_template":"qf-mod24-convcomb-weighted-point/{\"vectors\":[[-12,0],[-10,2]],\"weights\":[[1,3],[2,3]]}"},"fields":{"title":"An Exact Point from Qualified Convex Weights","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[-12, 0], [-10, 2]], weights=[1/3, 2/3]. Require qualified weights. Give (original vectors,normalized aligned weights,weighted point). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[-12, 0], [-10, 2]], [1/3, 2/3], [-32/3, 4/3])","explanation":"Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-12, 0], [-10, 2]], [1/3, 2/3], [-32/3, 4/3])."},"skills":["An Exact Point from Qualified Convex Weights"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0027","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af9fcc05775ce076bb25cf60b6aacdcd867c34945798487a70b7b427d4ec2115","family_id":"qf-mod24-convcomb-weighted-point","seed":202610245027,"semantic_template":"qf-mod24-convcomb-weighted-point/{\"vectors\":[[-11,0],[-9,2]],\"weights\":[[2,4],[2,4]]}"},"fields":{"title":"An Exact Point from Qualified Convex Weights","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[-11, 0], [-9, 2]], weights=[2/4, 2/4]. Require qualified weights. Give (original vectors,normalized aligned weights,weighted point). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[-11, 0], [-9, 2]], [1/2, 1/2], [-10, 1])","explanation":"Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-11, 0], [-9, 2]], [1/2, 1/2], [-10, 1])."},"skills":["An Exact Point from Qualified Convex Weights"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0028","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ac89b6e0c3954f8a6c464a0f19f1817ae688bd730a161de5d23144548fbddc4","family_id":"qf-mod24-convcomb-weighted-point","seed":202610245028,"semantic_template":"qf-mod24-convcomb-weighted-point/{\"vectors\":[[-10,0],[-8,2]],\"weights\":[[3,5],[2,5]]}"},"fields":{"title":"An Exact Point from Qualified Convex Weights","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[-10, 0], [-8, 2]], weights=[3/5, 2/5]. Require qualified weights. Give (original vectors,normalized aligned weights,weighted point). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[-10, 0], [-8, 2]], [3/5, 2/5], [-46/5, 4/5])","explanation":"Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-10, 0], [-8, 2]], [3/5, 2/5], [-46/5, 4/5])."},"skills":["An Exact Point from Qualified Convex Weights"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0029","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3639812ddc3d50e20b7a27f8f73832569324e845087095656716c4c8db63a968","family_id":"qf-mod24-convcomb-weighted-point","seed":202610245029,"semantic_template":"qf-mod24-convcomb-weighted-point/{\"vectors\":[[-9,0],[-7,2]],\"weights\":[[4,6],[2,6]]}"},"fields":{"title":"An Exact Point from Qualified Convex Weights","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[-9, 0], [-7, 2]], weights=[4/6, 2/6]. Require qualified weights. Give (original vectors,normalized aligned weights,weighted point). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[-9, 0], [-7, 2]], [2/3, 1/3], [-25/3, 2/3])","explanation":"Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-9, 0], [-7, 2]], [2/3, 1/3], [-25/3, 2/3])."},"skills":["An Exact Point from Qualified Convex Weights"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0030","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5499f9dcd25a0ab25dfe6eeb5db90281e342e53a63c0972a3b1bb4ceb32ca0eb","family_id":"qf-mod24-convcomb-weighted-point","seed":202610245030,"semantic_template":"qf-mod24-convcomb-weighted-point/{\"vectors\":[[-8,0],[-6,2]],\"weights\":[[5,7],[2,7]]}"},"fields":{"title":"An Exact Point from Qualified Convex Weights","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[-8, 0], [-6, 2]], weights=[5/7, 2/7]. Require qualified weights. Give (original vectors,normalized aligned weights,weighted point). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[-8, 0], [-6, 2]], [5/7, 2/7], [-52/7, 4/7])","explanation":"Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-8, 0], [-6, 2]], [5/7, 2/7], [-52/7, 4/7])."},"skills":["An Exact Point from Qualified Convex Weights"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0031","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0fe52ff16ca88503cf79addb2eb7e632a97c0d227441abedd50ccdcedb87d828","family_id":"qf-mod24-convcomb-weighted-point","seed":202610245031,"semantic_template":"qf-mod24-convcomb-weighted-point/{\"vectors\":[[-7,0],[-5,2]],\"weights\":[[6,8],[2,8]]}"},"fields":{"title":"An Exact Point from Qualified Convex Weights","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[-7, 0], [-5, 2]], weights=[6/8, 2/8]. Require qualified weights. Give (original vectors,normalized aligned weights,weighted point). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[-7, 0], [-5, 2]], [3/4, 1/4], [-13/2, 1/2])","explanation":"Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-7, 0], [-5, 2]], [3/4, 1/4], [-13/2, 1/2])."},"skills":["An Exact Point from Qualified Convex Weights"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0032","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d611dcd2eec2474255f8610fbe71748f968e19b7c0221dec8d1460464bcae05","family_id":"qf-mod24-convcomb-weighted-point","seed":202610245032,"semantic_template":"qf-mod24-convcomb-weighted-point/{\"vectors\":[[-6,0],[-4,2]],\"weights\":[[7,9],[2,9]]}"},"fields":{"title":"An Exact Point from Qualified Convex Weights","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[-6, 0], [-4, 2]], weights=[7/9, 2/9]. Require qualified weights. Give (original vectors,normalized aligned weights,weighted point). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[-6, 0], [-4, 2]], [7/9, 2/9], [-50/9, 4/9])","explanation":"Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-6, 0], [-4, 2]], [7/9, 2/9], [-50/9, 4/9])."},"skills":["An Exact Point from Qualified Convex Weights"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0033","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"234b13537d060b566dbaaed947a7d1e190a55519f43f91d46e57e960e4be5003","family_id":"qf-mod24-convcomb-weighted-point","seed":202610245033,"semantic_template":"qf-mod24-convcomb-weighted-point/{\"vectors\":[[-5,0],[-3,2]],\"weights\":[[8,10],[2,10]]}"},"fields":{"title":"An Exact Point from Qualified Convex Weights","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[-5, 0], [-3, 2]], weights=[8/10, 2/10]. Require qualified weights. Give (original vectors,normalized aligned weights,weighted point). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[-5, 0], [-3, 2]], [4/5, 1/5], [-23/5, 2/5])","explanation":"Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-5, 0], [-3, 2]], [4/5, 1/5], [-23/5, 2/5])."},"skills":["An Exact Point from Qualified Convex Weights"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0034","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6327c63c4e7f6e8e8dd4037e30360ac64dd4a896d85c4bbedbebfde016a6cb6","family_id":"qf-mod24-convcomb-weighted-point","seed":202610245034,"semantic_template":"qf-mod24-convcomb-weighted-point/{\"vectors\":[[-4,0],[-2,2]],\"weights\":[[9,11],[2,11]]}"},"fields":{"title":"An Exact Point from Qualified Convex Weights","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[-4, 0], [-2, 2]], weights=[9/11, 2/11]. Require qualified weights. Give (original vectors,normalized aligned weights,weighted point). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[-4, 0], [-2, 2]], [9/11, 2/11], [-40/11, 4/11])","explanation":"Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-4, 0], [-2, 2]], [9/11, 2/11], [-40/11, 4/11])."},"skills":["An Exact Point from Qualified Convex Weights"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0035","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6c6231a6bf210016df43de5998b02fe446c6b4c10e43a898d5ffd854a80e4e0","family_id":"qf-mod24-convcomb-weighted-point","seed":202610245035,"semantic_template":"qf-mod24-convcomb-weighted-point/{\"vectors\":[[-3,0],[-1,2]],\"weights\":[[10,12],[2,12]]}"},"fields":{"title":"An Exact Point from Qualified Convex Weights","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[-3, 0], [-1, 2]], weights=[10/12, 2/12]. Require qualified weights. Give (original vectors,normalized aligned weights,weighted point). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[-3, 0], [-1, 2]], [5/6, 1/6], [-8/3, 1/3])","explanation":"Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-3, 0], [-1, 2]], [5/6, 1/6], [-8/3, 1/3])."},"skills":["An Exact Point from Qualified Convex Weights"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0036","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1bf2c3257060ede990eb91aeb1e10a3f39c7bcf4bf2761383a5cdb0ecc50d8e1","family_id":"qf-mod24-convcomb-weighted-point","seed":202610245036,"semantic_template":"qf-mod24-convcomb-weighted-point/{\"vectors\":[[-2,0],[0,2]],\"weights\":[[11,13],[2,13]]}"},"fields":{"title":"An Exact Point from Qualified Convex Weights","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[-2, 0], [0, 2]], weights=[11/13, 2/13]. Require qualified weights. Give (original vectors,normalized aligned weights,weighted point). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[-2, 0], [0, 2]], [11/13, 2/13], [-22/13, 4/13])","explanation":"Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-2, 0], [0, 2]], [11/13, 2/13], [-22/13, 4/13])."},"skills":["An Exact Point from Qualified Convex Weights"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0037","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce805674ac1357381e2794bcf78d40ecb30d524c82a4b739bbe6873a5e22e87d","family_id":"qf-mod24-convcomb-weighted-point","seed":202610245037,"semantic_template":"qf-mod24-convcomb-weighted-point/{\"vectors\":[[-1,0],[1,2]],\"weights\":[[12,14],[2,14]]}"},"fields":{"title":"An Exact Point from Qualified Convex Weights","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[-1, 0], [1, 2]], weights=[12/14, 2/14]. Require qualified weights. Give (original vectors,normalized aligned weights,weighted point). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[-1, 0], [1, 2]], [6/7, 1/7], [-5/7, 2/7])","explanation":"Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[-1, 0], [1, 2]], [6/7, 1/7], [-5/7, 2/7])."},"skills":["An Exact Point from Qualified Convex Weights"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0038","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d80bcdb144d8f5b3f274fabfefe85a268a19789ba58342105f58d4494b8e3ab8","family_id":"qf-mod24-convcomb-weighted-point","seed":202610245038,"semantic_template":"qf-mod24-convcomb-weighted-point/{\"vectors\":[[0,0],[2,2]],\"weights\":[[13,15],[2,15]]}"},"fields":{"title":"An Exact Point from Qualified Convex Weights","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[0, 0], [2, 2]], weights=[13/15, 2/15]. Require qualified weights. Give (original vectors,normalized aligned weights,weighted point). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [2, 2]], [13/15, 2/15], [4/15, 4/15])","explanation":"Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [2, 2]], [13/15, 2/15], [4/15, 4/15])."},"skills":["An Exact Point from Qualified Convex Weights"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0039","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5668959d27c477a74376e79ebf8f6647d2c013957a85ae6cb8989216ef045e74","family_id":"qf-mod24-convcomb-weighted-point","seed":202610245039,"semantic_template":"qf-mod24-convcomb-weighted-point/{\"vectors\":[[1,0],[3,2]],\"weights\":[[14,16],[2,16]]}"},"fields":{"title":"An Exact Point from Qualified Convex Weights","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[1, 0], [3, 2]], weights=[14/16, 2/16]. Require qualified weights. Give (original vectors,normalized aligned weights,weighted point). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[1, 0], [3, 2]], [7/8, 1/8], [5/4, 1/4])","explanation":"Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[1, 0], [3, 2]], [7/8, 1/8], [5/4, 1/4])."},"skills":["An Exact Point from Qualified Convex Weights"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0040","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"555e1efa79cf6534d7519a5a4e814e49903f7ff61232beefbddfe2e71a7d9637","family_id":"qf-mod24-convcomb-weighted-point","seed":202610245040,"semantic_template":"qf-mod24-convcomb-weighted-point/{\"vectors\":[[2,0],[4,2]],\"weights\":[[15,17],[2,17]]}"},"fields":{"title":"An Exact Point from Qualified Convex Weights","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[2, 0], [4, 2]], weights=[15/17, 2/17]. Require qualified weights. Give (original vectors,normalized aligned weights,weighted point). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[2, 0], [4, 2]], [15/17, 2/17], [38/17, 4/17])","explanation":"Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[2, 0], [4, 2]], [15/17, 2/17], [38/17, 4/17])."},"skills":["An Exact Point from Qualified Convex Weights"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0041","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"030bc70c3a4294912dee7273266f078f0f696ec28583b14a161e060075a9ae3f","family_id":"qf-mod24-convcomb-weighted-point","seed":202610245041,"semantic_template":"qf-mod24-convcomb-weighted-point/{\"vectors\":[[3,0],[5,2]],\"weights\":[[16,18],[2,18]]}"},"fields":{"title":"An Exact Point from Qualified Convex Weights","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[3, 0], [5, 2]], weights=[16/18, 2/18]. Require qualified weights. Give (original vectors,normalized aligned weights,weighted point). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[3, 0], [5, 2]], [8/9, 1/9], [29/9, 2/9])","explanation":"Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[3, 0], [5, 2]], [8/9, 1/9], [29/9, 2/9])."},"skills":["An Exact Point from Qualified Convex Weights"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0042","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd865761839ed19b97f40b5b9c217d3e55b14553c62b3a9a2f508b0af53afc14","family_id":"qf-mod24-convcomb-weighted-point","seed":202610245042,"semantic_template":"qf-mod24-convcomb-weighted-point/{\"vectors\":[[4,0],[6,2]],\"weights\":[[17,19],[2,19]]}"},"fields":{"title":"An Exact Point from Qualified Convex Weights","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[4, 0], [6, 2]], weights=[17/19, 2/19]. Require qualified weights. Give (original vectors,normalized aligned weights,weighted point). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[4, 0], [6, 2]], [17/19, 2/19], [80/19, 4/19])","explanation":"Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[4, 0], [6, 2]], [17/19, 2/19], [80/19, 4/19])."},"skills":["An Exact Point from Qualified Convex Weights"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0043","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8c344f420cd9d1744a81baa53a149453fe628901709d842679b70ca78c0ac56","family_id":"qf-mod24-convcomb-weighted-point","seed":202610245043,"semantic_template":"qf-mod24-convcomb-weighted-point/{\"vectors\":[[5,0],[7,2]],\"weights\":[[18,20],[2,20]]}"},"fields":{"title":"An Exact Point from Qualified Convex Weights","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[5, 0], [7, 2]], weights=[18/20, 2/20]. Require qualified weights. Give (original vectors,normalized aligned weights,weighted point). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[5, 0], [7, 2]], [9/10, 1/10], [26/5, 1/5])","explanation":"Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[5, 0], [7, 2]], [9/10, 1/10], [26/5, 1/5])."},"skills":["An Exact Point from Qualified Convex Weights"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0044","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f109efd0ae53a6f104358db9613979e83a691b07b7e979c869f2ef5be07fcb29","family_id":"qf-mod24-convcomb-weighted-point","seed":202610245044,"semantic_template":"qf-mod24-convcomb-weighted-point/{\"vectors\":[[6,0],[8,2]],\"weights\":[[19,21],[2,21]]}"},"fields":{"title":"An Exact Point from Qualified Convex Weights","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[6, 0], [8, 2]], weights=[19/21, 2/21]. Require qualified weights. Give (original vectors,normalized aligned weights,weighted point). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[6, 0], [8, 2]], [19/21, 2/21], [130/21, 4/21])","explanation":"Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[6, 0], [8, 2]], [19/21, 2/21], [130/21, 4/21])."},"skills":["An Exact Point from Qualified Convex Weights"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0045","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00aa5248fe4b672dfa4dfa65c5896b4fd072bab7cc6c3e42bde8605499b13db5","family_id":"qf-mod24-convcomb-weighted-point","seed":202610245045,"semantic_template":"qf-mod24-convcomb-weighted-point/{\"vectors\":[[7,0],[9,2]],\"weights\":[[20,22],[2,22]]}"},"fields":{"title":"An Exact Point from Qualified Convex Weights","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[7, 0], [9, 2]], weights=[20/22, 2/22]. Require qualified weights. Give (original vectors,normalized aligned weights,weighted point). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[7, 0], [9, 2]], [10/11, 1/11], [79/11, 2/11])","explanation":"Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[7, 0], [9, 2]], [10/11, 1/11], [79/11, 2/11])."},"skills":["An Exact Point from Qualified Convex Weights"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0046","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"418b33859902fcb5ce7d2625b7945f729251c12ded800ad7604825fdb3f22adf","family_id":"qf-mod24-convcomb-weighted-point","seed":202610245046,"semantic_template":"qf-mod24-convcomb-weighted-point/{\"vectors\":[[8,0],[10,2]],\"weights\":[[21,23],[2,23]]}"},"fields":{"title":"An Exact Point from Qualified Convex Weights","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[8, 0], [10, 2]], weights=[21/23, 2/23]. Require qualified weights. Give (original vectors,normalized aligned weights,weighted point). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[8, 0], [10, 2]], [21/23, 2/23], [188/23, 4/23])","explanation":"Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[8, 0], [10, 2]], [21/23, 2/23], [188/23, 4/23])."},"skills":["An Exact Point from Qualified Convex Weights"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0047","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"559debdf2f800ef7a3da02f20c5a3312475f89171ed831d1c65c3f3dabad41d2","family_id":"qf-mod24-convcomb-weighted-point","seed":202610245047,"semantic_template":"qf-mod24-convcomb-weighted-point/{\"vectors\":[[9,0],[11,2]],\"weights\":[[22,24],[2,24]]}"},"fields":{"title":"An Exact Point from Qualified Convex Weights","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[9, 0], [11, 2]], weights=[22/24, 2/24]. Require qualified weights. Give (original vectors,normalized aligned weights,weighted point). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[9, 0], [11, 2]], [11/12, 1/12], [55/6, 1/6])","explanation":"Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[9, 0], [11, 2]], [11/12, 1/12], [55/6, 1/6])."},"skills":["An Exact Point from Qualified Convex Weights"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0048","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc91f7bf618256fed87327f6e6548029a4c965ac3a13b35b7450425f225336e1","family_id":"qf-mod24-convcomb-weighted-point","seed":202610245048,"semantic_template":"qf-mod24-convcomb-weighted-point/{\"vectors\":[[10,0],[12,2]],\"weights\":[[23,25],[2,25]]}"},"fields":{"title":"An Exact Point from Qualified Convex Weights","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[10, 0], [12, 2]], weights=[23/25, 2/25]. Require qualified weights. Give (original vectors,normalized aligned weights,weighted point). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[10, 0], [12, 2]], [23/25, 2/25], [254/25, 4/25])","explanation":"Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[10, 0], [12, 2]], [23/25, 2/25], [254/25, 4/25])."},"skills":["An Exact Point from Qualified Convex Weights"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0049","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f69976107a0003afbcae298b20d7f4dfa4d75a7bd3e671443b4d19d548c376f2","family_id":"qf-mod24-convcomb-weighted-point","seed":202610245049,"semantic_template":"qf-mod24-convcomb-weighted-point/{\"vectors\":[[11,0],[13,2]],\"weights\":[[24,26],[2,26]]}"},"fields":{"title":"An Exact Point from Qualified Convex Weights","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[11, 0], [13, 2]], weights=[24/26, 2/26]. Require qualified weights. Give (original vectors,normalized aligned weights,weighted point). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[11, 0], [13, 2]], [12/13, 1/13], [145/13, 2/13])","explanation":"Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[11, 0], [13, 2]], [12/13, 1/13], [145/13, 2/13])."},"skills":["An Exact Point from Qualified Convex Weights"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0050","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"193f1efb8af3f7d6932ad48ccab2b17b0e596d32ddd72263344222398c0f4ab1","family_id":"qf-mod24-convcomb-weighted-point","seed":202610245050,"semantic_template":"qf-mod24-convcomb-weighted-point/{\"vectors\":[[12,0],[14,2]],\"weights\":[[25,27],[2,27]]}"},"fields":{"title":"An Exact Point from Qualified Convex Weights","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[12, 0], [14, 2]], weights=[25/27, 2/27]. Require qualified weights. Give (original vectors,normalized aligned weights,weighted point). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[12, 0], [14, 2]], [25/27, 2/27], [328/27, 4/27])","explanation":"Retain zero weights, repeated vectors and original pairing when summing coordinates; no coordinate-uniqueness claim follows. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[12, 0], [14, 2]], [25/27, 2/27], [328/27, 4/27])."},"skills":["An Exact Point from Qualified Convex Weights"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0051","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa88ab92b4123721651875425a94fb973d5aebe2aa3c1c6574ad4f37c7cdbc08","family_id":"qf-mod24-convcomb-two-point","seed":202610245051,"semantic_template":"qf-mod24-convcomb-two-point/{\"lambda\":[1,3],\"left\":[-12,0],\"right\":[-9,3]}"},"fields":{"title":"Audit lambda without Clipping an Affine Extension","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. left=[-12, 0], right=[-9, 3], lambda=1/3. Give (left,right,normalized lambda,lambda in [0,1]?,lambda*left+(1−lambda)*right,is this a convex combination?). This audit still computes the affine expression when lambda lies outside [0,1], but both qualification flags must then be No. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([-12, 0], [-9, 3], 1/3, Yes, [-10, 2], Yes)","explanation":"Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-12, 0], [-9, 3], 1/3, Yes, [-10, 2], Yes)."},"skills":["Audit lambda without Clipping an Affine Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0052","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68166bc094dd5b119505759fd0634a5217a4ac4a62d6c3130c42fb3c7461ab1b","family_id":"qf-mod24-convcomb-two-point","seed":202610245052,"semantic_template":"qf-mod24-convcomb-two-point/{\"lambda\":[2,4],\"left\":[-11,0],\"right\":[-8,3]}"},"fields":{"title":"Audit lambda without Clipping an Affine Extension","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. left=[-11, 0], right=[-8, 3], lambda=2/4. Give (left,right,normalized lambda,lambda in [0,1]?,lambda*left+(1−lambda)*right,is this a convex combination?). This audit still computes the affine expression when lambda lies outside [0,1], but both qualification flags must then be No. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([-11, 0], [-8, 3], 1/2, Yes, [-19/2, 3/2], Yes)","explanation":"Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-11, 0], [-8, 3], 1/2, Yes, [-19/2, 3/2], Yes)."},"skills":["Audit lambda without Clipping an Affine Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0053","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"683f63c255720aad8f6172e9d0974673b6a4d26cd05dbd3502ba8e886f6f3eeb","family_id":"qf-mod24-convcomb-two-point","seed":202610245053,"semantic_template":"qf-mod24-convcomb-two-point/{\"lambda\":[3,5],\"left\":[-10,0],\"right\":[-7,3]}"},"fields":{"title":"Audit lambda without Clipping an Affine Extension","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. left=[-10, 0], right=[-7, 3], lambda=3/5. Give (left,right,normalized lambda,lambda in [0,1]?,lambda*left+(1−lambda)*right,is this a convex combination?). This audit still computes the affine expression when lambda lies outside [0,1], but both qualification flags must then be No. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([-10, 0], [-7, 3], 3/5, Yes, [-44/5, 6/5], Yes)","explanation":"Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 0], [-7, 3], 3/5, Yes, [-44/5, 6/5], Yes)."},"skills":["Audit lambda without Clipping an Affine Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0054","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"737fe6374fa7b83de030de3e7373ecdb5a1d0b66cf6b8b0d2659bdab6a6891af","family_id":"qf-mod24-convcomb-two-point","seed":202610245054,"semantic_template":"qf-mod24-convcomb-two-point/{\"lambda\":[4,6],\"left\":[-9,0],\"right\":[-6,3]}"},"fields":{"title":"Audit lambda without Clipping an Affine Extension","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. left=[-9, 0], right=[-6, 3], lambda=4/6. Give (left,right,normalized lambda,lambda in [0,1]?,lambda*left+(1−lambda)*right,is this a convex combination?). This audit still computes the affine expression when lambda lies outside [0,1], but both qualification flags must then be No. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([-9, 0], [-6, 3], 2/3, Yes, [-8, 1], Yes)","explanation":"Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 0], [-6, 3], 2/3, Yes, [-8, 1], Yes)."},"skills":["Audit lambda without Clipping an Affine Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0055","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72f076a83f6934d9da29c483c7cc87df3dcc6f29221c51d76f8c6d3c8bb89847","family_id":"qf-mod24-convcomb-two-point","seed":202610245055,"semantic_template":"qf-mod24-convcomb-two-point/{\"lambda\":[5,7],\"left\":[-8,0],\"right\":[-5,3]}"},"fields":{"title":"Audit lambda without Clipping an Affine Extension","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. left=[-8, 0], right=[-5, 3], lambda=5/7. Give (left,right,normalized lambda,lambda in [0,1]?,lambda*left+(1−lambda)*right,is this a convex combination?). This audit still computes the affine expression when lambda lies outside [0,1], but both qualification flags must then be No. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([-8, 0], [-5, 3], 5/7, Yes, [-50/7, 6/7], Yes)","explanation":"Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 0], [-5, 3], 5/7, Yes, [-50/7, 6/7], Yes)."},"skills":["Audit lambda without Clipping an Affine Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0056","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a0cbbb7c88304b0b1596ca9ab3d8665d17c65c8081305e924be76bf7c1d711da","family_id":"qf-mod24-convcomb-two-point","seed":202610245056,"semantic_template":"qf-mod24-convcomb-two-point/{\"lambda\":[6,8],\"left\":[-7,0],\"right\":[-4,3]}"},"fields":{"title":"Audit lambda without Clipping an Affine Extension","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. left=[-7, 0], right=[-4, 3], lambda=6/8. Give (left,right,normalized lambda,lambda in [0,1]?,lambda*left+(1−lambda)*right,is this a convex combination?). This audit still computes the affine expression when lambda lies outside [0,1], but both qualification flags must then be No. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([-7, 0], [-4, 3], 3/4, Yes, [-25/4, 3/4], Yes)","explanation":"Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-7, 0], [-4, 3], 3/4, Yes, [-25/4, 3/4], Yes)."},"skills":["Audit lambda without Clipping an Affine Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0057","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c92e5bbbec9449731327a5f9d52074f65cff3a4594c031c495992e4d1578523a","family_id":"qf-mod24-convcomb-two-point","seed":202610245057,"semantic_template":"qf-mod24-convcomb-two-point/{\"lambda\":[7,9],\"left\":[-6,0],\"right\":[-3,3]}"},"fields":{"title":"Audit lambda without Clipping an Affine Extension","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. left=[-6, 0], right=[-3, 3], lambda=7/9. Give (left,right,normalized lambda,lambda in [0,1]?,lambda*left+(1−lambda)*right,is this a convex combination?). This audit still computes the affine expression when lambda lies outside [0,1], but both qualification flags must then be No. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([-6, 0], [-3, 3], 7/9, Yes, [-16/3, 2/3], Yes)","explanation":"Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-6, 0], [-3, 3], 7/9, Yes, [-16/3, 2/3], Yes)."},"skills":["Audit lambda without Clipping an Affine Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0058","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"910f8b1bc4e66dbba85ff094751af67641f0856f7ad5bd362c4b6e5a91ce67a2","family_id":"qf-mod24-convcomb-two-point","seed":202610245058,"semantic_template":"qf-mod24-convcomb-two-point/{\"lambda\":[8,10],\"left\":[-5,0],\"right\":[-2,3]}"},"fields":{"title":"Audit lambda without Clipping an Affine Extension","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. left=[-5, 0], right=[-2, 3], lambda=8/10. Give (left,right,normalized lambda,lambda in [0,1]?,lambda*left+(1−lambda)*right,is this a convex combination?). This audit still computes the affine expression when lambda lies outside [0,1], but both qualification flags must then be No. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([-5, 0], [-2, 3], 4/5, Yes, [-22/5, 3/5], Yes)","explanation":"Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 0], [-2, 3], 4/5, Yes, [-22/5, 3/5], Yes)."},"skills":["Audit lambda without Clipping an Affine Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0059","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"493d53bc0f03f0c0c8481addab65b43620b97e1df634b258c116478b36cfc317","family_id":"qf-mod24-convcomb-two-point","seed":202610245059,"semantic_template":"qf-mod24-convcomb-two-point/{\"lambda\":[9,11],\"left\":[-4,0],\"right\":[-1,3]}"},"fields":{"title":"Audit lambda without Clipping an Affine Extension","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. left=[-4, 0], right=[-1, 3], lambda=9/11. Give (left,right,normalized lambda,lambda in [0,1]?,lambda*left+(1−lambda)*right,is this a convex combination?). This audit still computes the affine expression when lambda lies outside [0,1], but both qualification flags must then be No. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([-4, 0], [-1, 3], 9/11, Yes, [-38/11, 6/11], Yes)","explanation":"Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 0], [-1, 3], 9/11, Yes, [-38/11, 6/11], Yes)."},"skills":["Audit lambda without Clipping an Affine Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0060","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3fdd9533db09dac2192698917d2ba48ecc4309ade51c7f9d836c876920835d01","family_id":"qf-mod24-convcomb-two-point","seed":202610245060,"semantic_template":"qf-mod24-convcomb-two-point/{\"lambda\":[10,12],\"left\":[-3,0],\"right\":[0,3]}"},"fields":{"title":"Audit lambda without Clipping an Affine Extension","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. left=[-3, 0], right=[0, 3], lambda=10/12. Give (left,right,normalized lambda,lambda in [0,1]?,lambda*left+(1−lambda)*right,is this a convex combination?). This audit still computes the affine expression when lambda lies outside [0,1], but both qualification flags must then be No. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([-3, 0], [0, 3], 5/6, Yes, [-5/2, 1/2], Yes)","explanation":"Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 0], [0, 3], 5/6, Yes, [-5/2, 1/2], Yes)."},"skills":["Audit lambda without Clipping an Affine Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0061","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81f5f00f2b3c8642e139cdf002ee98c9312bf24ae2db6db1ef13e29f2f31cced","family_id":"qf-mod24-convcomb-two-point","seed":202610245061,"semantic_template":"qf-mod24-convcomb-two-point/{\"lambda\":[11,13],\"left\":[-2,0],\"right\":[1,3]}"},"fields":{"title":"Audit lambda without Clipping an Affine Extension","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. left=[-2, 0], right=[1, 3], lambda=11/13. Give (left,right,normalized lambda,lambda in [0,1]?,lambda*left+(1−lambda)*right,is this a convex combination?). This audit still computes the affine expression when lambda lies outside [0,1], but both qualification flags must then be No. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([-2, 0], [1, 3], 11/13, Yes, [-20/13, 6/13], Yes)","explanation":"Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-2, 0], [1, 3], 11/13, Yes, [-20/13, 6/13], Yes)."},"skills":["Audit lambda without Clipping an Affine Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0062","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd7d36279c69bf148648b713eed16b2ec76d96b7247cf5682e2fe3e7ce13fbc2","family_id":"qf-mod24-convcomb-two-point","seed":202610245062,"semantic_template":"qf-mod24-convcomb-two-point/{\"lambda\":[12,14],\"left\":[-1,0],\"right\":[2,3]}"},"fields":{"title":"Audit lambda without Clipping an Affine Extension","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. left=[-1, 0], right=[2, 3], lambda=12/14. Give (left,right,normalized lambda,lambda in [0,1]?,lambda*left+(1−lambda)*right,is this a convex combination?). This audit still computes the affine expression when lambda lies outside [0,1], but both qualification flags must then be No. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([-1, 0], [2, 3], 6/7, Yes, [-4/7, 3/7], Yes)","explanation":"Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-1, 0], [2, 3], 6/7, Yes, [-4/7, 3/7], Yes)."},"skills":["Audit lambda without Clipping an Affine Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0063","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81f35f585e81fe3669e70ee469d02daa31995b6b4ec58962f94a58516f182e84","family_id":"qf-mod24-convcomb-two-point","seed":202610245063,"semantic_template":"qf-mod24-convcomb-two-point/{\"lambda\":[13,15],\"left\":[0,0],\"right\":[3,3]}"},"fields":{"title":"Audit lambda without Clipping an Affine Extension","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. left=[0, 0], right=[3, 3], lambda=13/15. Give (left,right,normalized lambda,lambda in [0,1]?,lambda*left+(1−lambda)*right,is this a convex combination?). This audit still computes the affine expression when lambda lies outside [0,1], but both qualification flags must then be No. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([0, 0], [3, 3], 13/15, Yes, [2/5, 2/5], Yes)","explanation":"Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0], [3, 3], 13/15, Yes, [2/5, 2/5], Yes)."},"skills":["Audit lambda without Clipping an Affine Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0064","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0614996245ca91823313cf24e0ac85f66e61558d762a24eff41bfc75ed5321fc","family_id":"qf-mod24-convcomb-two-point","seed":202610245064,"semantic_template":"qf-mod24-convcomb-two-point/{\"lambda\":[14,16],\"left\":[1,0],\"right\":[4,3]}"},"fields":{"title":"Audit lambda without Clipping an Affine Extension","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. left=[1, 0], right=[4, 3], lambda=14/16. Give (left,right,normalized lambda,lambda in [0,1]?,lambda*left+(1−lambda)*right,is this a convex combination?). This audit still computes the affine expression when lambda lies outside [0,1], but both qualification flags must then be No. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([1, 0], [4, 3], 7/8, Yes, [11/8, 3/8], Yes)","explanation":"Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0], [4, 3], 7/8, Yes, [11/8, 3/8], Yes)."},"skills":["Audit lambda without Clipping an Affine Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0065","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"526e699aa4b5791e0fd43712e0f44a383c44237cf399611840fc3d53b90b99bc","family_id":"qf-mod24-convcomb-two-point","seed":202610245065,"semantic_template":"qf-mod24-convcomb-two-point/{\"lambda\":[15,17],\"left\":[2,0],\"right\":[5,3]}"},"fields":{"title":"Audit lambda without Clipping an Affine Extension","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. left=[2, 0], right=[5, 3], lambda=15/17. Give (left,right,normalized lambda,lambda in [0,1]?,lambda*left+(1−lambda)*right,is this a convex combination?). This audit still computes the affine expression when lambda lies outside [0,1], but both qualification flags must then be No. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([2, 0], [5, 3], 15/17, Yes, [40/17, 6/17], Yes)","explanation":"Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([2, 0], [5, 3], 15/17, Yes, [40/17, 6/17], Yes)."},"skills":["Audit lambda without Clipping an Affine Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0066","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae98b5b8daf827126b0cf97301235d3eefcf9a86dc5e33b5d3502bc25517aa38","family_id":"qf-mod24-convcomb-two-point","seed":202610245066,"semantic_template":"qf-mod24-convcomb-two-point/{\"lambda\":[16,18],\"left\":[3,0],\"right\":[6,3]}"},"fields":{"title":"Audit lambda without Clipping an Affine Extension","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. left=[3, 0], right=[6, 3], lambda=16/18. Give (left,right,normalized lambda,lambda in [0,1]?,lambda*left+(1−lambda)*right,is this a convex combination?). This audit still computes the affine expression when lambda lies outside [0,1], but both qualification flags must then be No. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([3, 0], [6, 3], 8/9, Yes, [10/3, 1/3], Yes)","explanation":"Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([3, 0], [6, 3], 8/9, Yes, [10/3, 1/3], Yes)."},"skills":["Audit lambda without Clipping an Affine Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0067","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11445eae0bf3f041b40976910d70fab7c3550b4d43e546ceb5954f9e9d30867c","family_id":"qf-mod24-convcomb-two-point","seed":202610245067,"semantic_template":"qf-mod24-convcomb-two-point/{\"lambda\":[17,19],\"left\":[4,0],\"right\":[7,3]}"},"fields":{"title":"Audit lambda without Clipping an Affine Extension","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. left=[4, 0], right=[7, 3], lambda=17/19. Give (left,right,normalized lambda,lambda in [0,1]?,lambda*left+(1−lambda)*right,is this a convex combination?). This audit still computes the affine expression when lambda lies outside [0,1], but both qualification flags must then be No. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([4, 0], [7, 3], 17/19, Yes, [82/19, 6/19], Yes)","explanation":"Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([4, 0], [7, 3], 17/19, Yes, [82/19, 6/19], Yes)."},"skills":["Audit lambda without Clipping an Affine Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0068","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4655daaed252c75041d372c0818ff62de4605a0d7f487285d30653ca0c55310","family_id":"qf-mod24-convcomb-two-point","seed":202610245068,"semantic_template":"qf-mod24-convcomb-two-point/{\"lambda\":[18,20],\"left\":[5,0],\"right\":[8,3]}"},"fields":{"title":"Audit lambda without Clipping an Affine Extension","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. left=[5, 0], right=[8, 3], lambda=18/20. Give (left,right,normalized lambda,lambda in [0,1]?,lambda*left+(1−lambda)*right,is this a convex combination?). This audit still computes the affine expression when lambda lies outside [0,1], but both qualification flags must then be No. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([5, 0], [8, 3], 9/10, Yes, [53/10, 3/10], Yes)","explanation":"Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0], [8, 3], 9/10, Yes, [53/10, 3/10], Yes)."},"skills":["Audit lambda without Clipping an Affine Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0069","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0598fbe00510758a33a2086d6377b0b417413b562ddf3dde48c7d6941f997318","family_id":"qf-mod24-convcomb-two-point","seed":202610245069,"semantic_template":"qf-mod24-convcomb-two-point/{\"lambda\":[19,21],\"left\":[6,0],\"right\":[9,3]}"},"fields":{"title":"Audit lambda without Clipping an Affine Extension","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. left=[6, 0], right=[9, 3], lambda=19/21. Give (left,right,normalized lambda,lambda in [0,1]?,lambda*left+(1−lambda)*right,is this a convex combination?). This audit still computes the affine expression when lambda lies outside [0,1], but both qualification flags must then be No. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([6, 0], [9, 3], 19/21, Yes, [44/7, 2/7], Yes)","explanation":"Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([6, 0], [9, 3], 19/21, Yes, [44/7, 2/7], Yes)."},"skills":["Audit lambda without Clipping an Affine Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0070","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e3eb2727804264bb1699e56009e15709eb754ea1f0653c2438d291f70ec798bf","family_id":"qf-mod24-convcomb-two-point","seed":202610245070,"semantic_template":"qf-mod24-convcomb-two-point/{\"lambda\":[20,22],\"left\":[7,0],\"right\":[10,3]}"},"fields":{"title":"Audit lambda without Clipping an Affine Extension","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. left=[7, 0], right=[10, 3], lambda=20/22. Give (left,right,normalized lambda,lambda in [0,1]?,lambda*left+(1−lambda)*right,is this a convex combination?). This audit still computes the affine expression when lambda lies outside [0,1], but both qualification flags must then be No. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([7, 0], [10, 3], 10/11, Yes, [80/11, 3/11], Yes)","explanation":"Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([7, 0], [10, 3], 10/11, Yes, [80/11, 3/11], Yes)."},"skills":["Audit lambda without Clipping an Affine Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0071","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc98e3834e1b69a874b579ac59f692eb55a7a9f9fc3b53904d6728eae5116248","family_id":"qf-mod24-convcomb-two-point","seed":202610245071,"semantic_template":"qf-mod24-convcomb-two-point/{\"lambda\":[21,23],\"left\":[8,0],\"right\":[11,3]}"},"fields":{"title":"Audit lambda without Clipping an Affine Extension","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. left=[8, 0], right=[11, 3], lambda=21/23. Give (left,right,normalized lambda,lambda in [0,1]?,lambda*left+(1−lambda)*right,is this a convex combination?). This audit still computes the affine expression when lambda lies outside [0,1], but both qualification flags must then be No. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([8, 0], [11, 3], 21/23, Yes, [190/23, 6/23], Yes)","explanation":"Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([8, 0], [11, 3], 21/23, Yes, [190/23, 6/23], Yes)."},"skills":["Audit lambda without Clipping an Affine Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0072","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8be09f2bb407048b5073a3e7ae7f762514176af167793ba2bc4f49c895d5273","family_id":"qf-mod24-convcomb-two-point","seed":202610245072,"semantic_template":"qf-mod24-convcomb-two-point/{\"lambda\":[22,24],\"left\":[9,0],\"right\":[12,3]}"},"fields":{"title":"Audit lambda without Clipping an Affine Extension","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. left=[9, 0], right=[12, 3], lambda=22/24. Give (left,right,normalized lambda,lambda in [0,1]?,lambda*left+(1−lambda)*right,is this a convex combination?). This audit still computes the affine expression when lambda lies outside [0,1], but both qualification flags must then be No. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([9, 0], [12, 3], 11/12, Yes, [37/4, 1/4], Yes)","explanation":"Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([9, 0], [12, 3], 11/12, Yes, [37/4, 1/4], Yes)."},"skills":["Audit lambda without Clipping an Affine Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0073","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d68d9ca68b429854cefd557f226bc248ba81a794b9b0896e3b067ecd82c3654","family_id":"qf-mod24-convcomb-two-point","seed":202610245073,"semantic_template":"qf-mod24-convcomb-two-point/{\"lambda\":[23,25],\"left\":[10,0],\"right\":[13,3]}"},"fields":{"title":"Audit lambda without Clipping an Affine Extension","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. left=[10, 0], right=[13, 3], lambda=23/25. Give (left,right,normalized lambda,lambda in [0,1]?,lambda*left+(1−lambda)*right,is this a convex combination?). This audit still computes the affine expression when lambda lies outside [0,1], but both qualification flags must then be No. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([10, 0], [13, 3], 23/25, Yes, [256/25, 6/25], Yes)","explanation":"Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([10, 0], [13, 3], 23/25, Yes, [256/25, 6/25], Yes)."},"skills":["Audit lambda without Clipping an Affine Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0074","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ff519b3e4750e20f32d43d96422f75f8721e8cec855ede13111f1abb89101a3","family_id":"qf-mod24-convcomb-two-point","seed":202610245074,"semantic_template":"qf-mod24-convcomb-two-point/{\"lambda\":[24,26],\"left\":[11,0],\"right\":[14,3]}"},"fields":{"title":"Audit lambda without Clipping an Affine Extension","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. left=[11, 0], right=[14, 3], lambda=24/26. Give (left,right,normalized lambda,lambda in [0,1]?,lambda*left+(1−lambda)*right,is this a convex combination?). This audit still computes the affine expression when lambda lies outside [0,1], but both qualification flags must then be No. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([11, 0], [14, 3], 12/13, Yes, [146/13, 3/13], Yes)","explanation":"Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([11, 0], [14, 3], 12/13, Yes, [146/13, 3/13], Yes)."},"skills":["Audit lambda without Clipping an Affine Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0075","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"600d39560b87f61a297e12e2df5d6f96ad6e2f48d301fdb9a9ec80653fb9130b","family_id":"qf-mod24-convcomb-two-point","seed":202610245075,"semantic_template":"qf-mod24-convcomb-two-point/{\"lambda\":[25,27],\"left\":[12,0],\"right\":[15,3]}"},"fields":{"title":"Audit lambda without Clipping an Affine Extension","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. left=[12, 0], right=[15, 3], lambda=25/27. Give (left,right,normalized lambda,lambda in [0,1]?,lambda*left+(1−lambda)*right,is this a convex combination?). This audit still computes the affine expression when lambda lies outside [0,1], but both qualification flags must then be No. Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([12, 0], [15, 3], 25/27, Yes, [110/9, 2/9], Yes)","explanation":"Keep the declared weighting direction: lambda=0 selects right, lambda=1 selects left. An outside value is not silently clipped to the segment. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([12, 0], [15, 3], 25/27, Yes, [110/9, 2/9], Yes)."},"skills":["Audit lambda without Clipping an Affine Extension"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0076","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77533b0b96880817b6c7e99d72dbf492d819896c0a8fbdb1f286ab8c8f12db3a","family_id":"qf-mod24-convcomb-witness-audit","seed":202610245076,"semantic_template":"qf-mod24-convcomb-witness-audit/{\"target\":[2,0],\"vectors\":[[0,0],[3,0]],\"weights\":[[1,3],[2,3]]}"},"fields":{"title":"A Qualified Convex Witness versus Its Target","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[0, 0], [3, 0]], weights=[1/3, 2/3], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [3, 0]], [1/3, 2/3], [2, 0], [2, 0], Yes)","explanation":"Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [3, 0]], [1/3, 2/3], [2, 0], [2, 0], Yes)."},"skills":["A Qualified Convex Witness versus Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0077","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19dd01787a78aa5a83677a69459c07d0cd37173de0bfb459f70e98909c41b77b","family_id":"qf-mod24-convcomb-witness-audit","seed":202610245077,"semantic_template":"qf-mod24-convcomb-witness-audit/{\"target\":[2,0],\"vectors\":[[0,0],[4,0]],\"weights\":[[2,4],[2,4]]}"},"fields":{"title":"A Qualified Convex Witness versus Its Target","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[0, 0], [4, 0]], weights=[2/4, 2/4], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [4, 0]], [1/2, 1/2], [2, 0], [2, 0], Yes)","explanation":"Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [4, 0]], [1/2, 1/2], [2, 0], [2, 0], Yes)."},"skills":["A Qualified Convex Witness versus Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0078","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e617d1512fced613804f2255c7cc70270df5d8b5af89cdf13e7db177d1b73460","family_id":"qf-mod24-convcomb-witness-audit","seed":202610245078,"semantic_template":"qf-mod24-convcomb-witness-audit/{\"target\":[2,0],\"vectors\":[[0,0],[5,0]],\"weights\":[[3,5],[2,5]]}"},"fields":{"title":"A Qualified Convex Witness versus Its Target","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[0, 0], [5, 0]], weights=[3/5, 2/5], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [5, 0]], [3/5, 2/5], [2, 0], [2, 0], Yes)","explanation":"Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [5, 0]], [3/5, 2/5], [2, 0], [2, 0], Yes)."},"skills":["A Qualified Convex Witness versus Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0079","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"411481b33f346b9b08d9ef82fa97d88ac55fa688fce7e882a746bf51f8d74c90","family_id":"qf-mod24-convcomb-witness-audit","seed":202610245079,"semantic_template":"qf-mod24-convcomb-witness-audit/{\"target\":[2,0],\"vectors\":[[0,0],[6,0]],\"weights\":[[4,6],[2,6]]}"},"fields":{"title":"A Qualified Convex Witness versus Its Target","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[0, 0], [6, 0]], weights=[4/6, 2/6], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [6, 0]], [2/3, 1/3], [2, 0], [2, 0], Yes)","explanation":"Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [6, 0]], [2/3, 1/3], [2, 0], [2, 0], Yes)."},"skills":["A Qualified Convex Witness versus Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0080","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9be4c8e231eb370ad08f301d3579bb3b237031a22f11338480dc5a7fd3255310","family_id":"qf-mod24-convcomb-witness-audit","seed":202610245080,"semantic_template":"qf-mod24-convcomb-witness-audit/{\"target\":[2,0],\"vectors\":[[0,0],[7,0]],\"weights\":[[5,7],[2,7]]}"},"fields":{"title":"A Qualified Convex Witness versus Its Target","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[0, 0], [7, 0]], weights=[5/7, 2/7], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [7, 0]], [5/7, 2/7], [2, 0], [2, 0], Yes)","explanation":"Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [7, 0]], [5/7, 2/7], [2, 0], [2, 0], Yes)."},"skills":["A Qualified Convex Witness versus Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0081","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e38f5689d823c8fc11109cc0227fbfca2c300ccaa05ef3d5f8af64d4bfd4da3","family_id":"qf-mod24-convcomb-witness-audit","seed":202610245081,"semantic_template":"qf-mod24-convcomb-witness-audit/{\"target\":[2,0],\"vectors\":[[0,0],[8,0]],\"weights\":[[6,8],[2,8]]}"},"fields":{"title":"A Qualified Convex Witness versus Its Target","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[0, 0], [8, 0]], weights=[6/8, 2/8], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [8, 0]], [3/4, 1/4], [2, 0], [2, 0], Yes)","explanation":"Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [8, 0]], [3/4, 1/4], [2, 0], [2, 0], Yes)."},"skills":["A Qualified Convex Witness versus Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0082","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cdff8f7036e9919498e4c5b615ba40cd7fb8d7e177bfcc3aa00459fb86a24bc1","family_id":"qf-mod24-convcomb-witness-audit","seed":202610245082,"semantic_template":"qf-mod24-convcomb-witness-audit/{\"target\":[2,0],\"vectors\":[[0,0],[9,0]],\"weights\":[[7,9],[2,9]]}"},"fields":{"title":"A Qualified Convex Witness versus Its Target","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[0, 0], [9, 0]], weights=[7/9, 2/9], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [9, 0]], [7/9, 2/9], [2, 0], [2, 0], Yes)","explanation":"Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [9, 0]], [7/9, 2/9], [2, 0], [2, 0], Yes)."},"skills":["A Qualified Convex Witness versus Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0083","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62fceaf873703d2a2741b6797196f8e0236a39e2826aa43a1f5dfb83e2f0f600","family_id":"qf-mod24-convcomb-witness-audit","seed":202610245083,"semantic_template":"qf-mod24-convcomb-witness-audit/{\"target\":[2,0],\"vectors\":[[0,0],[10,0]],\"weights\":[[8,10],[2,10]]}"},"fields":{"title":"A Qualified Convex Witness versus Its Target","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[0, 0], [10, 0]], weights=[8/10, 2/10], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [10, 0]], [4/5, 1/5], [2, 0], [2, 0], Yes)","explanation":"Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [10, 0]], [4/5, 1/5], [2, 0], [2, 0], Yes)."},"skills":["A Qualified Convex Witness versus Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0084","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"688829f932ed8bc7c959ec6927f5261be4dcb76b5c6f950d47721815d7b1629a","family_id":"qf-mod24-convcomb-witness-audit","seed":202610245084,"semantic_template":"qf-mod24-convcomb-witness-audit/{\"target\":[2,0],\"vectors\":[[0,0],[11,0]],\"weights\":[[9,11],[2,11]]}"},"fields":{"title":"A Qualified Convex Witness versus Its Target","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[0, 0], [11, 0]], weights=[9/11, 2/11], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [11, 0]], [9/11, 2/11], [2, 0], [2, 0], Yes)","explanation":"Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [11, 0]], [9/11, 2/11], [2, 0], [2, 0], Yes)."},"skills":["A Qualified Convex Witness versus Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0085","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb334695fafeb7e57cb5490fdf8214e03e2f9cccb237750b69339effe6798c2b","family_id":"qf-mod24-convcomb-witness-audit","seed":202610245085,"semantic_template":"qf-mod24-convcomb-witness-audit/{\"target\":[2,0],\"vectors\":[[0,0],[12,0]],\"weights\":[[10,12],[2,12]]}"},"fields":{"title":"A Qualified Convex Witness versus Its Target","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[0, 0], [12, 0]], weights=[10/12, 2/12], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [12, 0]], [5/6, 1/6], [2, 0], [2, 0], Yes)","explanation":"Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [12, 0]], [5/6, 1/6], [2, 0], [2, 0], Yes)."},"skills":["A Qualified Convex Witness versus Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0086","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e039be9560948a971cfecf753aba39caf5104d97fcb00e5ad543f8c99385edba","family_id":"qf-mod24-convcomb-witness-audit","seed":202610245086,"semantic_template":"qf-mod24-convcomb-witness-audit/{\"target\":[2,0],\"vectors\":[[0,0],[13,0]],\"weights\":[[11,13],[2,13]]}"},"fields":{"title":"A Qualified Convex Witness versus Its Target","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[0, 0], [13, 0]], weights=[11/13, 2/13], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [13, 0]], [11/13, 2/13], [2, 0], [2, 0], Yes)","explanation":"Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [13, 0]], [11/13, 2/13], [2, 0], [2, 0], Yes)."},"skills":["A Qualified Convex Witness versus Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0087","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e810f2f11e6dea0e90cf8c8fe24ceb9216a8742f1977f24b46a670fa8cfb4f9","family_id":"qf-mod24-convcomb-witness-audit","seed":202610245087,"semantic_template":"qf-mod24-convcomb-witness-audit/{\"target\":[2,0],\"vectors\":[[0,0],[14,0]],\"weights\":[[12,14],[2,14]]}"},"fields":{"title":"A Qualified Convex Witness versus Its Target","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[0, 0], [14, 0]], weights=[12/14, 2/14], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [14, 0]], [6/7, 1/7], [2, 0], [2, 0], Yes)","explanation":"Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [14, 0]], [6/7, 1/7], [2, 0], [2, 0], Yes)."},"skills":["A Qualified Convex Witness versus Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0088","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2440fe6eaa74ea160707a322483c3c0a6031e42f1183d235b0c77b2bdd2309a9","family_id":"qf-mod24-convcomb-witness-audit","seed":202610245088,"semantic_template":"qf-mod24-convcomb-witness-audit/{\"target\":[2,0],\"vectors\":[[0,0],[15,0]],\"weights\":[[13,15],[2,15]]}"},"fields":{"title":"A Qualified Convex Witness versus Its Target","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[0, 0], [15, 0]], weights=[13/15, 2/15], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [15, 0]], [13/15, 2/15], [2, 0], [2, 0], Yes)","explanation":"Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [15, 0]], [13/15, 2/15], [2, 0], [2, 0], Yes)."},"skills":["A Qualified Convex Witness versus Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0089","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7f221f2fc8c5fedb880bd04daf2a52a2993a4f7cff04521d01cc82d8d062006","family_id":"qf-mod24-convcomb-witness-audit","seed":202610245089,"semantic_template":"qf-mod24-convcomb-witness-audit/{\"target\":[2,0],\"vectors\":[[0,0],[16,0]],\"weights\":[[14,16],[2,16]]}"},"fields":{"title":"A Qualified Convex Witness versus Its Target","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[0, 0], [16, 0]], weights=[14/16, 2/16], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [16, 0]], [7/8, 1/8], [2, 0], [2, 0], Yes)","explanation":"Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [16, 0]], [7/8, 1/8], [2, 0], [2, 0], Yes)."},"skills":["A Qualified Convex Witness versus Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0090","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cbcd029a23389fca5c337e9c0be886d3438a5171d15aecbdf94016e0dc39b4ed","family_id":"qf-mod24-convcomb-witness-audit","seed":202610245090,"semantic_template":"qf-mod24-convcomb-witness-audit/{\"target\":[2,0],\"vectors\":[[0,0],[17,0]],\"weights\":[[15,17],[2,17]]}"},"fields":{"title":"A Qualified Convex Witness versus Its Target","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[0, 0], [17, 0]], weights=[15/17, 2/17], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [17, 0]], [15/17, 2/17], [2, 0], [2, 0], Yes)","explanation":"Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [17, 0]], [15/17, 2/17], [2, 0], [2, 0], Yes)."},"skills":["A Qualified Convex Witness versus Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0091","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ed18123fd69bd320ed42c1dea9280ea1d4f23e4bf306e78b796c0b9a622135d","family_id":"qf-mod24-convcomb-witness-audit","seed":202610245091,"semantic_template":"qf-mod24-convcomb-witness-audit/{\"target\":[2,0],\"vectors\":[[0,0],[18,0]],\"weights\":[[16,18],[2,18]]}"},"fields":{"title":"A Qualified Convex Witness versus Its Target","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[0, 0], [18, 0]], weights=[16/18, 2/18], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [18, 0]], [8/9, 1/9], [2, 0], [2, 0], Yes)","explanation":"Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [18, 0]], [8/9, 1/9], [2, 0], [2, 0], Yes)."},"skills":["A Qualified Convex Witness versus Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0092","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eae8dab93d254f6026336326ab650bb3e2d8d4939fa7de2356c95b1ba8acb4dd","family_id":"qf-mod24-convcomb-witness-audit","seed":202610245092,"semantic_template":"qf-mod24-convcomb-witness-audit/{\"target\":[2,0],\"vectors\":[[0,0],[19,0]],\"weights\":[[17,19],[2,19]]}"},"fields":{"title":"A Qualified Convex Witness versus Its Target","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[0, 0], [19, 0]], weights=[17/19, 2/19], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [19, 0]], [17/19, 2/19], [2, 0], [2, 0], Yes)","explanation":"Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [19, 0]], [17/19, 2/19], [2, 0], [2, 0], Yes)."},"skills":["A Qualified Convex Witness versus Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0093","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"663dc599f727130b2107c297494a0d424df6779d0a3e494ba0da8d67514cd8c4","family_id":"qf-mod24-convcomb-witness-audit","seed":202610245093,"semantic_template":"qf-mod24-convcomb-witness-audit/{\"target\":[2,0],\"vectors\":[[0,0],[20,0]],\"weights\":[[18,20],[2,20]]}"},"fields":{"title":"A Qualified Convex Witness versus Its Target","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[0, 0], [20, 0]], weights=[18/20, 2/20], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [20, 0]], [9/10, 1/10], [2, 0], [2, 0], Yes)","explanation":"Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [20, 0]], [9/10, 1/10], [2, 0], [2, 0], Yes)."},"skills":["A Qualified Convex Witness versus Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0094","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f976c0a1c64461fc2f67d7e33a290d2a9347c23bdfa7f5fa125b8a81b4237298","family_id":"qf-mod24-convcomb-witness-audit","seed":202610245094,"semantic_template":"qf-mod24-convcomb-witness-audit/{\"target\":[2,0],\"vectors\":[[0,0],[21,0]],\"weights\":[[19,21],[2,21]]}"},"fields":{"title":"A Qualified Convex Witness versus Its Target","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[0, 0], [21, 0]], weights=[19/21, 2/21], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [21, 0]], [19/21, 2/21], [2, 0], [2, 0], Yes)","explanation":"Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [21, 0]], [19/21, 2/21], [2, 0], [2, 0], Yes)."},"skills":["A Qualified Convex Witness versus Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0095","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14df61919f72f38d6088d7b4d8b33f4a257ed336c84bf1554e5b79cf2a3eb341","family_id":"qf-mod24-convcomb-witness-audit","seed":202610245095,"semantic_template":"qf-mod24-convcomb-witness-audit/{\"target\":[2,0],\"vectors\":[[0,0],[22,0]],\"weights\":[[20,22],[2,22]]}"},"fields":{"title":"A Qualified Convex Witness versus Its Target","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[0, 0], [22, 0]], weights=[20/22, 2/22], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [22, 0]], [10/11, 1/11], [2, 0], [2, 0], Yes)","explanation":"Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [22, 0]], [10/11, 1/11], [2, 0], [2, 0], Yes)."},"skills":["A Qualified Convex Witness versus Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0096","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7e6a241b2463ff5877b6f4d5697df69c2f5ae83cb45faeeace3262f0f16267a","family_id":"qf-mod24-convcomb-witness-audit","seed":202610245096,"semantic_template":"qf-mod24-convcomb-witness-audit/{\"target\":[2,0],\"vectors\":[[0,0],[23,0]],\"weights\":[[21,23],[2,23]]}"},"fields":{"title":"A Qualified Convex Witness versus Its Target","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[0, 0], [23, 0]], weights=[21/23, 2/23], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [23, 0]], [21/23, 2/23], [2, 0], [2, 0], Yes)","explanation":"Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [23, 0]], [21/23, 2/23], [2, 0], [2, 0], Yes)."},"skills":["A Qualified Convex Witness versus Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0097","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"708655dcffac2e4ff2b49a75b88083d08cbf4912138e7862db7dcb864706c201","family_id":"qf-mod24-convcomb-witness-audit","seed":202610245097,"semantic_template":"qf-mod24-convcomb-witness-audit/{\"target\":[2,0],\"vectors\":[[0,0],[24,0]],\"weights\":[[22,24],[2,24]]}"},"fields":{"title":"A Qualified Convex Witness versus Its Target","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[0, 0], [24, 0]], weights=[22/24, 2/24], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [24, 0]], [11/12, 1/12], [2, 0], [2, 0], Yes)","explanation":"Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [24, 0]], [11/12, 1/12], [2, 0], [2, 0], Yes)."},"skills":["A Qualified Convex Witness versus Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0098","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ce442faa36c4abe70debc8faf3cdfdbab561a396e6e1a4951f4f56ff00549d7","family_id":"qf-mod24-convcomb-witness-audit","seed":202610245098,"semantic_template":"qf-mod24-convcomb-witness-audit/{\"target\":[2,0],\"vectors\":[[0,0],[25,0]],\"weights\":[[23,25],[2,25]]}"},"fields":{"title":"A Qualified Convex Witness versus Its Target","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[0, 0], [25, 0]], weights=[23/25, 2/25], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [25, 0]], [23/25, 2/25], [2, 0], [2, 0], Yes)","explanation":"Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [25, 0]], [23/25, 2/25], [2, 0], [2, 0], Yes)."},"skills":["A Qualified Convex Witness versus Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0099","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37318c177bcea2268fa0721fcbb81c8c797c74d5c5887a36d01ee4870f7fba9a","family_id":"qf-mod24-convcomb-witness-audit","seed":202610245099,"semantic_template":"qf-mod24-convcomb-witness-audit/{\"target\":[2,0],\"vectors\":[[0,0],[26,0]],\"weights\":[[24,26],[2,26]]}"},"fields":{"title":"A Qualified Convex Witness versus Its Target","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[0, 0], [26, 0]], weights=[24/26, 2/26], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [26, 0]], [12/13, 1/13], [2, 0], [2, 0], Yes)","explanation":"Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [26, 0]], [12/13, 1/13], [2, 0], [2, 0], Yes)."},"skills":["A Qualified Convex Witness versus Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261024-r05-finite-convex-combination-0100","batch_id":"batch-20261013-advance-20261024-r05-finite-convex-combination-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"600a98e0ea023405913833d503e4796c9b24600bdac0db84588d006921035382","family_id":"qf-mod24-convcomb-witness-audit","seed":202610245100,"semantic_template":"qf-mod24-convcomb-witness-audit/{\"target\":[2,0],\"vectors\":[[0,0],[27,0]],\"weights\":[[25,27],[2,27]]}"},"fields":{"title":"A Qualified Convex Witness versus Its Target","stem":"Use a finite nonempty ordered vector list and aligned exact rational weights. Convex weights may be zero but must be nonnegative and sum exactly to 1, not merely approximately. A convex-combination witness establishes its weighted point, not unique coordinates without separate affine independence. vectors=[[0, 0], [27, 0]], weights=[25/27, 2/27], target=[2, 0]. Require qualified weights and matching dimension. Give (vectors,weights,target,reconstructed point,equal?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.","answer":{"kind":"expression","value":"([[0, 0], [27, 0]], [25/27, 2/27], [2, 0], [2, 0], Yes)","explanation":"Reconstruct the point with exact aligned weights; successful reconstruction does not by itself make these weights unique. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [27, 0]], [25/27, 2/27], [2, 0], [2, 0], Yes)."},"skills":["A Qualified Convex Witness versus Its Target"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}