{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0001","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"657ee21786a8cabef64e93e5aa5f386652977d29ca09c7693b6bb6213d3da365","family_id":"qf-mod40-totalexp-partition-audit","seed":202611094001,"semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,10],[2,10],[7,10],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[0,1,3,99]}"},"fields":{"title":"A Complete Disjoint Partition and Global Mean Use the Same Law","stem":"Use the complete normalized finite law masses=[1/10, 2/10, 7/10, 0/1], aligned values X=[0, 1, 3, 99] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 3/10], [1, 7/10]], Yes, 23/10)","explanation":"Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 3/10], [1, 7/10]], Yes, 23/10)."},"skills":["A Complete Disjoint Partition and Global Mean Use the Same Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0002","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"669bb7ddf124fbe40164701d05f6a86392e76a29b6a44df7514df5c72e19c5db","family_id":"qf-mod40-totalexp-partition-audit","seed":202611094002,"semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,11],[2,11],[8,11],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[1,2,4,100]}"},"fields":{"title":"A Complete Disjoint Partition and Global Mean Use the Same Law","stem":"Use the complete normalized finite law masses=[1/11, 2/11, 8/11, 0/1], aligned values X=[1, 2, 4, 100] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 3/11], [1, 8/11]], Yes, 37/11)","explanation":"Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 3/11], [1, 8/11]], Yes, 37/11)."},"skills":["A Complete Disjoint Partition and Global Mean Use the Same Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0003","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68f69bf3d5323d54895b8cde7c528bd39d57da15b898ab168db8f3015ba5e357","family_id":"qf-mod40-totalexp-partition-audit","seed":202611094003,"semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,12],[2,12],[9,12],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[2,3,5,101]}"},"fields":{"title":"A Complete Disjoint Partition and Global Mean Use the Same Law","stem":"Use the complete normalized finite law masses=[1/12, 2/12, 9/12, 0/1], aligned values X=[2, 3, 5, 101] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 1/4], [1, 3/4]], Yes, 53/12)","explanation":"Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 1/4], [1, 3/4]], Yes, 53/12)."},"skills":["A Complete Disjoint Partition and Global Mean Use the Same Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0004","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3cf7bed29cdeab683f6d845f374b53befc9b592c4f442829d6bce3a1c4d0cf74","family_id":"qf-mod40-totalexp-partition-audit","seed":202611094004,"semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,13],[2,13],[10,13],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[3,4,6,102]}"},"fields":{"title":"A Complete Disjoint Partition and Global Mean Use the Same Law","stem":"Use the complete normalized finite law masses=[1/13, 2/13, 10/13, 0/1], aligned values X=[3, 4, 6, 102] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 3/13], [1, 10/13]], Yes, 71/13)","explanation":"Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 3/13], [1, 10/13]], Yes, 71/13)."},"skills":["A Complete Disjoint Partition and Global Mean Use the Same Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0005","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b380627c60e507be64d9ce3302e96f0dbc6f5286d571d2d7fb3edacc23fa930","family_id":"qf-mod40-totalexp-partition-audit","seed":202611094005,"semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,14],[2,14],[11,14],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[4,5,7,103]}"},"fields":{"title":"A Complete Disjoint Partition and Global Mean Use the Same Law","stem":"Use the complete normalized finite law masses=[1/14, 2/14, 11/14, 0/1], aligned values X=[4, 5, 7, 103] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 3/14], [1, 11/14]], Yes, 13/2)","explanation":"Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 3/14], [1, 11/14]], Yes, 13/2)."},"skills":["A Complete Disjoint Partition and Global Mean Use the Same Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0006","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8cddf38bfc1c399aecde2438aee54b80ccb1cb936aab1c15c155b00a602ee25","family_id":"qf-mod40-totalexp-partition-audit","seed":202611094006,"semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,15],[2,15],[12,15],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[5,6,8,104]}"},"fields":{"title":"A Complete Disjoint Partition and Global Mean Use the Same Law","stem":"Use the complete normalized finite law masses=[1/15, 2/15, 12/15, 0/1], aligned values X=[5, 6, 8, 104] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 1/5], [1, 4/5]], Yes, 113/15)","explanation":"Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 1/5], [1, 4/5]], Yes, 113/15)."},"skills":["A Complete Disjoint Partition and Global Mean Use the Same Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0007","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc0328d540af4fe69bde34d8ccbd18ee8df9af8fa516124bb30fc2ca91990828","family_id":"qf-mod40-totalexp-partition-audit","seed":202611094007,"semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,16],[2,16],[13,16],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[6,7,9,105]}"},"fields":{"title":"A Complete Disjoint Partition and Global Mean Use the Same Law","stem":"Use the complete normalized finite law masses=[1/16, 2/16, 13/16, 0/1], aligned values X=[6, 7, 9, 105] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 3/16], [1, 13/16]], Yes, 137/16)","explanation":"Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 3/16], [1, 13/16]], Yes, 137/16)."},"skills":["A Complete Disjoint Partition and Global Mean Use the Same Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0008","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f72c895132379eede8e3db89e430218788a561de93e4b558e829035c51dc1f72","family_id":"qf-mod40-totalexp-partition-audit","seed":202611094008,"semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,17],[2,17],[14,17],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[7,8,10,106]}"},"fields":{"title":"A Complete Disjoint Partition and Global Mean Use the Same Law","stem":"Use the complete normalized finite law masses=[1/17, 2/17, 14/17, 0/1], aligned values X=[7, 8, 10, 106] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 3/17], [1, 14/17]], Yes, 163/17)","explanation":"Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 3/17], [1, 14/17]], Yes, 163/17)."},"skills":["A Complete Disjoint Partition and Global Mean Use the Same Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0009","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd0727e625bafcc3ca1dc5b3f26cd83eee410ebbe29b0c8a2b3fb85790b6b313","family_id":"qf-mod40-totalexp-partition-audit","seed":202611094009,"semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,18],[2,18],[15,18],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[8,9,11,107]}"},"fields":{"title":"A Complete Disjoint Partition and Global Mean Use the Same Law","stem":"Use the complete normalized finite law masses=[1/18, 2/18, 15/18, 0/1], aligned values X=[8, 9, 11, 107] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 1/6], [1, 5/6]], Yes, 191/18)","explanation":"Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 1/6], [1, 5/6]], Yes, 191/18)."},"skills":["A Complete Disjoint Partition and Global Mean Use the Same Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0010","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06eaa6a284169148bdac99a5384eb5de5fda8e07060106992999ef222307edf4","family_id":"qf-mod40-totalexp-partition-audit","seed":202611094010,"semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,19],[2,19],[16,19],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[9,10,12,108]}"},"fields":{"title":"A Complete Disjoint Partition and Global Mean Use the Same Law","stem":"Use the complete normalized finite law masses=[1/19, 2/19, 16/19, 0/1], aligned values X=[9, 10, 12, 108] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 3/19], [1, 16/19]], Yes, 221/19)","explanation":"Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 3/19], [1, 16/19]], Yes, 221/19)."},"skills":["A Complete Disjoint Partition and Global Mean Use the Same Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0011","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d790b5ac02c7971303b2dbe37714cd41950677bc97974405999c98ce3892efd","family_id":"qf-mod40-totalexp-partition-audit","seed":202611094011,"semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,20],[2,20],[17,20],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[10,11,13,109]}"},"fields":{"title":"A Complete Disjoint Partition and Global Mean Use the Same Law","stem":"Use the complete normalized finite law masses=[1/20, 2/20, 17/20, 0/1], aligned values X=[10, 11, 13, 109] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 3/20], [1, 17/20]], Yes, 253/20)","explanation":"Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 3/20], [1, 17/20]], Yes, 253/20)."},"skills":["A Complete Disjoint Partition and Global Mean Use the Same Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0012","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ffe9e645db12251d16c72fb600ffd8366daf5308ab5f32a8c62213afadd1ca5f","family_id":"qf-mod40-totalexp-partition-audit","seed":202611094012,"semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,21],[2,21],[18,21],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[11,12,14,110]}"},"fields":{"title":"A Complete Disjoint Partition and Global Mean Use the Same Law","stem":"Use the complete normalized finite law masses=[1/21, 2/21, 18/21, 0/1], aligned values X=[11, 12, 14, 110] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 1/7], [1, 6/7]], Yes, 41/3)","explanation":"Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 1/7], [1, 6/7]], Yes, 41/3)."},"skills":["A Complete Disjoint Partition and Global Mean Use the Same Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0013","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4366f67b37d8080295bba94c84d7356a5644332e35995c6a3bd87e7ee2cf279","family_id":"qf-mod40-totalexp-partition-audit","seed":202611094013,"semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,22],[2,22],[19,22],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[12,13,15,111]}"},"fields":{"title":"A Complete Disjoint Partition and Global Mean Use the Same Law","stem":"Use the complete normalized finite law masses=[1/22, 2/22, 19/22, 0/1], aligned values X=[12, 13, 15, 111] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 3/22], [1, 19/22]], Yes, 323/22)","explanation":"Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 3/22], [1, 19/22]], Yes, 323/22)."},"skills":["A Complete Disjoint Partition and Global Mean Use the Same Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0014","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"293a2f200f07db5aa68e8f67993f10856f5c3ed9dc38d7fdf5414a4d07f1c879","family_id":"qf-mod40-totalexp-partition-audit","seed":202611094014,"semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,23],[2,23],[20,23],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[13,14,16,112]}"},"fields":{"title":"A Complete Disjoint Partition and Global Mean Use the Same Law","stem":"Use the complete normalized finite law masses=[1/23, 2/23, 20/23, 0/1], aligned values X=[13, 14, 16, 112] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 3/23], [1, 20/23]], Yes, 361/23)","explanation":"Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 3/23], [1, 20/23]], Yes, 361/23)."},"skills":["A Complete Disjoint Partition and Global Mean Use the Same Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0015","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81eea622b68f53382ef2d824f1c82cc3678c689dfe17d85543e9588eaee23c03","family_id":"qf-mod40-totalexp-partition-audit","seed":202611094015,"semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,24],[2,24],[21,24],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[14,15,17,113]}"},"fields":{"title":"A Complete Disjoint Partition and Global Mean Use the Same Law","stem":"Use the complete normalized finite law masses=[1/24, 2/24, 21/24, 0/1], aligned values X=[14, 15, 17, 113] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 1/8], [1, 7/8]], Yes, 401/24)","explanation":"Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 1/8], [1, 7/8]], Yes, 401/24)."},"skills":["A Complete Disjoint Partition and Global Mean Use the Same Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0016","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"812b20a4bf8053513b8bf0487ae83f079c050927e8d93418683a76a8faf981ae","family_id":"qf-mod40-totalexp-partition-audit","seed":202611094016,"semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,25],[2,25],[22,25],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[15,16,18,114]}"},"fields":{"title":"A Complete Disjoint Partition and Global Mean Use the Same Law","stem":"Use the complete normalized finite law masses=[1/25, 2/25, 22/25, 0/1], aligned values X=[15, 16, 18, 114] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 3/25], [1, 22/25]], Yes, 443/25)","explanation":"Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 3/25], [1, 22/25]], Yes, 443/25)."},"skills":["A Complete Disjoint Partition and Global Mean Use the Same Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0017","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e18bd4feeab9a3812a74a226120082d74446a3b658054f9a15f48c42b304c4d","family_id":"qf-mod40-totalexp-partition-audit","seed":202611094017,"semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,26],[2,26],[23,26],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[16,17,19,115]}"},"fields":{"title":"A Complete Disjoint Partition and Global Mean Use the Same Law","stem":"Use the complete normalized finite law masses=[1/26, 2/26, 23/26, 0/1], aligned values X=[16, 17, 19, 115] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 3/26], [1, 23/26]], Yes, 487/26)","explanation":"Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 3/26], [1, 23/26]], Yes, 487/26)."},"skills":["A Complete Disjoint Partition and Global Mean Use the Same Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0018","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec2e8ba7d5f7fae4439af142734a36227d7f18096b5dd175979e7ad036e53b5a","family_id":"qf-mod40-totalexp-partition-audit","seed":202611094018,"semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,27],[2,27],[24,27],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[17,18,20,116]}"},"fields":{"title":"A Complete Disjoint Partition and Global Mean Use the Same Law","stem":"Use the complete normalized finite law masses=[1/27, 2/27, 24/27, 0/1], aligned values X=[17, 18, 20, 116] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 1/9], [1, 8/9]], Yes, 533/27)","explanation":"Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 1/9], [1, 8/9]], Yes, 533/27)."},"skills":["A Complete Disjoint Partition and Global Mean Use the Same Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0019","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97286dc11a655e38c5f57f2310b0c2a69529964674340c166771b51094f3eb46","family_id":"qf-mod40-totalexp-partition-audit","seed":202611094019,"semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,28],[2,28],[25,28],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[18,19,21,117]}"},"fields":{"title":"A Complete Disjoint Partition and Global Mean Use the Same Law","stem":"Use the complete normalized finite law masses=[1/28, 2/28, 25/28, 0/1], aligned values X=[18, 19, 21, 117] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 3/28], [1, 25/28]], Yes, 83/4)","explanation":"Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 3/28], [1, 25/28]], Yes, 83/4)."},"skills":["A Complete Disjoint Partition and Global Mean Use the Same Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0020","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4d26eb7e6fdda352a637fddb7ceb452b851e5ad3ca198f03dc1d24228761162","family_id":"qf-mod40-totalexp-partition-audit","seed":202611094020,"semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,29],[2,29],[26,29],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[19,20,22,118]}"},"fields":{"title":"A Complete Disjoint Partition and Global Mean Use the Same Law","stem":"Use the complete normalized finite law masses=[1/29, 2/29, 26/29, 0/1], aligned values X=[19, 20, 22, 118] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 3/29], [1, 26/29]], Yes, 631/29)","explanation":"Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 3/29], [1, 26/29]], Yes, 631/29)."},"skills":["A Complete Disjoint Partition and Global Mean Use the Same Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0021","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d5494a326fc47cf94022789304ab093c0220b9ca6fd68ad6953e0c59c9d59e3","family_id":"qf-mod40-totalexp-partition-audit","seed":202611094021,"semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,30],[2,30],[27,30],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[20,21,23,119]}"},"fields":{"title":"A Complete Disjoint Partition and Global Mean Use the Same Law","stem":"Use the complete normalized finite law masses=[1/30, 2/30, 27/30, 0/1], aligned values X=[20, 21, 23, 119] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 1/10], [1, 9/10]], Yes, 683/30)","explanation":"Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 1/10], [1, 9/10]], Yes, 683/30)."},"skills":["A Complete Disjoint Partition and Global Mean Use the Same Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0022","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18ea7dbc15edcbc3e2a10ae33958b96060ec7d9e4871111f70b26682031aa91e","family_id":"qf-mod40-totalexp-partition-audit","seed":202611094022,"semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,31],[2,31],[28,31],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[21,22,24,120]}"},"fields":{"title":"A Complete Disjoint Partition and Global Mean Use the Same Law","stem":"Use the complete normalized finite law masses=[1/31, 2/31, 28/31, 0/1], aligned values X=[21, 22, 24, 120] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 3/31], [1, 28/31]], Yes, 737/31)","explanation":"Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 3/31], [1, 28/31]], Yes, 737/31)."},"skills":["A Complete Disjoint Partition and Global Mean Use the Same Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0023","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63d1f393cca384c7f1b660196eef6c9091bf725410537c22604956a1735eef27","family_id":"qf-mod40-totalexp-partition-audit","seed":202611094023,"semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,32],[2,32],[29,32],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[22,23,25,121]}"},"fields":{"title":"A Complete Disjoint Partition and Global Mean Use the Same Law","stem":"Use the complete normalized finite law masses=[1/32, 2/32, 29/32, 0/1], aligned values X=[22, 23, 25, 121] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 3/32], [1, 29/32]], Yes, 793/32)","explanation":"Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 3/32], [1, 29/32]], Yes, 793/32)."},"skills":["A Complete Disjoint Partition and Global Mean Use the Same Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0024","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a0736067bcd465898fba882273c1b1c10ccf54f56c9b8c6a401456c5e210866e","family_id":"qf-mod40-totalexp-partition-audit","seed":202611094024,"semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,33],[2,33],[30,33],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[23,24,26,122]}"},"fields":{"title":"A Complete Disjoint Partition and Global Mean Use the Same Law","stem":"Use the complete normalized finite law masses=[1/33, 2/33, 30/33, 0/1], aligned values X=[23, 24, 26, 122] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 1/11], [1, 10/11]], Yes, 851/33)","explanation":"Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 1/11], [1, 10/11]], Yes, 851/33)."},"skills":["A Complete Disjoint Partition and Global Mean Use the Same Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0025","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d34ee14b43837211dbe770b37bd0d1cd2f2209ad0750e4d799647da813c0facd","family_id":"qf-mod40-totalexp-partition-audit","seed":202611094025,"semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,34],[2,34],[31,34],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[24,25,27,123]}"},"fields":{"title":"A Complete Disjoint Partition and Global Mean Use the Same Law","stem":"Use the complete normalized finite law masses=[1/34, 2/34, 31/34, 0/1], aligned values X=[24, 25, 27, 123] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 3/34], [1, 31/34]], Yes, 911/34)","explanation":"Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 3/34], [1, 31/34]], Yes, 911/34)."},"skills":["A Complete Disjoint Partition and Global Mean Use the Same Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0026","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5de3118dff3a1a00da61df52fc9c183cf4324d1cd49b1af0016f684f0d4ad50","family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094026,"semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,10],[2,10],[7,10],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[0,1,3,99]}"},"fields":{"title":"Weight Every Positive Cell Mean by Its Own Cell Probability","stem":"Use the complete normalized finite law masses=[1/10, 2/10, 7/10, 0/1], aligned values X=[0, 1, 3, 99] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).","answer":{"kind":"expression","value":"([[0, 3/10, 2/3, 1/5], [1, 7/10, 3, 21/10]], 23/10, 23/10, Yes)","explanation":"Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 3/10, 2/3, 1/5], [1, 7/10, 3, 21/10]], 23/10, 23/10, Yes)."},"skills":["Weight Every Positive Cell Mean by Its Own Cell Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0027","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ddc3ff7a28fd9c99eed8e9c614d1cac36434c2deeee869efd66f06e9db409e67","family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094027,"semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,11],[2,11],[8,11],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[1,2,4,100]}"},"fields":{"title":"Weight Every Positive Cell Mean by Its Own Cell Probability","stem":"Use the complete normalized finite law masses=[1/11, 2/11, 8/11, 0/1], aligned values X=[1, 2, 4, 100] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).","answer":{"kind":"expression","value":"([[0, 3/11, 5/3, 5/11], [1, 8/11, 4, 32/11]], 37/11, 37/11, Yes)","explanation":"Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 3/11, 5/3, 5/11], [1, 8/11, 4, 32/11]], 37/11, 37/11, Yes)."},"skills":["Weight Every Positive Cell Mean by Its Own Cell Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0028","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aebfdc2ec598e21e174da5743f7c325ea64f31b477cf4e6fc59135f82e2d2466","family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094028,"semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,12],[2,12],[9,12],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[2,3,5,101]}"},"fields":{"title":"Weight Every Positive Cell Mean by Its Own Cell Probability","stem":"Use the complete normalized finite law masses=[1/12, 2/12, 9/12, 0/1], aligned values X=[2, 3, 5, 101] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).","answer":{"kind":"expression","value":"([[0, 1/4, 8/3, 2/3], [1, 3/4, 5, 15/4]], 53/12, 53/12, Yes)","explanation":"Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 1/4, 8/3, 2/3], [1, 3/4, 5, 15/4]], 53/12, 53/12, Yes)."},"skills":["Weight Every Positive Cell Mean by Its Own Cell Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0029","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d5ff37d986a02523b5e58c95672bc74b31ae189c4ebaec4538432afb139213d","family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094029,"semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,13],[2,13],[10,13],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[3,4,6,102]}"},"fields":{"title":"Weight Every Positive Cell Mean by Its Own Cell Probability","stem":"Use the complete normalized finite law masses=[1/13, 2/13, 10/13, 0/1], aligned values X=[3, 4, 6, 102] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).","answer":{"kind":"expression","value":"([[0, 3/13, 11/3, 11/13], [1, 10/13, 6, 60/13]], 71/13, 71/13, Yes)","explanation":"Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 3/13, 11/3, 11/13], [1, 10/13, 6, 60/13]], 71/13, 71/13, Yes)."},"skills":["Weight Every Positive Cell Mean by Its Own Cell Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0030","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e837fe541bf4506566ee70aed83992e2603dfd54582fb0d1bf78fd292ca298f2","family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094030,"semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,14],[2,14],[11,14],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[4,5,7,103]}"},"fields":{"title":"Weight Every Positive Cell Mean by Its Own Cell Probability","stem":"Use the complete normalized finite law masses=[1/14, 2/14, 11/14, 0/1], aligned values X=[4, 5, 7, 103] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).","answer":{"kind":"expression","value":"([[0, 3/14, 14/3, 1], [1, 11/14, 7, 11/2]], 13/2, 13/2, Yes)","explanation":"Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 3/14, 14/3, 1], [1, 11/14, 7, 11/2]], 13/2, 13/2, Yes)."},"skills":["Weight Every Positive Cell Mean by Its Own Cell Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0031","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1590bb6447e7bb5357aa058eef41c4de775ada6ee70cbbe86a777a4995c8f44","family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094031,"semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,15],[2,15],[12,15],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[5,6,8,104]}"},"fields":{"title":"Weight Every Positive Cell Mean by Its Own Cell Probability","stem":"Use the complete normalized finite law masses=[1/15, 2/15, 12/15, 0/1], aligned values X=[5, 6, 8, 104] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).","answer":{"kind":"expression","value":"([[0, 1/5, 17/3, 17/15], [1, 4/5, 8, 32/5]], 113/15, 113/15, Yes)","explanation":"Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 1/5, 17/3, 17/15], [1, 4/5, 8, 32/5]], 113/15, 113/15, Yes)."},"skills":["Weight Every Positive Cell Mean by Its Own Cell Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0032","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f38fd3ccfb224f3de65ed52e68471f235534f2fb98911921a0b6426666b7631","family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094032,"semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,16],[2,16],[13,16],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[6,7,9,105]}"},"fields":{"title":"Weight Every Positive Cell Mean by Its Own Cell Probability","stem":"Use the complete normalized finite law masses=[1/16, 2/16, 13/16, 0/1], aligned values X=[6, 7, 9, 105] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).","answer":{"kind":"expression","value":"([[0, 3/16, 20/3, 5/4], [1, 13/16, 9, 117/16]], 137/16, 137/16, Yes)","explanation":"Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 3/16, 20/3, 5/4], [1, 13/16, 9, 117/16]], 137/16, 137/16, Yes)."},"skills":["Weight Every Positive Cell Mean by Its Own Cell Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0033","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"306f2e54ff3690c7b7788a8e9165cd3b353ed52f8e36e2177caa019cf2b1da07","family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094033,"semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,17],[2,17],[14,17],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[7,8,10,106]}"},"fields":{"title":"Weight Every Positive Cell Mean by Its Own Cell Probability","stem":"Use the complete normalized finite law masses=[1/17, 2/17, 14/17, 0/1], aligned values X=[7, 8, 10, 106] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).","answer":{"kind":"expression","value":"([[0, 3/17, 23/3, 23/17], [1, 14/17, 10, 140/17]], 163/17, 163/17, Yes)","explanation":"Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 3/17, 23/3, 23/17], [1, 14/17, 10, 140/17]], 163/17, 163/17, Yes)."},"skills":["Weight Every Positive Cell Mean by Its Own Cell Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0034","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1223a03291a469ced7dad2127735a1139e778bce033163a7bd10f73fe46e7e7","family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094034,"semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,18],[2,18],[15,18],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[8,9,11,107]}"},"fields":{"title":"Weight Every Positive Cell Mean by Its Own Cell Probability","stem":"Use the complete normalized finite law masses=[1/18, 2/18, 15/18, 0/1], aligned values X=[8, 9, 11, 107] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).","answer":{"kind":"expression","value":"([[0, 1/6, 26/3, 13/9], [1, 5/6, 11, 55/6]], 191/18, 191/18, Yes)","explanation":"Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 1/6, 26/3, 13/9], [1, 5/6, 11, 55/6]], 191/18, 191/18, Yes)."},"skills":["Weight Every Positive Cell Mean by Its Own Cell Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0035","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9be9e3b9be0ac1340616295c0e1c45e956830ced9800fff29add62ce4677d06","family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094035,"semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,19],[2,19],[16,19],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[9,10,12,108]}"},"fields":{"title":"Weight Every Positive Cell Mean by Its Own Cell Probability","stem":"Use the complete normalized finite law masses=[1/19, 2/19, 16/19, 0/1], aligned values X=[9, 10, 12, 108] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).","answer":{"kind":"expression","value":"([[0, 3/19, 29/3, 29/19], [1, 16/19, 12, 192/19]], 221/19, 221/19, Yes)","explanation":"Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 3/19, 29/3, 29/19], [1, 16/19, 12, 192/19]], 221/19, 221/19, Yes)."},"skills":["Weight Every Positive Cell Mean by Its Own Cell Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0036","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ccd772c2d0cded4b960ae3eec76eba14bbc76a7cbcb97f1882476fc3be4f594d","family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094036,"semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,20],[2,20],[17,20],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[10,11,13,109]}"},"fields":{"title":"Weight Every Positive Cell Mean by Its Own Cell Probability","stem":"Use the complete normalized finite law masses=[1/20, 2/20, 17/20, 0/1], aligned values X=[10, 11, 13, 109] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).","answer":{"kind":"expression","value":"([[0, 3/20, 32/3, 8/5], [1, 17/20, 13, 221/20]], 253/20, 253/20, Yes)","explanation":"Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 3/20, 32/3, 8/5], [1, 17/20, 13, 221/20]], 253/20, 253/20, Yes)."},"skills":["Weight Every Positive Cell Mean by Its Own Cell Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0037","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58c598020aa1a39fc6bf26ac094962dd2d1547fd47d859d1797c67634695c7fe","family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094037,"semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,21],[2,21],[18,21],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[11,12,14,110]}"},"fields":{"title":"Weight Every Positive Cell Mean by Its Own Cell Probability","stem":"Use the complete normalized finite law masses=[1/21, 2/21, 18/21, 0/1], aligned values X=[11, 12, 14, 110] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).","answer":{"kind":"expression","value":"([[0, 1/7, 35/3, 5/3], [1, 6/7, 14, 12]], 41/3, 41/3, Yes)","explanation":"Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 1/7, 35/3, 5/3], [1, 6/7, 14, 12]], 41/3, 41/3, Yes)."},"skills":["Weight Every Positive Cell Mean by Its Own Cell Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0038","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1f4a1b9dce370f602250ff6c69010deedd2acae16ced9b3065edaf5b65b89b6","family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094038,"semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,22],[2,22],[19,22],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[12,13,15,111]}"},"fields":{"title":"Weight Every Positive Cell Mean by Its Own Cell Probability","stem":"Use the complete normalized finite law masses=[1/22, 2/22, 19/22, 0/1], aligned values X=[12, 13, 15, 111] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).","answer":{"kind":"expression","value":"([[0, 3/22, 38/3, 19/11], [1, 19/22, 15, 285/22]], 323/22, 323/22, Yes)","explanation":"Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 3/22, 38/3, 19/11], [1, 19/22, 15, 285/22]], 323/22, 323/22, Yes)."},"skills":["Weight Every Positive Cell Mean by Its Own Cell Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0039","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41ac2f5a3b84f730f44115d00bbdb152fd6ec370c51f3b2971921d4fc97c3b89","family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094039,"semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,23],[2,23],[20,23],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[13,14,16,112]}"},"fields":{"title":"Weight Every Positive Cell Mean by Its Own Cell Probability","stem":"Use the complete normalized finite law masses=[1/23, 2/23, 20/23, 0/1], aligned values X=[13, 14, 16, 112] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).","answer":{"kind":"expression","value":"([[0, 3/23, 41/3, 41/23], [1, 20/23, 16, 320/23]], 361/23, 361/23, Yes)","explanation":"Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 3/23, 41/3, 41/23], [1, 20/23, 16, 320/23]], 361/23, 361/23, Yes)."},"skills":["Weight Every Positive Cell Mean by Its Own Cell Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0040","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a1b90adc426711703b48aa217b1d2b88517268b21251e79744da590754617f8","family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094040,"semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,24],[2,24],[21,24],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[14,15,17,113]}"},"fields":{"title":"Weight Every Positive Cell Mean by Its Own Cell Probability","stem":"Use the complete normalized finite law masses=[1/24, 2/24, 21/24, 0/1], aligned values X=[14, 15, 17, 113] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).","answer":{"kind":"expression","value":"([[0, 1/8, 44/3, 11/6], [1, 7/8, 17, 119/8]], 401/24, 401/24, Yes)","explanation":"Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 1/8, 44/3, 11/6], [1, 7/8, 17, 119/8]], 401/24, 401/24, Yes)."},"skills":["Weight Every Positive Cell Mean by Its Own Cell Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0041","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca5e12fd217a2fed23d30957560373a94504bfdd46f5b7b58e67db7678ee6457","family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094041,"semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,25],[2,25],[22,25],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[15,16,18,114]}"},"fields":{"title":"Weight Every Positive Cell Mean by Its Own Cell Probability","stem":"Use the complete normalized finite law masses=[1/25, 2/25, 22/25, 0/1], aligned values X=[15, 16, 18, 114] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).","answer":{"kind":"expression","value":"([[0, 3/25, 47/3, 47/25], [1, 22/25, 18, 396/25]], 443/25, 443/25, Yes)","explanation":"Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 3/25, 47/3, 47/25], [1, 22/25, 18, 396/25]], 443/25, 443/25, Yes)."},"skills":["Weight Every Positive Cell Mean by Its Own Cell Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0042","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c91f017c91c8d331c3418c97ded1e19867b1e01c2a875a0b888c535aca7e969","family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094042,"semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,26],[2,26],[23,26],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[16,17,19,115]}"},"fields":{"title":"Weight Every Positive Cell Mean by Its Own Cell Probability","stem":"Use the complete normalized finite law masses=[1/26, 2/26, 23/26, 0/1], aligned values X=[16, 17, 19, 115] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).","answer":{"kind":"expression","value":"([[0, 3/26, 50/3, 25/13], [1, 23/26, 19, 437/26]], 487/26, 487/26, Yes)","explanation":"Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 3/26, 50/3, 25/13], [1, 23/26, 19, 437/26]], 487/26, 487/26, Yes)."},"skills":["Weight Every Positive Cell Mean by Its Own Cell Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0043","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c34a016598f5cfc191d332286c40b49e45045b524d5e0a2a99d4b16cb3110d71","family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094043,"semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,27],[2,27],[24,27],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[17,18,20,116]}"},"fields":{"title":"Weight Every Positive Cell Mean by Its Own Cell Probability","stem":"Use the complete normalized finite law masses=[1/27, 2/27, 24/27, 0/1], aligned values X=[17, 18, 20, 116] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).","answer":{"kind":"expression","value":"([[0, 1/9, 53/3, 53/27], [1, 8/9, 20, 160/9]], 533/27, 533/27, Yes)","explanation":"Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 1/9, 53/3, 53/27], [1, 8/9, 20, 160/9]], 533/27, 533/27, Yes)."},"skills":["Weight Every Positive Cell Mean by Its Own Cell Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0044","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c47743aa00e2ab7d75673498447596d681df69332b45f52911b0c99456f0ee3","family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094044,"semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,28],[2,28],[25,28],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[18,19,21,117]}"},"fields":{"title":"Weight Every Positive Cell Mean by Its Own Cell Probability","stem":"Use the complete normalized finite law masses=[1/28, 2/28, 25/28, 0/1], aligned values X=[18, 19, 21, 117] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).","answer":{"kind":"expression","value":"([[0, 3/28, 56/3, 2], [1, 25/28, 21, 75/4]], 83/4, 83/4, Yes)","explanation":"Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 3/28, 56/3, 2], [1, 25/28, 21, 75/4]], 83/4, 83/4, Yes)."},"skills":["Weight Every Positive Cell Mean by Its Own Cell Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0045","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60eedb007c62eb73e675bf51ef745ea2d495b1c39adbaf26c3b40dbda0294784","family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094045,"semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,29],[2,29],[26,29],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[19,20,22,118]}"},"fields":{"title":"Weight Every Positive Cell Mean by Its Own Cell Probability","stem":"Use the complete normalized finite law masses=[1/29, 2/29, 26/29, 0/1], aligned values X=[19, 20, 22, 118] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).","answer":{"kind":"expression","value":"([[0, 3/29, 59/3, 59/29], [1, 26/29, 22, 572/29]], 631/29, 631/29, Yes)","explanation":"Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 3/29, 59/3, 59/29], [1, 26/29, 22, 572/29]], 631/29, 631/29, Yes)."},"skills":["Weight Every Positive Cell Mean by Its Own Cell Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0046","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1976790ad53a6ac34e102830952bf619050ff3c4de61d1bd35805f2137bede00","family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094046,"semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,30],[2,30],[27,30],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[20,21,23,119]}"},"fields":{"title":"Weight Every Positive Cell Mean by Its Own Cell Probability","stem":"Use the complete normalized finite law masses=[1/30, 2/30, 27/30, 0/1], aligned values X=[20, 21, 23, 119] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).","answer":{"kind":"expression","value":"([[0, 1/10, 62/3, 31/15], [1, 9/10, 23, 207/10]], 683/30, 683/30, Yes)","explanation":"Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 1/10, 62/3, 31/15], [1, 9/10, 23, 207/10]], 683/30, 683/30, Yes)."},"skills":["Weight Every Positive Cell Mean by Its Own Cell Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0047","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93fa2737fb31c6e26db2de806893bf7587bb60ec6595669c98e6616b3e965a6f","family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094047,"semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,31],[2,31],[28,31],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[21,22,24,120]}"},"fields":{"title":"Weight Every Positive Cell Mean by Its Own Cell Probability","stem":"Use the complete normalized finite law masses=[1/31, 2/31, 28/31, 0/1], aligned values X=[21, 22, 24, 120] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).","answer":{"kind":"expression","value":"([[0, 3/31, 65/3, 65/31], [1, 28/31, 24, 672/31]], 737/31, 737/31, Yes)","explanation":"Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 3/31, 65/3, 65/31], [1, 28/31, 24, 672/31]], 737/31, 737/31, Yes)."},"skills":["Weight Every Positive Cell Mean by Its Own Cell Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0048","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7cf3de513ab795a74664dfeafba2472c481dd769aa4f51f2fcac4eecfcbf496f","family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094048,"semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,32],[2,32],[29,32],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[22,23,25,121]}"},"fields":{"title":"Weight Every Positive Cell Mean by Its Own Cell Probability","stem":"Use the complete normalized finite law masses=[1/32, 2/32, 29/32, 0/1], aligned values X=[22, 23, 25, 121] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).","answer":{"kind":"expression","value":"([[0, 3/32, 68/3, 17/8], [1, 29/32, 25, 725/32]], 793/32, 793/32, Yes)","explanation":"Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 3/32, 68/3, 17/8], [1, 29/32, 25, 725/32]], 793/32, 793/32, Yes)."},"skills":["Weight Every Positive Cell Mean by Its Own Cell Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0049","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb0ad0fda371ff31525e51e521658ea0b0908a9106717a384225ff743b80aa29","family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094049,"semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,33],[2,33],[30,33],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[23,24,26,122]}"},"fields":{"title":"Weight Every Positive Cell Mean by Its Own Cell Probability","stem":"Use the complete normalized finite law masses=[1/33, 2/33, 30/33, 0/1], aligned values X=[23, 24, 26, 122] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).","answer":{"kind":"expression","value":"([[0, 1/11, 71/3, 71/33], [1, 10/11, 26, 260/11]], 851/33, 851/33, Yes)","explanation":"Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 1/11, 71/3, 71/33], [1, 10/11, 26, 260/11]], 851/33, 851/33, Yes)."},"skills":["Weight Every Positive Cell Mean by Its Own Cell Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0050","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3fa2682eaa2b0ee87603a1bfdac46e14971f47b783a369288c9bd1c545b1c1f","family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094050,"semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,34],[2,34],[31,34],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[24,25,27,123]}"},"fields":{"title":"Weight Every Positive Cell Mean by Its Own Cell Probability","stem":"Use the complete normalized finite law masses=[1/34, 2/34, 31/34, 0/1], aligned values X=[24, 25, 27, 123] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).","answer":{"kind":"expression","value":"([[0, 3/34, 74/3, 37/17], [1, 31/34, 27, 837/34]], 911/34, 911/34, Yes)","explanation":"Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 3/34, 74/3, 37/17], [1, 31/34, 27, 837/34]], 911/34, 911/34, Yes)."},"skills":["Weight Every Positive Cell Mean by Its Own Cell Probability"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0051","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa920e47e289295117085c370959a0deeebdb10a6fd0ee992d8b1b4182810c59","family_id":"qf-mod40-totalexp-tower-identity","seed":202611094051,"semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,10],[2,10],[7,10],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[0,1,3,99]}"},"fields":{"title":"A Cell-Constant Conditional Version Has the Same Expectation","stem":"Use the complete normalized finite law masses=[1/10, 2/10, 7/10, 0/1], aligned values X=[0, 1, 3, 99] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).","answer":{"kind":"expression","value":"([2/3, 2/3, 3, 3], 23/10, 23/10, Yes, conditional_expectation_constant_on_partition_cells)","explanation":"Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([2/3, 2/3, 3, 3], 23/10, 23/10, Yes, conditional_expectation_constant_on_partition_cells)."},"skills":["A Cell-Constant Conditional Version Has the Same Expectation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0052","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f394d4943f1b23b6b2899dca09c3e333246085d3dedf764ff38c65d00fb5ef7","family_id":"qf-mod40-totalexp-tower-identity","seed":202611094052,"semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,11],[2,11],[8,11],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[1,2,4,100]}"},"fields":{"title":"A Cell-Constant Conditional Version Has the Same Expectation","stem":"Use the complete normalized finite law masses=[1/11, 2/11, 8/11, 0/1], aligned values X=[1, 2, 4, 100] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).","answer":{"kind":"expression","value":"([5/3, 5/3, 4, 4], 37/11, 37/11, Yes, conditional_expectation_constant_on_partition_cells)","explanation":"Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([5/3, 5/3, 4, 4], 37/11, 37/11, Yes, conditional_expectation_constant_on_partition_cells)."},"skills":["A Cell-Constant Conditional Version Has the Same Expectation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0053","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0572afc0b0904337ea2f2506a7317d29700cfab4b8e52f094e9a2395f47a71c8","family_id":"qf-mod40-totalexp-tower-identity","seed":202611094053,"semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,12],[2,12],[9,12],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[2,3,5,101]}"},"fields":{"title":"A Cell-Constant Conditional Version Has the Same Expectation","stem":"Use the complete normalized finite law masses=[1/12, 2/12, 9/12, 0/1], aligned values X=[2, 3, 5, 101] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).","answer":{"kind":"expression","value":"([8/3, 8/3, 5, 5], 53/12, 53/12, Yes, conditional_expectation_constant_on_partition_cells)","explanation":"Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([8/3, 8/3, 5, 5], 53/12, 53/12, Yes, conditional_expectation_constant_on_partition_cells)."},"skills":["A Cell-Constant Conditional Version Has the Same Expectation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0054","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d48512e11ccc6a125cb6c7afb734c7e2ed7db157b6a1cd8f8723e85382bb7af0","family_id":"qf-mod40-totalexp-tower-identity","seed":202611094054,"semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,13],[2,13],[10,13],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[3,4,6,102]}"},"fields":{"title":"A Cell-Constant Conditional Version Has the Same Expectation","stem":"Use the complete normalized finite law masses=[1/13, 2/13, 10/13, 0/1], aligned values X=[3, 4, 6, 102] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).","answer":{"kind":"expression","value":"([11/3, 11/3, 6, 6], 71/13, 71/13, Yes, conditional_expectation_constant_on_partition_cells)","explanation":"Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([11/3, 11/3, 6, 6], 71/13, 71/13, Yes, conditional_expectation_constant_on_partition_cells)."},"skills":["A Cell-Constant Conditional Version Has the Same Expectation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0055","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d57807a8c870827b139fd2baa4a3219b2ff4ca1bfe95c88106f6be84be13cc3","family_id":"qf-mod40-totalexp-tower-identity","seed":202611094055,"semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,14],[2,14],[11,14],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[4,5,7,103]}"},"fields":{"title":"A Cell-Constant Conditional Version Has the Same Expectation","stem":"Use the complete normalized finite law masses=[1/14, 2/14, 11/14, 0/1], aligned values X=[4, 5, 7, 103] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).","answer":{"kind":"expression","value":"([14/3, 14/3, 7, 7], 13/2, 13/2, Yes, conditional_expectation_constant_on_partition_cells)","explanation":"Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([14/3, 14/3, 7, 7], 13/2, 13/2, Yes, conditional_expectation_constant_on_partition_cells)."},"skills":["A Cell-Constant Conditional Version Has the Same Expectation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0056","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95cc4ea4f8e05d2e8e799dbee1a4c394c97de3ce69898b6d0208d01bea094152","family_id":"qf-mod40-totalexp-tower-identity","seed":202611094056,"semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,15],[2,15],[12,15],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[5,6,8,104]}"},"fields":{"title":"A Cell-Constant Conditional Version Has the Same Expectation","stem":"Use the complete normalized finite law masses=[1/15, 2/15, 12/15, 0/1], aligned values X=[5, 6, 8, 104] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).","answer":{"kind":"expression","value":"([17/3, 17/3, 8, 8], 113/15, 113/15, Yes, conditional_expectation_constant_on_partition_cells)","explanation":"Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([17/3, 17/3, 8, 8], 113/15, 113/15, Yes, conditional_expectation_constant_on_partition_cells)."},"skills":["A Cell-Constant Conditional Version Has the Same Expectation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0057","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e294b529072524b9f646dfd97bcd085e54cbbd323835eff17814f9745d0b601e","family_id":"qf-mod40-totalexp-tower-identity","seed":202611094057,"semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,16],[2,16],[13,16],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[6,7,9,105]}"},"fields":{"title":"A Cell-Constant Conditional Version Has the Same Expectation","stem":"Use the complete normalized finite law masses=[1/16, 2/16, 13/16, 0/1], aligned values X=[6, 7, 9, 105] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).","answer":{"kind":"expression","value":"([20/3, 20/3, 9, 9], 137/16, 137/16, Yes, conditional_expectation_constant_on_partition_cells)","explanation":"Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([20/3, 20/3, 9, 9], 137/16, 137/16, Yes, conditional_expectation_constant_on_partition_cells)."},"skills":["A Cell-Constant Conditional Version Has the Same Expectation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0058","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e2274f95e34847570787fda885e7cca4b03d3915b066d47c18c2fb1f917150e","family_id":"qf-mod40-totalexp-tower-identity","seed":202611094058,"semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,17],[2,17],[14,17],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[7,8,10,106]}"},"fields":{"title":"A Cell-Constant Conditional Version Has the Same Expectation","stem":"Use the complete normalized finite law masses=[1/17, 2/17, 14/17, 0/1], aligned values X=[7, 8, 10, 106] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).","answer":{"kind":"expression","value":"([23/3, 23/3, 10, 10], 163/17, 163/17, Yes, conditional_expectation_constant_on_partition_cells)","explanation":"Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([23/3, 23/3, 10, 10], 163/17, 163/17, Yes, conditional_expectation_constant_on_partition_cells)."},"skills":["A Cell-Constant Conditional Version Has the Same Expectation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0059","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2954984cde5de91905dc89241ab3866ef35563cb3ddc2d2c099a6f9add568dc6","family_id":"qf-mod40-totalexp-tower-identity","seed":202611094059,"semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,18],[2,18],[15,18],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[8,9,11,107]}"},"fields":{"title":"A Cell-Constant Conditional Version Has the Same Expectation","stem":"Use the complete normalized finite law masses=[1/18, 2/18, 15/18, 0/1], aligned values X=[8, 9, 11, 107] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).","answer":{"kind":"expression","value":"([26/3, 26/3, 11, 11], 191/18, 191/18, Yes, conditional_expectation_constant_on_partition_cells)","explanation":"Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([26/3, 26/3, 11, 11], 191/18, 191/18, Yes, conditional_expectation_constant_on_partition_cells)."},"skills":["A Cell-Constant Conditional Version Has the Same Expectation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0060","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"422c9287409e51e84d6cd2a33bcdfc4cbb26f9d04b00e403c7e04df3234ea3f9","family_id":"qf-mod40-totalexp-tower-identity","seed":202611094060,"semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,19],[2,19],[16,19],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[9,10,12,108]}"},"fields":{"title":"A Cell-Constant Conditional Version Has the Same Expectation","stem":"Use the complete normalized finite law masses=[1/19, 2/19, 16/19, 0/1], aligned values X=[9, 10, 12, 108] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).","answer":{"kind":"expression","value":"([29/3, 29/3, 12, 12], 221/19, 221/19, Yes, conditional_expectation_constant_on_partition_cells)","explanation":"Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([29/3, 29/3, 12, 12], 221/19, 221/19, Yes, conditional_expectation_constant_on_partition_cells)."},"skills":["A Cell-Constant Conditional Version Has the Same Expectation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0061","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8941a3096a01dd97453de3660fc83a492b7546bf1a83a9d42cef7b2c63cef98c","family_id":"qf-mod40-totalexp-tower-identity","seed":202611094061,"semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,20],[2,20],[17,20],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[10,11,13,109]}"},"fields":{"title":"A Cell-Constant Conditional Version Has the Same Expectation","stem":"Use the complete normalized finite law masses=[1/20, 2/20, 17/20, 0/1], aligned values X=[10, 11, 13, 109] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).","answer":{"kind":"expression","value":"([32/3, 32/3, 13, 13], 253/20, 253/20, Yes, conditional_expectation_constant_on_partition_cells)","explanation":"Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([32/3, 32/3, 13, 13], 253/20, 253/20, Yes, conditional_expectation_constant_on_partition_cells)."},"skills":["A Cell-Constant Conditional Version Has the Same Expectation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0062","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4eacfad87fcea50268d3e1c71f1fc5b5b4e99712f1525710d198b256bdf19fd","family_id":"qf-mod40-totalexp-tower-identity","seed":202611094062,"semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,21],[2,21],[18,21],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[11,12,14,110]}"},"fields":{"title":"A Cell-Constant Conditional Version Has the Same Expectation","stem":"Use the complete normalized finite law masses=[1/21, 2/21, 18/21, 0/1], aligned values X=[11, 12, 14, 110] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).","answer":{"kind":"expression","value":"([35/3, 35/3, 14, 14], 41/3, 41/3, Yes, conditional_expectation_constant_on_partition_cells)","explanation":"Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([35/3, 35/3, 14, 14], 41/3, 41/3, Yes, conditional_expectation_constant_on_partition_cells)."},"skills":["A Cell-Constant Conditional Version Has the Same Expectation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0063","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"455214829f89137e9cb2d163398c3b6f50f0c07f190a9b89b453911156451253","family_id":"qf-mod40-totalexp-tower-identity","seed":202611094063,"semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,22],[2,22],[19,22],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[12,13,15,111]}"},"fields":{"title":"A Cell-Constant Conditional Version Has the Same Expectation","stem":"Use the complete normalized finite law masses=[1/22, 2/22, 19/22, 0/1], aligned values X=[12, 13, 15, 111] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).","answer":{"kind":"expression","value":"([38/3, 38/3, 15, 15], 323/22, 323/22, Yes, conditional_expectation_constant_on_partition_cells)","explanation":"Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([38/3, 38/3, 15, 15], 323/22, 323/22, Yes, conditional_expectation_constant_on_partition_cells)."},"skills":["A Cell-Constant Conditional Version Has the Same Expectation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0064","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2268eaa7e18a32a58ed73937ffa4b4bafd02378b4119ca1d87a57b1136e75eb9","family_id":"qf-mod40-totalexp-tower-identity","seed":202611094064,"semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,23],[2,23],[20,23],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[13,14,16,112]}"},"fields":{"title":"A Cell-Constant Conditional Version Has the Same Expectation","stem":"Use the complete normalized finite law masses=[1/23, 2/23, 20/23, 0/1], aligned values X=[13, 14, 16, 112] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).","answer":{"kind":"expression","value":"([41/3, 41/3, 16, 16], 361/23, 361/23, Yes, conditional_expectation_constant_on_partition_cells)","explanation":"Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([41/3, 41/3, 16, 16], 361/23, 361/23, Yes, conditional_expectation_constant_on_partition_cells)."},"skills":["A Cell-Constant Conditional Version Has the Same Expectation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0065","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a2d7f3bf1ea4ef68852e49c836bb02cdd609419e20ab137cf54c8b4e8ffadca","family_id":"qf-mod40-totalexp-tower-identity","seed":202611094065,"semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,24],[2,24],[21,24],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[14,15,17,113]}"},"fields":{"title":"A Cell-Constant Conditional Version Has the Same Expectation","stem":"Use the complete normalized finite law masses=[1/24, 2/24, 21/24, 0/1], aligned values X=[14, 15, 17, 113] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).","answer":{"kind":"expression","value":"([44/3, 44/3, 17, 17], 401/24, 401/24, Yes, conditional_expectation_constant_on_partition_cells)","explanation":"Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([44/3, 44/3, 17, 17], 401/24, 401/24, Yes, conditional_expectation_constant_on_partition_cells)."},"skills":["A Cell-Constant Conditional Version Has the Same Expectation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0066","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"553428ca5ba4b99109554c4c1b4015161f1531a56668449454614557154865d7","family_id":"qf-mod40-totalexp-tower-identity","seed":202611094066,"semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,25],[2,25],[22,25],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[15,16,18,114]}"},"fields":{"title":"A Cell-Constant Conditional Version Has the Same Expectation","stem":"Use the complete normalized finite law masses=[1/25, 2/25, 22/25, 0/1], aligned values X=[15, 16, 18, 114] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).","answer":{"kind":"expression","value":"([47/3, 47/3, 18, 18], 443/25, 443/25, Yes, conditional_expectation_constant_on_partition_cells)","explanation":"Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([47/3, 47/3, 18, 18], 443/25, 443/25, Yes, conditional_expectation_constant_on_partition_cells)."},"skills":["A Cell-Constant Conditional Version Has the Same Expectation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0067","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b028a12b5b3950519dfc2eeb9540e2ca38150e895afae5955770fec58fd48f05","family_id":"qf-mod40-totalexp-tower-identity","seed":202611094067,"semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,26],[2,26],[23,26],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[16,17,19,115]}"},"fields":{"title":"A Cell-Constant Conditional Version Has the Same Expectation","stem":"Use the complete normalized finite law masses=[1/26, 2/26, 23/26, 0/1], aligned values X=[16, 17, 19, 115] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).","answer":{"kind":"expression","value":"([50/3, 50/3, 19, 19], 487/26, 487/26, Yes, conditional_expectation_constant_on_partition_cells)","explanation":"Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([50/3, 50/3, 19, 19], 487/26, 487/26, Yes, conditional_expectation_constant_on_partition_cells)."},"skills":["A Cell-Constant Conditional Version Has the Same Expectation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0068","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70b67609086c27da09f2eb183c48eb3e8622b5cfcd30120559fd8a4b8621f6b9","family_id":"qf-mod40-totalexp-tower-identity","seed":202611094068,"semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,27],[2,27],[24,27],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[17,18,20,116]}"},"fields":{"title":"A Cell-Constant Conditional Version Has the Same Expectation","stem":"Use the complete normalized finite law masses=[1/27, 2/27, 24/27, 0/1], aligned values X=[17, 18, 20, 116] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).","answer":{"kind":"expression","value":"([53/3, 53/3, 20, 20], 533/27, 533/27, Yes, conditional_expectation_constant_on_partition_cells)","explanation":"Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([53/3, 53/3, 20, 20], 533/27, 533/27, Yes, conditional_expectation_constant_on_partition_cells)."},"skills":["A Cell-Constant Conditional Version Has the Same Expectation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0069","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30ecf6d3ef799b1a26c37cc6e563ce33be7460b0d8b11504d5d98b49f2d9cb4a","family_id":"qf-mod40-totalexp-tower-identity","seed":202611094069,"semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,28],[2,28],[25,28],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[18,19,21,117]}"},"fields":{"title":"A Cell-Constant Conditional Version Has the Same Expectation","stem":"Use the complete normalized finite law masses=[1/28, 2/28, 25/28, 0/1], aligned values X=[18, 19, 21, 117] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).","answer":{"kind":"expression","value":"([56/3, 56/3, 21, 21], 83/4, 83/4, Yes, conditional_expectation_constant_on_partition_cells)","explanation":"Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([56/3, 56/3, 21, 21], 83/4, 83/4, Yes, conditional_expectation_constant_on_partition_cells)."},"skills":["A Cell-Constant Conditional Version Has the Same Expectation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0070","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f6ce181ae6aad6b22c08604756e42060c179ee364c63810d87cde054c88e94b","family_id":"qf-mod40-totalexp-tower-identity","seed":202611094070,"semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,29],[2,29],[26,29],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[19,20,22,118]}"},"fields":{"title":"A Cell-Constant Conditional Version Has the Same Expectation","stem":"Use the complete normalized finite law masses=[1/29, 2/29, 26/29, 0/1], aligned values X=[19, 20, 22, 118] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).","answer":{"kind":"expression","value":"([59/3, 59/3, 22, 22], 631/29, 631/29, Yes, conditional_expectation_constant_on_partition_cells)","explanation":"Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([59/3, 59/3, 22, 22], 631/29, 631/29, Yes, conditional_expectation_constant_on_partition_cells)."},"skills":["A Cell-Constant Conditional Version Has the Same Expectation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0071","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3eaf5c389da1622b762134aa1679d944a708366dad25f4fe0eaa5a90c8c757c1","family_id":"qf-mod40-totalexp-tower-identity","seed":202611094071,"semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,30],[2,30],[27,30],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[20,21,23,119]}"},"fields":{"title":"A Cell-Constant Conditional Version Has the Same Expectation","stem":"Use the complete normalized finite law masses=[1/30, 2/30, 27/30, 0/1], aligned values X=[20, 21, 23, 119] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).","answer":{"kind":"expression","value":"([62/3, 62/3, 23, 23], 683/30, 683/30, Yes, conditional_expectation_constant_on_partition_cells)","explanation":"Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([62/3, 62/3, 23, 23], 683/30, 683/30, Yes, conditional_expectation_constant_on_partition_cells)."},"skills":["A Cell-Constant Conditional Version Has the Same Expectation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0072","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b43e0ccfd67f7580d572da0433181556e85661d78f65552c37aec9a78b2e4194","family_id":"qf-mod40-totalexp-tower-identity","seed":202611094072,"semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,31],[2,31],[28,31],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[21,22,24,120]}"},"fields":{"title":"A Cell-Constant Conditional Version Has the Same Expectation","stem":"Use the complete normalized finite law masses=[1/31, 2/31, 28/31, 0/1], aligned values X=[21, 22, 24, 120] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).","answer":{"kind":"expression","value":"([65/3, 65/3, 24, 24], 737/31, 737/31, Yes, conditional_expectation_constant_on_partition_cells)","explanation":"Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([65/3, 65/3, 24, 24], 737/31, 737/31, Yes, conditional_expectation_constant_on_partition_cells)."},"skills":["A Cell-Constant Conditional Version Has the Same Expectation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0073","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc9efa5cc386a21adee86741a62a07be8a6d2c1abfd128864eac63ba5621cba3","family_id":"qf-mod40-totalexp-tower-identity","seed":202611094073,"semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,32],[2,32],[29,32],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[22,23,25,121]}"},"fields":{"title":"A Cell-Constant Conditional Version Has the Same Expectation","stem":"Use the complete normalized finite law masses=[1/32, 2/32, 29/32, 0/1], aligned values X=[22, 23, 25, 121] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).","answer":{"kind":"expression","value":"([68/3, 68/3, 25, 25], 793/32, 793/32, Yes, conditional_expectation_constant_on_partition_cells)","explanation":"Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([68/3, 68/3, 25, 25], 793/32, 793/32, Yes, conditional_expectation_constant_on_partition_cells)."},"skills":["A Cell-Constant Conditional Version Has the Same Expectation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0074","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3f18d10af258c929f8cd7ead649d6a05b63b82cb7ebd6ea2b45f03c6c08b0d5","family_id":"qf-mod40-totalexp-tower-identity","seed":202611094074,"semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,33],[2,33],[30,33],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[23,24,26,122]}"},"fields":{"title":"A Cell-Constant Conditional Version Has the Same Expectation","stem":"Use the complete normalized finite law masses=[1/33, 2/33, 30/33, 0/1], aligned values X=[23, 24, 26, 122] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).","answer":{"kind":"expression","value":"([71/3, 71/3, 26, 26], 851/33, 851/33, Yes, conditional_expectation_constant_on_partition_cells)","explanation":"Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([71/3, 71/3, 26, 26], 851/33, 851/33, Yes, conditional_expectation_constant_on_partition_cells)."},"skills":["A Cell-Constant Conditional Version Has the Same Expectation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0075","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74dd5fc03418dec9c1d3c65866145b516f192356797c52ca6f952a0007318eae","family_id":"qf-mod40-totalexp-tower-identity","seed":202611094075,"semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,34],[2,34],[31,34],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[24,25,27,123]}"},"fields":{"title":"A Cell-Constant Conditional Version Has the Same Expectation","stem":"Use the complete normalized finite law masses=[1/34, 2/34, 31/34, 0/1], aligned values X=[24, 25, 27, 123] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).","answer":{"kind":"expression","value":"([74/3, 74/3, 27, 27], 911/34, 911/34, Yes, conditional_expectation_constant_on_partition_cells)","explanation":"Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([74/3, 74/3, 27, 27], 911/34, 911/34, Yes, conditional_expectation_constant_on_partition_cells)."},"skills":["A Cell-Constant Conditional Version Has the Same Expectation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0076","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"793a9c0c48f0cf4993573f7a1124901bfe33812e59262f4ec624e17d7d83ea5d","family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094076,"semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,10],[2,10],[7,10],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[0,1,3,99]}"},"fields":{"title":"Null Cells Retain Undefined Quotients and Zero Weighted Contributions","stem":"Use the complete normalized finite law masses=[1/10, 2/10, 7/10, 0/1], aligned values X=[0, 1, 3, 99] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).","answer":{"kind":"expression","value":"([[0, 1, 23/10], [1, 0, null]], 23/10, 23/10, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation":"Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 23/10], [1, 0, null]], 23/10, 23/10, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)."},"skills":["Null Cells Retain Undefined Quotients and Zero Weighted Contributions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0077","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a99f53f17099517275ac0bb818dc5b71581889cd3e35549a7b3b9a620e56887a","family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094077,"semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,11],[2,11],[8,11],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[1,2,4,100]}"},"fields":{"title":"Null Cells Retain Undefined Quotients and Zero Weighted Contributions","stem":"Use the complete normalized finite law masses=[1/11, 2/11, 8/11, 0/1], aligned values X=[1, 2, 4, 100] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).","answer":{"kind":"expression","value":"([[0, 1, 37/11], [1, 0, null]], 37/11, 37/11, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation":"Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 37/11], [1, 0, null]], 37/11, 37/11, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)."},"skills":["Null Cells Retain Undefined Quotients and Zero Weighted Contributions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0078","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e759f2da60c4b0f17dfd65521faaac40d8b0f45328e7d8cd32a34ad474312597","family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094078,"semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,12],[2,12],[9,12],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[2,3,5,101]}"},"fields":{"title":"Null Cells Retain Undefined Quotients and Zero Weighted Contributions","stem":"Use the complete normalized finite law masses=[1/12, 2/12, 9/12, 0/1], aligned values X=[2, 3, 5, 101] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).","answer":{"kind":"expression","value":"([[0, 1, 53/12], [1, 0, null]], 53/12, 53/12, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation":"Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 53/12], [1, 0, null]], 53/12, 53/12, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)."},"skills":["Null Cells Retain Undefined Quotients and Zero Weighted Contributions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0079","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc882dbfd771f0d568af3bbb83c553900f75ebea48772a01fdc964e9dd5fd48d","family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094079,"semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,13],[2,13],[10,13],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[3,4,6,102]}"},"fields":{"title":"Null Cells Retain Undefined Quotients and Zero Weighted Contributions","stem":"Use the complete normalized finite law masses=[1/13, 2/13, 10/13, 0/1], aligned values X=[3, 4, 6, 102] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).","answer":{"kind":"expression","value":"([[0, 1, 71/13], [1, 0, null]], 71/13, 71/13, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation":"Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 71/13], [1, 0, null]], 71/13, 71/13, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)."},"skills":["Null Cells Retain Undefined Quotients and Zero Weighted Contributions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0080","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93dec73b4260fde5917e812208429b4e43cc3736167e05f59e4f4f1253bb021b","family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094080,"semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,14],[2,14],[11,14],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[4,5,7,103]}"},"fields":{"title":"Null Cells Retain Undefined Quotients and Zero Weighted Contributions","stem":"Use the complete normalized finite law masses=[1/14, 2/14, 11/14, 0/1], aligned values X=[4, 5, 7, 103] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).","answer":{"kind":"expression","value":"([[0, 1, 13/2], [1, 0, null]], 13/2, 13/2, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation":"Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 13/2], [1, 0, null]], 13/2, 13/2, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)."},"skills":["Null Cells Retain Undefined Quotients and Zero Weighted Contributions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0081","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c6e97e04f8c38fc4b115f2458f598a42669c56855ad8d560d19e9642517bc60","family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094081,"semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,15],[2,15],[12,15],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[5,6,8,104]}"},"fields":{"title":"Null Cells Retain Undefined Quotients and Zero Weighted Contributions","stem":"Use the complete normalized finite law masses=[1/15, 2/15, 12/15, 0/1], aligned values X=[5, 6, 8, 104] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).","answer":{"kind":"expression","value":"([[0, 1, 113/15], [1, 0, null]], 113/15, 113/15, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation":"Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 113/15], [1, 0, null]], 113/15, 113/15, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)."},"skills":["Null Cells Retain Undefined Quotients and Zero Weighted Contributions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0082","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b308ff7d506c76554e499722247bda9a58cf66c804ecf98bf96674d51273ecfe","family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094082,"semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,16],[2,16],[13,16],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[6,7,9,105]}"},"fields":{"title":"Null Cells Retain Undefined Quotients and Zero Weighted Contributions","stem":"Use the complete normalized finite law masses=[1/16, 2/16, 13/16, 0/1], aligned values X=[6, 7, 9, 105] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).","answer":{"kind":"expression","value":"([[0, 1, 137/16], [1, 0, null]], 137/16, 137/16, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation":"Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 137/16], [1, 0, null]], 137/16, 137/16, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)."},"skills":["Null Cells Retain Undefined Quotients and Zero Weighted Contributions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0083","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c535b714e0b7dc7477a94289247ccff007a37054a99df9c720aae77ec42c3d79","family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094083,"semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,17],[2,17],[14,17],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[7,8,10,106]}"},"fields":{"title":"Null Cells Retain Undefined Quotients and Zero Weighted Contributions","stem":"Use the complete normalized finite law masses=[1/17, 2/17, 14/17, 0/1], aligned values X=[7, 8, 10, 106] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).","answer":{"kind":"expression","value":"([[0, 1, 163/17], [1, 0, null]], 163/17, 163/17, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation":"Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 163/17], [1, 0, null]], 163/17, 163/17, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)."},"skills":["Null Cells Retain Undefined Quotients and Zero Weighted Contributions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0084","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6860032802bb5d62fcf7b7aa6469920b8d6be659592505aabc5c149f6696004b","family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094084,"semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,18],[2,18],[15,18],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[8,9,11,107]}"},"fields":{"title":"Null Cells Retain Undefined Quotients and Zero Weighted Contributions","stem":"Use the complete normalized finite law masses=[1/18, 2/18, 15/18, 0/1], aligned values X=[8, 9, 11, 107] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).","answer":{"kind":"expression","value":"([[0, 1, 191/18], [1, 0, null]], 191/18, 191/18, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation":"Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 191/18], [1, 0, null]], 191/18, 191/18, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)."},"skills":["Null Cells Retain Undefined Quotients and Zero Weighted Contributions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0085","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fab20ab2f9e2a5191515fe1034b07372746194701e7b912e6a9cc704ba57b583","family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094085,"semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,19],[2,19],[16,19],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[9,10,12,108]}"},"fields":{"title":"Null Cells Retain Undefined Quotients and Zero Weighted Contributions","stem":"Use the complete normalized finite law masses=[1/19, 2/19, 16/19, 0/1], aligned values X=[9, 10, 12, 108] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).","answer":{"kind":"expression","value":"([[0, 1, 221/19], [1, 0, null]], 221/19, 221/19, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation":"Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 221/19], [1, 0, null]], 221/19, 221/19, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)."},"skills":["Null Cells Retain Undefined Quotients and Zero Weighted Contributions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0086","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7c07507d8680b86fddd05add37f9cd1a79dba6ae01fc998be956b2a365d5980f","family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094086,"semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,20],[2,20],[17,20],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[10,11,13,109]}"},"fields":{"title":"Null Cells Retain Undefined Quotients and Zero Weighted Contributions","stem":"Use the complete normalized finite law masses=[1/20, 2/20, 17/20, 0/1], aligned values X=[10, 11, 13, 109] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).","answer":{"kind":"expression","value":"([[0, 1, 253/20], [1, 0, null]], 253/20, 253/20, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation":"Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 253/20], [1, 0, null]], 253/20, 253/20, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)."},"skills":["Null Cells Retain Undefined Quotients and Zero Weighted Contributions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0087","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5565d3a672318a08c949592c7b6d16de842a968301672f2b245b7b9e4fb81aa2","family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094087,"semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,21],[2,21],[18,21],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[11,12,14,110]}"},"fields":{"title":"Null Cells Retain Undefined Quotients and Zero Weighted Contributions","stem":"Use the complete normalized finite law masses=[1/21, 2/21, 18/21, 0/1], aligned values X=[11, 12, 14, 110] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).","answer":{"kind":"expression","value":"([[0, 1, 41/3], [1, 0, null]], 41/3, 41/3, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation":"Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 41/3], [1, 0, null]], 41/3, 41/3, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)."},"skills":["Null Cells Retain Undefined Quotients and Zero Weighted Contributions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0088","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"beeae0a1b369b13f8b64f2f64b86b024de1a7be0a4adb51586da02667af83800","family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094088,"semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,22],[2,22],[19,22],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[12,13,15,111]}"},"fields":{"title":"Null Cells Retain Undefined Quotients and Zero Weighted Contributions","stem":"Use the complete normalized finite law masses=[1/22, 2/22, 19/22, 0/1], aligned values X=[12, 13, 15, 111] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).","answer":{"kind":"expression","value":"([[0, 1, 323/22], [1, 0, null]], 323/22, 323/22, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation":"Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 323/22], [1, 0, null]], 323/22, 323/22, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)."},"skills":["Null Cells Retain Undefined Quotients and Zero Weighted Contributions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0089","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"150974b34aa9d8fc74af70db9dbbf4af5ae220ff3b2bfa0a22eb74cd100f06b5","family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094089,"semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,23],[2,23],[20,23],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[13,14,16,112]}"},"fields":{"title":"Null Cells Retain Undefined Quotients and Zero Weighted Contributions","stem":"Use the complete normalized finite law masses=[1/23, 2/23, 20/23, 0/1], aligned values X=[13, 14, 16, 112] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).","answer":{"kind":"expression","value":"([[0, 1, 361/23], [1, 0, null]], 361/23, 361/23, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation":"Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 361/23], [1, 0, null]], 361/23, 361/23, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)."},"skills":["Null Cells Retain Undefined Quotients and Zero Weighted Contributions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0090","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"604ad49fc221d3d581658079c76ab930f5dea3247a4bf09d123f44e5b6959d70","family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094090,"semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,24],[2,24],[21,24],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[14,15,17,113]}"},"fields":{"title":"Null Cells Retain Undefined Quotients and Zero Weighted Contributions","stem":"Use the complete normalized finite law masses=[1/24, 2/24, 21/24, 0/1], aligned values X=[14, 15, 17, 113] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).","answer":{"kind":"expression","value":"([[0, 1, 401/24], [1, 0, null]], 401/24, 401/24, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation":"Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 401/24], [1, 0, null]], 401/24, 401/24, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)."},"skills":["Null Cells Retain Undefined Quotients and Zero Weighted Contributions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0091","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ada24bfe610f7e17b6638259cc30fb48c92b72d32c49b207fc4d2612c0cb6ec9","family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094091,"semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,25],[2,25],[22,25],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[15,16,18,114]}"},"fields":{"title":"Null Cells Retain Undefined Quotients and Zero Weighted Contributions","stem":"Use the complete normalized finite law masses=[1/25, 2/25, 22/25, 0/1], aligned values X=[15, 16, 18, 114] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).","answer":{"kind":"expression","value":"([[0, 1, 443/25], [1, 0, null]], 443/25, 443/25, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation":"Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 443/25], [1, 0, null]], 443/25, 443/25, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)."},"skills":["Null Cells Retain Undefined Quotients and Zero Weighted Contributions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0092","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c15a49cc1dd4764412bf9e453e8ecd436b7db63f94dee9a1c538ecfde5e1e6f","family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094092,"semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,26],[2,26],[23,26],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[16,17,19,115]}"},"fields":{"title":"Null Cells Retain Undefined Quotients and Zero Weighted Contributions","stem":"Use the complete normalized finite law masses=[1/26, 2/26, 23/26, 0/1], aligned values X=[16, 17, 19, 115] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).","answer":{"kind":"expression","value":"([[0, 1, 487/26], [1, 0, null]], 487/26, 487/26, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation":"Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 487/26], [1, 0, null]], 487/26, 487/26, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)."},"skills":["Null Cells Retain Undefined Quotients and Zero Weighted Contributions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0093","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e3f7295721831a025d4c54657f4eb1f95e714acc6b95e92ded5dc0c59247a484","family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094093,"semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,27],[2,27],[24,27],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[17,18,20,116]}"},"fields":{"title":"Null Cells Retain Undefined Quotients and Zero Weighted Contributions","stem":"Use the complete normalized finite law masses=[1/27, 2/27, 24/27, 0/1], aligned values X=[17, 18, 20, 116] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).","answer":{"kind":"expression","value":"([[0, 1, 533/27], [1, 0, null]], 533/27, 533/27, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation":"Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 533/27], [1, 0, null]], 533/27, 533/27, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)."},"skills":["Null Cells Retain Undefined Quotients and Zero Weighted Contributions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0094","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1f1682fb791ff7ab0e745c6c4f565642b2b808d70c2b72857e1a38f68c78964","family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094094,"semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,28],[2,28],[25,28],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[18,19,21,117]}"},"fields":{"title":"Null Cells Retain Undefined Quotients and Zero Weighted Contributions","stem":"Use the complete normalized finite law masses=[1/28, 2/28, 25/28, 0/1], aligned values X=[18, 19, 21, 117] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).","answer":{"kind":"expression","value":"([[0, 1, 83/4], [1, 0, null]], 83/4, 83/4, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation":"Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 83/4], [1, 0, null]], 83/4, 83/4, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)."},"skills":["Null Cells Retain Undefined Quotients and Zero Weighted Contributions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0095","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff5dac96fafb73535d3e102147dbd66694ea757739c6692c006aae9215240db8","family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094095,"semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,29],[2,29],[26,29],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[19,20,22,118]}"},"fields":{"title":"Null Cells Retain Undefined Quotients and Zero Weighted Contributions","stem":"Use the complete normalized finite law masses=[1/29, 2/29, 26/29, 0/1], aligned values X=[19, 20, 22, 118] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).","answer":{"kind":"expression","value":"([[0, 1, 631/29], [1, 0, null]], 631/29, 631/29, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation":"Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 631/29], [1, 0, null]], 631/29, 631/29, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)."},"skills":["Null Cells Retain Undefined Quotients and Zero Weighted Contributions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0096","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77dd88009c91dc4917b6a7bb50089945c2c47689175042dc9769f63b48acc9a2","family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094096,"semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,30],[2,30],[27,30],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[20,21,23,119]}"},"fields":{"title":"Null Cells Retain Undefined Quotients and Zero Weighted Contributions","stem":"Use the complete normalized finite law masses=[1/30, 2/30, 27/30, 0/1], aligned values X=[20, 21, 23, 119] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).","answer":{"kind":"expression","value":"([[0, 1, 683/30], [1, 0, null]], 683/30, 683/30, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation":"Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 683/30], [1, 0, null]], 683/30, 683/30, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)."},"skills":["Null Cells Retain Undefined Quotients and Zero Weighted Contributions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0097","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"514b3e2f0942fe76ce33b5b0dc27e11d2894787bcb9fb56fd899f04f6f499f20","family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094097,"semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,31],[2,31],[28,31],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[21,22,24,120]}"},"fields":{"title":"Null Cells Retain Undefined Quotients and Zero Weighted Contributions","stem":"Use the complete normalized finite law masses=[1/31, 2/31, 28/31, 0/1], aligned values X=[21, 22, 24, 120] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).","answer":{"kind":"expression","value":"([[0, 1, 737/31], [1, 0, null]], 737/31, 737/31, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation":"Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 737/31], [1, 0, null]], 737/31, 737/31, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)."},"skills":["Null Cells Retain Undefined Quotients and Zero Weighted Contributions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0098","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ab507b9b75b002dfa7872ca6b48c81cc2c6e25d00f79a6c754738671f6c4868","family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094098,"semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,32],[2,32],[29,32],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[22,23,25,121]}"},"fields":{"title":"Null Cells Retain Undefined Quotients and Zero Weighted Contributions","stem":"Use the complete normalized finite law masses=[1/32, 2/32, 29/32, 0/1], aligned values X=[22, 23, 25, 121] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).","answer":{"kind":"expression","value":"([[0, 1, 793/32], [1, 0, null]], 793/32, 793/32, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation":"Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 793/32], [1, 0, null]], 793/32, 793/32, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)."},"skills":["Null Cells Retain Undefined Quotients and Zero Weighted Contributions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0099","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4e62a26e74e8a8ad7db7c7f06767ab65be857ec1c5d73080f95579f9652263b","family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094099,"semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,33],[2,33],[30,33],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[23,24,26,122]}"},"fields":{"title":"Null Cells Retain Undefined Quotients and Zero Weighted Contributions","stem":"Use the complete normalized finite law masses=[1/33, 2/33, 30/33, 0/1], aligned values X=[23, 24, 26, 122] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).","answer":{"kind":"expression","value":"([[0, 1, 851/33], [1, 0, null]], 851/33, 851/33, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation":"Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 851/33], [1, 0, null]], 851/33, 851/33, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)."},"skills":["Null Cells Retain Undefined Quotients and Zero Weighted Contributions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0100","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"44ba110d507ddcb030797e3efb11e5bef75fe3a0da5a13ce7298265e3aaf88a7","family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094100,"semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,34],[2,34],[31,34],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[24,25,27,123]}"},"fields":{"title":"Null Cells Retain Undefined Quotients and Zero Weighted Contributions","stem":"Use the complete normalized finite law masses=[1/34, 2/34, 31/34, 0/1], aligned values X=[24, 25, 27, 123] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).","answer":{"kind":"expression","value":"([[0, 1, 911/34], [1, 0, null]], 911/34, 911/34, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation":"Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 911/34], [1, 0, null]], 911/34, 911/34, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)."},"skills":["Null Cells Retain Undefined Quotients and Zero Weighted Contributions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}