{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0001","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"762b7f9f6cd5821d743900c58fd2ad474131b98316bf197dc738c3d7316c90c3","family_id":"qf-mod25-measdeep-preimage-complement-law","seed":202610255001,"semantic_template":"qf-mod25-measdeep-preimage-complement-law/{\"event\":[0,1],\"mapping\":[0,0,0,0],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"Complement under a Complete Total-Map Preimage","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[0, 0, 0, 0], target event=[0, 1] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.","answer":{"kind":"expression","value":"([0, 0, 0, 0], [0, 1], [2, 3], [0, 1, 2, 3], [], [], Yes)","explanation":"Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 0, 0], [0, 1], [2, 3], [0, 1, 2, 3], [], [], Yes)."},"skills":["Complement under a Complete Total-Map Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0002","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08316cff0618287d4e53329c647c2505fa71b104589b9fd7ebe14f305c3f3545","family_id":"qf-mod25-measdeep-preimage-complement-law","seed":202610255002,"semantic_template":"qf-mod25-measdeep-preimage-complement-law/{\"event\":[1,2],\"mapping\":[1,0,0,0],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"Complement under a Complete Total-Map Preimage","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[1, 0, 0, 0], target event=[1, 2] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.","answer":{"kind":"expression","value":"([1, 0, 0, 0], [1, 2], [0, 3], [0], [1, 2, 3], [1, 2, 3], Yes)","explanation":"Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 0, 0], [1, 2], [0, 3], [0], [1, 2, 3], [1, 2, 3], Yes)."},"skills":["Complement under a Complete Total-Map Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0003","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"898af8b33e57606017f234674fc53e4e14c0c79d13739429ddd2fc7a4ae11277","family_id":"qf-mod25-measdeep-preimage-complement-law","seed":202610255003,"semantic_template":"qf-mod25-measdeep-preimage-complement-law/{\"event\":[2,3],\"mapping\":[2,0,0,0],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"Complement under a Complete Total-Map Preimage","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[2, 0, 0, 0], target event=[2, 3] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.","answer":{"kind":"expression","value":"([2, 0, 0, 0], [2, 3], [0, 1], [0], [1, 2, 3], [1, 2, 3], Yes)","explanation":"Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 0, 0, 0], [2, 3], [0, 1], [0], [1, 2, 3], [1, 2, 3], Yes)."},"skills":["Complement under a Complete Total-Map Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0004","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef2217dbe23f5a39029cb2a6de6afd2bd09bb3e74617001cf27a00b7687058cb","family_id":"qf-mod25-measdeep-preimage-complement-law","seed":202610255004,"semantic_template":"qf-mod25-measdeep-preimage-complement-law/{\"event\":[0,3],\"mapping\":[3,0,0,0],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"Complement under a Complete Total-Map Preimage","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[3, 0, 0, 0], target event=[0, 3] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.","answer":{"kind":"expression","value":"([3, 0, 0, 0], [0, 3], [1, 2], [0, 1, 2, 3], [], [], Yes)","explanation":"Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 0, 0, 0], [0, 3], [1, 2], [0, 1, 2, 3], [], [], Yes)."},"skills":["Complement under a Complete Total-Map Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0005","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b2903aa74fc16875b570d360b7b02a02a461139096a1312e57230efad2820b9","family_id":"qf-mod25-measdeep-preimage-complement-law","seed":202610255005,"semantic_template":"qf-mod25-measdeep-preimage-complement-law/{\"event\":[0,1],\"mapping\":[0,1,0,0],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"Complement under a Complete Total-Map Preimage","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[0, 1, 0, 0], target event=[0, 1] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.","answer":{"kind":"expression","value":"([0, 1, 0, 0], [0, 1], [2, 3], [0, 1, 2, 3], [], [], Yes)","explanation":"Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0, 0], [0, 1], [2, 3], [0, 1, 2, 3], [], [], Yes)."},"skills":["Complement under a Complete Total-Map Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0006","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4628f4608eee81b3ba1afd0a20349b2bed5679332edc3dc762f88f58ab015d84","family_id":"qf-mod25-measdeep-preimage-complement-law","seed":202610255006,"semantic_template":"qf-mod25-measdeep-preimage-complement-law/{\"event\":[1,2],\"mapping\":[1,1,0,0],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"Complement under a Complete Total-Map Preimage","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[1, 1, 0, 0], target event=[1, 2] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.","answer":{"kind":"expression","value":"([1, 1, 0, 0], [1, 2], [0, 3], [0, 1], [2, 3], [2, 3], Yes)","explanation":"Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 0, 0], [1, 2], [0, 3], [0, 1], [2, 3], [2, 3], Yes)."},"skills":["Complement under a Complete Total-Map Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0007","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7f58900e17cc4f82933306a7fbb440415472bc947984598f57943087e758245","family_id":"qf-mod25-measdeep-preimage-complement-law","seed":202610255007,"semantic_template":"qf-mod25-measdeep-preimage-complement-law/{\"event\":[2,3],\"mapping\":[2,1,0,0],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"Complement under a Complete Total-Map Preimage","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[2, 1, 0, 0], target event=[2, 3] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.","answer":{"kind":"expression","value":"([2, 1, 0, 0], [2, 3], [0, 1], [0], [1, 2, 3], [1, 2, 3], Yes)","explanation":"Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 1, 0, 0], [2, 3], [0, 1], [0], [1, 2, 3], [1, 2, 3], Yes)."},"skills":["Complement under a Complete Total-Map Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0008","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2fd39dcfb02ba0af32ea440f3dae07f5c704dd2a68f7459d30f685ee4a6f5d76","family_id":"qf-mod25-measdeep-preimage-complement-law","seed":202610255008,"semantic_template":"qf-mod25-measdeep-preimage-complement-law/{\"event\":[0,3],\"mapping\":[3,1,0,0],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"Complement under a Complete Total-Map Preimage","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[3, 1, 0, 0], target event=[0, 3] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.","answer":{"kind":"expression","value":"([3, 1, 0, 0], [0, 3], [1, 2], [0, 2, 3], [1], [1], Yes)","explanation":"Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 1, 0, 0], [0, 3], [1, 2], [0, 2, 3], [1], [1], Yes)."},"skills":["Complement under a Complete Total-Map Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0009","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76e9f04420e284ef176c5dfe7b5e11dd78182dc4d7c7b1ff65d4e4f552336856","family_id":"qf-mod25-measdeep-preimage-complement-law","seed":202610255009,"semantic_template":"qf-mod25-measdeep-preimage-complement-law/{\"event\":[0,1],\"mapping\":[0,2,0,0],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"Complement under a Complete Total-Map Preimage","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[0, 2, 0, 0], target event=[0, 1] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.","answer":{"kind":"expression","value":"([0, 2, 0, 0], [0, 1], [2, 3], [0, 2, 3], [1], [1], Yes)","explanation":"Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 2, 0, 0], [0, 1], [2, 3], [0, 2, 3], [1], [1], Yes)."},"skills":["Complement under a Complete Total-Map Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0010","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2c6ef602f026fb63a72da0ddac944b6b397145dc52a4f676efa5fb980ebf71c","family_id":"qf-mod25-measdeep-preimage-complement-law","seed":202610255010,"semantic_template":"qf-mod25-measdeep-preimage-complement-law/{\"event\":[1,2],\"mapping\":[1,2,0,0],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"Complement under a Complete Total-Map Preimage","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[1, 2, 0, 0], target event=[1, 2] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.","answer":{"kind":"expression","value":"([1, 2, 0, 0], [1, 2], [0, 3], [0, 1], [2, 3], [2, 3], Yes)","explanation":"Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 2, 0, 0], [1, 2], [0, 3], [0, 1], [2, 3], [2, 3], Yes)."},"skills":["Complement under a Complete Total-Map Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0011","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dcff5317f841b400fd517de38ab6b1c67664e1dc92b9ec9e1f43a1cffef915b9","family_id":"qf-mod25-measdeep-preimage-complement-law","seed":202610255011,"semantic_template":"qf-mod25-measdeep-preimage-complement-law/{\"event\":[2,3],\"mapping\":[2,2,0,0],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"Complement under a Complete Total-Map Preimage","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[2, 2, 0, 0], target event=[2, 3] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.","answer":{"kind":"expression","value":"([2, 2, 0, 0], [2, 3], [0, 1], [0, 1], [2, 3], [2, 3], Yes)","explanation":"Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, 0, 0], [2, 3], [0, 1], [0, 1], [2, 3], [2, 3], Yes)."},"skills":["Complement under a Complete Total-Map Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0012","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0bfb75dbe9d13060a35f6e28c3594b145f4035667576dd27a815fc8ec291962f","family_id":"qf-mod25-measdeep-preimage-complement-law","seed":202610255012,"semantic_template":"qf-mod25-measdeep-preimage-complement-law/{\"event\":[0,3],\"mapping\":[3,2,0,0],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"Complement under a Complete Total-Map Preimage","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[3, 2, 0, 0], target event=[0, 3] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.","answer":{"kind":"expression","value":"([3, 2, 0, 0], [0, 3], [1, 2], [0, 2, 3], [1], [1], Yes)","explanation":"Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 0, 0], [0, 3], [1, 2], [0, 2, 3], [1], [1], Yes)."},"skills":["Complement under a Complete Total-Map Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0013","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f70ff1ca4dd30e5d333b3439f347026cb9213eb9ad85eea3bbccdab9a5956e5b","family_id":"qf-mod25-measdeep-preimage-complement-law","seed":202610255013,"semantic_template":"qf-mod25-measdeep-preimage-complement-law/{\"event\":[0,1],\"mapping\":[0,3,0,0],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"Complement under a Complete Total-Map Preimage","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[0, 3, 0, 0], target event=[0, 1] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.","answer":{"kind":"expression","value":"([0, 3, 0, 0], [0, 1], [2, 3], [0, 2, 3], [1], [1], Yes)","explanation":"Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 3, 0, 0], [0, 1], [2, 3], [0, 2, 3], [1], [1], Yes)."},"skills":["Complement under a Complete Total-Map Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0014","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc56c297038b391904fb01be9a8ebc276da9066e93b7f8d8efa8044d5f5b30e8","family_id":"qf-mod25-measdeep-preimage-complement-law","seed":202610255014,"semantic_template":"qf-mod25-measdeep-preimage-complement-law/{\"event\":[1,2],\"mapping\":[1,3,0,0],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"Complement under a Complete Total-Map Preimage","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[1, 3, 0, 0], target event=[1, 2] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.","answer":{"kind":"expression","value":"([1, 3, 0, 0], [1, 2], [0, 3], [0], [1, 2, 3], [1, 2, 3], Yes)","explanation":"Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 3, 0, 0], [1, 2], [0, 3], [0], [1, 2, 3], [1, 2, 3], Yes)."},"skills":["Complement under a Complete Total-Map Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0015","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e37e83807c60533bb7a8cfc2c13c1077c81fa893d61c217551fc610a142757b3","family_id":"qf-mod25-measdeep-preimage-complement-law","seed":202610255015,"semantic_template":"qf-mod25-measdeep-preimage-complement-law/{\"event\":[2,3],\"mapping\":[2,3,0,0],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"Complement under a Complete Total-Map Preimage","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[2, 3, 0, 0], target event=[2, 3] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.","answer":{"kind":"expression","value":"([2, 3, 0, 0], [2, 3], [0, 1], [0, 1], [2, 3], [2, 3], Yes)","explanation":"Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 3, 0, 0], [2, 3], [0, 1], [0, 1], [2, 3], [2, 3], Yes)."},"skills":["Complement under a Complete Total-Map Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0016","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"674f779a17beb7ece50832c42d50db72db543679bf9caab36326b47f80093f5d","family_id":"qf-mod25-measdeep-preimage-complement-law","seed":202610255016,"semantic_template":"qf-mod25-measdeep-preimage-complement-law/{\"event\":[0,3],\"mapping\":[3,3,0,0],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"Complement under a Complete Total-Map Preimage","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[3, 3, 0, 0], target event=[0, 3] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.","answer":{"kind":"expression","value":"([3, 3, 0, 0], [0, 3], [1, 2], [0, 1, 2, 3], [], [], Yes)","explanation":"Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 3, 0, 0], [0, 3], [1, 2], [0, 1, 2, 3], [], [], Yes)."},"skills":["Complement under a Complete Total-Map Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0017","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74e3f37fff32aeae03fe01f4429dce523f05a80c6f9891fc6eb3bf27de5618b6","family_id":"qf-mod25-measdeep-preimage-complement-law","seed":202610255017,"semantic_template":"qf-mod25-measdeep-preimage-complement-law/{\"event\":[0,1],\"mapping\":[0,0,1,0],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"Complement under a Complete Total-Map Preimage","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[0, 0, 1, 0], target event=[0, 1] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.","answer":{"kind":"expression","value":"([0, 0, 1, 0], [0, 1], [2, 3], [0, 1, 2, 3], [], [], Yes)","explanation":"Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 1, 0], [0, 1], [2, 3], [0, 1, 2, 3], [], [], Yes)."},"skills":["Complement under a Complete Total-Map Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0018","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d36af47bc06ec4d5eb31cc78b3ba3de37771ed017656bc8e00fd41e5d97f7805","family_id":"qf-mod25-measdeep-preimage-complement-law","seed":202610255018,"semantic_template":"qf-mod25-measdeep-preimage-complement-law/{\"event\":[1,2],\"mapping\":[1,0,1,0],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"Complement under a Complete Total-Map Preimage","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[1, 0, 1, 0], target event=[1, 2] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.","answer":{"kind":"expression","value":"([1, 0, 1, 0], [1, 2], [0, 3], [0, 2], [1, 3], [1, 3], Yes)","explanation":"Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1, 0], [1, 2], [0, 3], [0, 2], [1, 3], [1, 3], Yes)."},"skills":["Complement under a Complete Total-Map Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0019","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73635e7a49bb9093f767f1ee8bb1397827f369b1949fc748c6d716036bf6c808","family_id":"qf-mod25-measdeep-preimage-complement-law","seed":202610255019,"semantic_template":"qf-mod25-measdeep-preimage-complement-law/{\"event\":[2,3],\"mapping\":[2,0,1,0],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"Complement under a Complete Total-Map Preimage","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[2, 0, 1, 0], target event=[2, 3] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.","answer":{"kind":"expression","value":"([2, 0, 1, 0], [2, 3], [0, 1], [0], [1, 2, 3], [1, 2, 3], Yes)","explanation":"Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 0, 1, 0], [2, 3], [0, 1], [0], [1, 2, 3], [1, 2, 3], Yes)."},"skills":["Complement under a Complete Total-Map Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0020","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aaafc1d04f70a3f511872e948af60987d383313e3ad7aa12b0fa62302279a5c7","family_id":"qf-mod25-measdeep-preimage-complement-law","seed":202610255020,"semantic_template":"qf-mod25-measdeep-preimage-complement-law/{\"event\":[0,3],\"mapping\":[3,0,1,0],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"Complement under a Complete Total-Map Preimage","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[3, 0, 1, 0], target event=[0, 3] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.","answer":{"kind":"expression","value":"([3, 0, 1, 0], [0, 3], [1, 2], [0, 1, 3], [2], [2], Yes)","explanation":"Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 0, 1, 0], [0, 3], [1, 2], [0, 1, 3], [2], [2], Yes)."},"skills":["Complement under a Complete Total-Map Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0021","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe7fc13c7ad3986ba4510ee5fe5bebd05fd12423a2474a23eefe4b793fd15333","family_id":"qf-mod25-measdeep-preimage-complement-law","seed":202610255021,"semantic_template":"qf-mod25-measdeep-preimage-complement-law/{\"event\":[0,1],\"mapping\":[0,1,1,0],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"Complement under a Complete Total-Map Preimage","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[0, 1, 1, 0], target event=[0, 1] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.","answer":{"kind":"expression","value":"([0, 1, 1, 0], [0, 1], [2, 3], [0, 1, 2, 3], [], [], Yes)","explanation":"Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 1, 0], [0, 1], [2, 3], [0, 1, 2, 3], [], [], Yes)."},"skills":["Complement under a Complete Total-Map Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0022","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2489e2ac51349eb4b2b162aabb21dca595dbc67e035cd47e61dcaf02628c9cf","family_id":"qf-mod25-measdeep-preimage-complement-law","seed":202610255022,"semantic_template":"qf-mod25-measdeep-preimage-complement-law/{\"event\":[1,2],\"mapping\":[1,1,1,0],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"Complement under a Complete Total-Map Preimage","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[1, 1, 1, 0], target event=[1, 2] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.","answer":{"kind":"expression","value":"([1, 1, 1, 0], [1, 2], [0, 3], [0, 1, 2], [3], [3], Yes)","explanation":"Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 1, 0], [1, 2], [0, 3], [0, 1, 2], [3], [3], Yes)."},"skills":["Complement under a Complete Total-Map Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0023","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"807a6977f426175c1bd112c7b96e722ff7eb5a470a4484651784620acb13d685","family_id":"qf-mod25-measdeep-preimage-complement-law","seed":202610255023,"semantic_template":"qf-mod25-measdeep-preimage-complement-law/{\"event\":[2,3],\"mapping\":[2,1,1,0],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"Complement under a Complete Total-Map Preimage","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[2, 1, 1, 0], target event=[2, 3] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.","answer":{"kind":"expression","value":"([2, 1, 1, 0], [2, 3], [0, 1], [0], [1, 2, 3], [1, 2, 3], Yes)","explanation":"Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 1, 1, 0], [2, 3], [0, 1], [0], [1, 2, 3], [1, 2, 3], Yes)."},"skills":["Complement under a Complete Total-Map Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0024","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3b95119ebfd4fe8008dc282be437e54ad6a9d30bdb20ac6fb8cb8b9034cc756","family_id":"qf-mod25-measdeep-preimage-complement-law","seed":202610255024,"semantic_template":"qf-mod25-measdeep-preimage-complement-law/{\"event\":[0,3],\"mapping\":[3,1,1,0],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"Complement under a Complete Total-Map Preimage","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[3, 1, 1, 0], target event=[0, 3] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.","answer":{"kind":"expression","value":"([3, 1, 1, 0], [0, 3], [1, 2], [0, 3], [1, 2], [1, 2], Yes)","explanation":"Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 1, 1, 0], [0, 3], [1, 2], [0, 3], [1, 2], [1, 2], Yes)."},"skills":["Complement under a Complete Total-Map Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0025","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"603d23120401c89cf5ac3625232ed4c39347c7a423d62c8006dea3b08a5eecc5","family_id":"qf-mod25-measdeep-preimage-complement-law","seed":202610255025,"semantic_template":"qf-mod25-measdeep-preimage-complement-law/{\"event\":[0,1],\"mapping\":[0,2,1,0],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"Complement under a Complete Total-Map Preimage","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[0, 2, 1, 0], target event=[0, 1] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.","answer":{"kind":"expression","value":"([0, 2, 1, 0], [0, 1], [2, 3], [0, 2, 3], [1], [1], Yes)","explanation":"Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 2, 1, 0], [0, 1], [2, 3], [0, 2, 3], [1], [1], Yes)."},"skills":["Complement under a Complete Total-Map Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0026","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41b8557f16d89aa70fa3d19635170406afac960efb57d075b2b7e4cf2d2d0f5e","family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255026,"semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[0],\"mapping\":[0,0,0,0],\"right_event\":[1,2],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"The Preimage of a Union Equals the Union of Preimages","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[0, 0, 0, 0], target events left=[0], right=[1, 2], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.","answer":{"kind":"expression","value":"([0, 0, 0, 0], [0], [1, 2], [0, 1, 2], [0, 1, 2, 3], [], [0, 1, 2, 3], [0, 1, 2, 3], Yes)","explanation":"Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 0, 0], [0], [1, 2], [0, 1, 2], [0, 1, 2, 3], [], [0, 1, 2, 3], [0, 1, 2, 3], Yes)."},"skills":["The Preimage of a Union Equals the Union of Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0027","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d31e02eb932e53a25d9de44b228df410481fe781405fbba730bf486ff7f9548a","family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255027,"semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[1],\"mapping\":[1,0,0,0],\"right_event\":[2,3],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"The Preimage of a Union Equals the Union of Preimages","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[1, 0, 0, 0], target events left=[1], right=[2, 3], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.","answer":{"kind":"expression","value":"([1, 0, 0, 0], [1], [2, 3], [1, 2, 3], [0], [], [0], [0], Yes)","explanation":"Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 0, 0], [1], [2, 3], [1, 2, 3], [0], [], [0], [0], Yes)."},"skills":["The Preimage of a Union Equals the Union of Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0028","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0f0c67fd145149d0cb6063efbe768bb2352449d7b5a856e36c92fdd7eba0d79","family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255028,"semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[2],\"mapping\":[2,0,0,0],\"right_event\":[0,3],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"The Preimage of a Union Equals the Union of Preimages","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[2, 0, 0, 0], target events left=[2], right=[0, 3], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.","answer":{"kind":"expression","value":"([2, 0, 0, 0], [2], [0, 3], [0, 2, 3], [0], [1, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3], Yes)","explanation":"Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 0, 0, 0], [2], [0, 3], [0, 2, 3], [0], [1, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3], Yes)."},"skills":["The Preimage of a Union Equals the Union of Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0029","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"afbb98cfc8cca3f2a4d46c720c042611fc75e48f298ec1dea7bdbaaadc559035","family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255029,"semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[3],\"mapping\":[3,0,0,0],\"right_event\":[0,1],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"The Preimage of a Union Equals the Union of Preimages","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[3, 0, 0, 0], target events left=[3], right=[0, 1], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.","answer":{"kind":"expression","value":"([3, 0, 0, 0], [3], [0, 1], [0, 1, 3], [0], [1, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3], Yes)","explanation":"Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 0, 0, 0], [3], [0, 1], [0, 1, 3], [0], [1, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3], Yes)."},"skills":["The Preimage of a Union Equals the Union of Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0030","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"965a6a1fc5f5b0f5c632d28a96eeed7ba417eb9bf8fe58571ca88730d3a03c30","family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255030,"semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[0],\"mapping\":[0,1,0,0],\"right_event\":[1,2],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"The Preimage of a Union Equals the Union of Preimages","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[0, 1, 0, 0], target events left=[0], right=[1, 2], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.","answer":{"kind":"expression","value":"([0, 1, 0, 0], [0], [1, 2], [0, 1, 2], [0, 2, 3], [1], [0, 1, 2, 3], [0, 1, 2, 3], Yes)","explanation":"Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0, 0], [0], [1, 2], [0, 1, 2], [0, 2, 3], [1], [0, 1, 2, 3], [0, 1, 2, 3], Yes)."},"skills":["The Preimage of a Union Equals the Union of Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0031","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac69be5e703cbc4b4887ac580050b46072ddd40e836226d7ac7d614e8b988546","family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255031,"semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[1],\"mapping\":[1,1,0,0],\"right_event\":[2,3],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"The Preimage of a Union Equals the Union of Preimages","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[1, 1, 0, 0], target events left=[1], right=[2, 3], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.","answer":{"kind":"expression","value":"([1, 1, 0, 0], [1], [2, 3], [1, 2, 3], [0, 1], [], [0, 1], [0, 1], Yes)","explanation":"Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 0, 0], [1], [2, 3], [1, 2, 3], [0, 1], [], [0, 1], [0, 1], Yes)."},"skills":["The Preimage of a Union Equals the Union of Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0032","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3251c895294955af1e90e04ef6f9f95e90c4d8c9200dbc2600547a451c626df9","family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255032,"semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[2],\"mapping\":[2,1,0,0],\"right_event\":[0,3],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"The Preimage of a Union Equals the Union of Preimages","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[2, 1, 0, 0], target events left=[2], right=[0, 3], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.","answer":{"kind":"expression","value":"([2, 1, 0, 0], [2], [0, 3], [0, 2, 3], [0], [2, 3], [0, 2, 3], [0, 2, 3], Yes)","explanation":"Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 1, 0, 0], [2], [0, 3], [0, 2, 3], [0], [2, 3], [0, 2, 3], [0, 2, 3], Yes)."},"skills":["The Preimage of a Union Equals the Union of Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0033","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a87e98189bcebd7c74974e34db8437cd3c76b0bce2d8ca2795eda1d3277dafe9","family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255033,"semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[3],\"mapping\":[3,1,0,0],\"right_event\":[0,1],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"The Preimage of a Union Equals the Union of Preimages","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[3, 1, 0, 0], target events left=[3], right=[0, 1], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.","answer":{"kind":"expression","value":"([3, 1, 0, 0], [3], [0, 1], [0, 1, 3], [0], [1, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3], Yes)","explanation":"Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 1, 0, 0], [3], [0, 1], [0, 1, 3], [0], [1, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3], Yes)."},"skills":["The Preimage of a Union Equals the Union of Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0034","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee820467a8b715fa36d529175835478dd2c68327f0a8442f98d299f09d733973","family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255034,"semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[0],\"mapping\":[0,2,0,0],\"right_event\":[1,2],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"The Preimage of a Union Equals the Union of Preimages","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[0, 2, 0, 0], target events left=[0], right=[1, 2], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.","answer":{"kind":"expression","value":"([0, 2, 0, 0], [0], [1, 2], [0, 1, 2], [0, 2, 3], [1], [0, 1, 2, 3], [0, 1, 2, 3], Yes)","explanation":"Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 2, 0, 0], [0], [1, 2], [0, 1, 2], [0, 2, 3], [1], [0, 1, 2, 3], [0, 1, 2, 3], Yes)."},"skills":["The Preimage of a Union Equals the Union of Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0035","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4eb9b4b64503f503aac9927556d091725b1ec1aee3ec242bcb0a5f2d0979cfb6","family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255035,"semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[1],\"mapping\":[1,2,0,0],\"right_event\":[2,3],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"The Preimage of a Union Equals the Union of Preimages","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[1, 2, 0, 0], target events left=[1], right=[2, 3], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.","answer":{"kind":"expression","value":"([1, 2, 0, 0], [1], [2, 3], [1, 2, 3], [0], [1], [0, 1], [0, 1], Yes)","explanation":"Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 2, 0, 0], [1], [2, 3], [1, 2, 3], [0], [1], [0, 1], [0, 1], Yes)."},"skills":["The Preimage of a Union Equals the Union of Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0036","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8591a3a05aaeba1d05399d30e3ca83d20b41ae26174c9901419d13a95beb39cc","family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255036,"semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[2],\"mapping\":[2,2,0,0],\"right_event\":[0,3],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"The Preimage of a Union Equals the Union of Preimages","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[2, 2, 0, 0], target events left=[2], right=[0, 3], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.","answer":{"kind":"expression","value":"([2, 2, 0, 0], [2], [0, 3], [0, 2, 3], [0, 1], [2, 3], [0, 1, 2, 3], [0, 1, 2, 3], Yes)","explanation":"Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, 0, 0], [2], [0, 3], [0, 2, 3], [0, 1], [2, 3], [0, 1, 2, 3], [0, 1, 2, 3], Yes)."},"skills":["The Preimage of a Union Equals the Union of Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0037","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1de2447e61f7b196c1bc672971bbbfc6d864539b3b341e05f371d4f75e006404","family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255037,"semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[3],\"mapping\":[3,2,0,0],\"right_event\":[0,1],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"The Preimage of a Union Equals the Union of Preimages","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[3, 2, 0, 0], target events left=[3], right=[0, 1], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.","answer":{"kind":"expression","value":"([3, 2, 0, 0], [3], [0, 1], [0, 1, 3], [0], [2, 3], [0, 2, 3], [0, 2, 3], Yes)","explanation":"Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 0, 0], [3], [0, 1], [0, 1, 3], [0], [2, 3], [0, 2, 3], [0, 2, 3], Yes)."},"skills":["The Preimage of a Union Equals the Union of Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0038","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb31d603319449dc0861dd397d322387491d4ce255f9a7cc78eba621e361fb1a","family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255038,"semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[0],\"mapping\":[0,3,0,0],\"right_event\":[1,2],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"The Preimage of a Union Equals the Union of Preimages","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[0, 3, 0, 0], target events left=[0], right=[1, 2], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.","answer":{"kind":"expression","value":"([0, 3, 0, 0], [0], [1, 2], [0, 1, 2], [0, 2, 3], [], [0, 2, 3], [0, 2, 3], Yes)","explanation":"Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 3, 0, 0], [0], [1, 2], [0, 1, 2], [0, 2, 3], [], [0, 2, 3], [0, 2, 3], Yes)."},"skills":["The Preimage of a Union Equals the Union of Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0039","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0cfe8da638eb955c3110878b2632e08d29e99693955a7ebfa2f500f66926b613","family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255039,"semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[1],\"mapping\":[1,3,0,0],\"right_event\":[2,3],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"The Preimage of a Union Equals the Union of Preimages","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[1, 3, 0, 0], target events left=[1], right=[2, 3], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.","answer":{"kind":"expression","value":"([1, 3, 0, 0], [1], [2, 3], [1, 2, 3], [0], [1], [0, 1], [0, 1], Yes)","explanation":"Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 3, 0, 0], [1], [2, 3], [1, 2, 3], [0], [1], [0, 1], [0, 1], Yes)."},"skills":["The Preimage of a Union Equals the Union of Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0040","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1bf83c4cb15725e2a8d966dc1d7d426f5b56ba2dc0b264fa81ad78df70e476b","family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255040,"semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[2],\"mapping\":[2,3,0,0],\"right_event\":[0,3],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"The Preimage of a Union Equals the Union of Preimages","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[2, 3, 0, 0], target events left=[2], right=[0, 3], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.","answer":{"kind":"expression","value":"([2, 3, 0, 0], [2], [0, 3], [0, 2, 3], [0], [1, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3], Yes)","explanation":"Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 3, 0, 0], [2], [0, 3], [0, 2, 3], [0], [1, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3], Yes)."},"skills":["The Preimage of a Union Equals the Union of Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0041","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"09720e10e5070fb4d99fd84c69ccd87a148609ea6b7d51567f48fc1eb06afdf5","family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255041,"semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[3],\"mapping\":[3,3,0,0],\"right_event\":[0,1],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"The Preimage of a Union Equals the Union of Preimages","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[3, 3, 0, 0], target events left=[3], right=[0, 1], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.","answer":{"kind":"expression","value":"([3, 3, 0, 0], [3], [0, 1], [0, 1, 3], [0, 1], [2, 3], [0, 1, 2, 3], [0, 1, 2, 3], Yes)","explanation":"Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 3, 0, 0], [3], [0, 1], [0, 1, 3], [0, 1], [2, 3], [0, 1, 2, 3], [0, 1, 2, 3], Yes)."},"skills":["The Preimage of a Union Equals the Union of Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0042","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64cb310119165152c4888216780c7632125491681c68231556c33f102c316d83","family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255042,"semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[0],\"mapping\":[0,0,1,0],\"right_event\":[1,2],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"The Preimage of a Union Equals the Union of Preimages","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[0, 0, 1, 0], target events left=[0], right=[1, 2], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.","answer":{"kind":"expression","value":"([0, 0, 1, 0], [0], [1, 2], [0, 1, 2], [0, 1, 3], [2], [0, 1, 2, 3], [0, 1, 2, 3], Yes)","explanation":"Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 1, 0], [0], [1, 2], [0, 1, 2], [0, 1, 3], [2], [0, 1, 2, 3], [0, 1, 2, 3], Yes)."},"skills":["The Preimage of a Union Equals the Union of Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0043","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67dac28a703dc4d68ff7f47f13b61933c9880607e46ab3359c19da9daec54292","family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255043,"semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[1],\"mapping\":[1,0,1,0],\"right_event\":[2,3],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"The Preimage of a Union Equals the Union of Preimages","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[1, 0, 1, 0], target events left=[1], right=[2, 3], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.","answer":{"kind":"expression","value":"([1, 0, 1, 0], [1], [2, 3], [1, 2, 3], [0, 2], [], [0, 2], [0, 2], Yes)","explanation":"Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1, 0], [1], [2, 3], [1, 2, 3], [0, 2], [], [0, 2], [0, 2], Yes)."},"skills":["The Preimage of a Union Equals the Union of Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0044","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c098bdfd6d4d1539c75d85617141a095c9f2edd92d467919a78af27526d37329","family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255044,"semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[2],\"mapping\":[2,0,1,0],\"right_event\":[0,3],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"The Preimage of a Union Equals the Union of Preimages","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[2, 0, 1, 0], target events left=[2], right=[0, 3], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.","answer":{"kind":"expression","value":"([2, 0, 1, 0], [2], [0, 3], [0, 2, 3], [0], [1, 3], [0, 1, 3], [0, 1, 3], Yes)","explanation":"Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 0, 1, 0], [2], [0, 3], [0, 2, 3], [0], [1, 3], [0, 1, 3], [0, 1, 3], Yes)."},"skills":["The Preimage of a Union Equals the Union of Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0045","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8aacb9ed16e03480981a39ed6aeb99b19d9ae43163e744cb3db0d31c71c5305","family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255045,"semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[3],\"mapping\":[3,0,1,0],\"right_event\":[0,1],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"The Preimage of a Union Equals the Union of Preimages","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[3, 0, 1, 0], target events left=[3], right=[0, 1], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.","answer":{"kind":"expression","value":"([3, 0, 1, 0], [3], [0, 1], [0, 1, 3], [0], [1, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3], Yes)","explanation":"Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 0, 1, 0], [3], [0, 1], [0, 1, 3], [0], [1, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3], Yes)."},"skills":["The Preimage of a Union Equals the Union of Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0046","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"afa3160ee44cb48efb1941f3578ea5a71db26d53e4cc8fe9d7860a04d97194fd","family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255046,"semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[0],\"mapping\":[0,1,1,0],\"right_event\":[1,2],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"The Preimage of a Union Equals the Union of Preimages","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[0, 1, 1, 0], target events left=[0], right=[1, 2], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.","answer":{"kind":"expression","value":"([0, 1, 1, 0], [0], [1, 2], [0, 1, 2], [0, 3], [1, 2], [0, 1, 2, 3], [0, 1, 2, 3], Yes)","explanation":"Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 1, 0], [0], [1, 2], [0, 1, 2], [0, 3], [1, 2], [0, 1, 2, 3], [0, 1, 2, 3], Yes)."},"skills":["The Preimage of a Union Equals the Union of Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0047","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3965126c4a0b236ead32f4c73b2b88d01058e0ec6b7255cd73ad6a1aa5d71684","family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255047,"semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[1],\"mapping\":[1,1,1,0],\"right_event\":[2,3],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"The Preimage of a Union Equals the Union of Preimages","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[1, 1, 1, 0], target events left=[1], right=[2, 3], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.","answer":{"kind":"expression","value":"([1, 1, 1, 0], [1], [2, 3], [1, 2, 3], [0, 1, 2], [], [0, 1, 2], [0, 1, 2], Yes)","explanation":"Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 1, 0], [1], [2, 3], [1, 2, 3], [0, 1, 2], [], [0, 1, 2], [0, 1, 2], Yes)."},"skills":["The Preimage of a Union Equals the Union of Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0048","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f995c6f8d4d941f4f263297252c79b42e2ea1479f11076bf114afc95e94a70b","family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255048,"semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[2],\"mapping\":[2,1,1,0],\"right_event\":[0,3],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"The Preimage of a Union Equals the Union of Preimages","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[2, 1, 1, 0], target events left=[2], right=[0, 3], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.","answer":{"kind":"expression","value":"([2, 1, 1, 0], [2], [0, 3], [0, 2, 3], [0], [3], [0, 3], [0, 3], Yes)","explanation":"Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 1, 1, 0], [2], [0, 3], [0, 2, 3], [0], [3], [0, 3], [0, 3], Yes)."},"skills":["The Preimage of a Union Equals the Union of Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0049","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"148db4e3aa4c486d8b133ad171aa112e0eec899dccceee4933f39856fa3b2f76","family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255049,"semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[3],\"mapping\":[3,1,1,0],\"right_event\":[0,1],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"The Preimage of a Union Equals the Union of Preimages","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[3, 1, 1, 0], target events left=[3], right=[0, 1], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.","answer":{"kind":"expression","value":"([3, 1, 1, 0], [3], [0, 1], [0, 1, 3], [0], [1, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3], Yes)","explanation":"Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 1, 1, 0], [3], [0, 1], [0, 1, 3], [0], [1, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3], Yes)."},"skills":["The Preimage of a Union Equals the Union of Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0050","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"038c2ed8dc42ee82eea1893c2ece16bd374dd090f814bbc535eeb2a9c13c3295","family_id":"qf-mod25-measdeep-preimage-union-law","seed":202610255050,"semantic_template":"qf-mod25-measdeep-preimage-union-law/{\"left_event\":[0],\"mapping\":[0,2,1,0],\"right_event\":[1,2],\"source_n\":4,\"target_n\":4}"},"fields":{"title":"The Preimage of a Union Equals the Union of Preimages","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[0, 2, 1, 0], target events left=[0], right=[1, 2], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.","answer":{"kind":"expression","value":"([0, 2, 1, 0], [0], [1, 2], [0, 1, 2], [0, 3], [1, 2], [0, 1, 2, 3], [0, 1, 2, 3], Yes)","explanation":"Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 2, 1, 0], [0], [1, 2], [0, 1, 2], [0, 3], [1, 2], [0, 1, 2, 3], [0, 1, 2, 3], Yes)."},"skills":["The Preimage of a Union Equals the Union of Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0051","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ce49239b5fbfaaf482b4a16715f74ccc04af45373457e415d496f5cf62a76cf","family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255051,"semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0],[1,2,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":0}"},"fields":{"title":"All Preimages of a Qualified Constant Finite Map","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [0], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], constant target value=0. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).","answer":{"kind":"expression","value":"(4, 4, 0, [0, 0, 0, 0], [[[], [], Yes], [[0], [0, 1, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 0, [0, 0, 0, 0], [[[], [], Yes], [[0], [0, 1, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["All Preimages of a Qualified Constant Finite Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0052","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8542fb5ff1d445de803c84b3ca9a1acdadad45c31c9fdb8641595f9357748c5","family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255052,"semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,2,3],[1]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":1}"},"fields":{"title":"All Preimages of a Qualified Constant Finite Map","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [1], [0, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], constant target value=1. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).","answer":{"kind":"expression","value":"(4, 4, 1, [1, 1, 1, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 1, 2, 3], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 1, [1, 1, 1, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 1, 2, 3], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["All Preimages of a Qualified Constant Finite Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0053","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84902d03834053bed3b1e1509f4d85066aa66c98278fa4b4ca2f5e7e2de41aa4","family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255053,"semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1],[2,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":2}"},"fields":{"title":"All Preimages of a Qualified Constant Finite Map","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [0, 1], [2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], constant target value=2. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).","answer":{"kind":"expression","value":"(4, 4, 2, [2, 2, 2, 2], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [0, 1, 2, 3], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [], Yes], [[0, 1, 3], [], Yes], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 2, [2, 2, 2, 2], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [0, 1, 2, 3], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [], Yes], [[0, 1, 3], [], Yes], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["All Preimages of a Qualified Constant Finite Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0054","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a47ec2a8475f72f62d235f59011cc46e78ea233b3b164566cbc5116b48847d5","family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255054,"semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1,3],[2]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":3}"},"fields":{"title":"All Preimages of a Qualified Constant Finite Map","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [2], [0, 1, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], constant target value=3. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).","answer":{"kind":"expression","value":"(4, 4, 3, [3, 3, 3, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [], Yes], [[0, 1, 2], [], Yes], [[3], [0, 1, 2, 3], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 3, [3, 3, 3, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [], Yes], [[0, 1, 2], [], Yes], [[3], [0, 1, 2, 3], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["All Preimages of a Qualified Constant Finite Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0055","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95eb41bccbb46cfdb1ed9b14c706df7e248b9af98522f75f999e3689717bfc7b","family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255055,"semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,2],[1,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":0}"},"fields":{"title":"All Preimages of a Qualified Constant Finite Map","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [0, 2], [1, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], constant target value=0. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).","answer":{"kind":"expression","value":"(4, 4, 0, [0, 0, 0, 0], [[[], [], Yes], [[0], [0, 1, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 0, [0, 0, 0, 0], [[[], [], Yes], [[0], [0, 1, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["All Preimages of a Qualified Constant Finite Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0056","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ee5ec3880c3929f5c2da9ee4dabc613783878bb352d5d1cefce194b904c4a16","family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255056,"semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,3],[1,2]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":1}"},"fields":{"title":"All Preimages of a Qualified Constant Finite Map","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [1, 2], [0, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], constant target value=1. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).","answer":{"kind":"expression","value":"(4, 4, 1, [1, 1, 1, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 1, 2, 3], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 1, [1, 1, 1, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 1, 2, 3], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["All Preimages of a Qualified Constant Finite Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0057","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92e41d845354fe9a158507841c8f6baccd1881231404029173e3d7c6a330936e","family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255057,"semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1,2],[3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":2}"},"fields":{"title":"All Preimages of a Qualified Constant Finite Map","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [0, 1, 2], [3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], constant target value=2. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).","answer":{"kind":"expression","value":"(4, 4, 2, [2, 2, 2, 2], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [0, 1, 2, 3], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [], Yes], [[0, 1, 3], [], Yes], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 2, [2, 2, 2, 2], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [0, 1, 2, 3], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [], Yes], [[0, 1, 3], [], Yes], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["All Preimages of a Qualified Constant Finite Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0058","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"164057ec5bd0cc739e942a367a6e33d74f62cce3cf2e2d95100a0e2766431a5b","family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255058,"semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0],[1,2,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":3}"},"fields":{"title":"All Preimages of a Qualified Constant Finite Map","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [0], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], constant target value=3. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).","answer":{"kind":"expression","value":"(4, 4, 3, [3, 3, 3, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [], Yes], [[0, 1, 2], [], Yes], [[3], [0, 1, 2, 3], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 3, [3, 3, 3, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [], Yes], [[0, 1, 2], [], Yes], [[3], [0, 1, 2, 3], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["All Preimages of a Qualified Constant Finite Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0059","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c435d032c442ea1256fc40b7279dffae61ff5addd0ed81db9b6eaae1bbf3092c","family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255059,"semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,2,3],[1]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":0}"},"fields":{"title":"All Preimages of a Qualified Constant Finite Map","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [1], [0, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], constant target value=0. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).","answer":{"kind":"expression","value":"(4, 4, 0, [0, 0, 0, 0], [[[], [], Yes], [[0], [0, 1, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 0, [0, 0, 0, 0], [[[], [], Yes], [[0], [0, 1, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["All Preimages of a Qualified Constant Finite Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0060","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f410b9b92af95928340075a1071edae39279ee8a4aa878d59fb4aaa405ab3cf","family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255060,"semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1],[2,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":1}"},"fields":{"title":"All Preimages of a Qualified Constant Finite Map","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [0, 1], [2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], constant target value=1. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).","answer":{"kind":"expression","value":"(4, 4, 1, [1, 1, 1, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 1, 2, 3], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 1, [1, 1, 1, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 1, 2, 3], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["All Preimages of a Qualified Constant Finite Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0061","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17aa3fe88c62a934889f51724873a8350f34d40de3dff96d106ac59de44bc8dd","family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255061,"semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1,3],[2]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":2}"},"fields":{"title":"All Preimages of a Qualified Constant Finite Map","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [2], [0, 1, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], constant target value=2. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).","answer":{"kind":"expression","value":"(4, 4, 2, [2, 2, 2, 2], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [0, 1, 2, 3], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [], Yes], [[0, 1, 3], [], Yes], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 2, [2, 2, 2, 2], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [0, 1, 2, 3], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [], Yes], [[0, 1, 3], [], Yes], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["All Preimages of a Qualified Constant Finite Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0062","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"793612f520f05e8317bde5db54ce74d84eeac5b3725924041e94b7f4079bfce5","family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255062,"semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,2],[1,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":3}"},"fields":{"title":"All Preimages of a Qualified Constant Finite Map","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [0, 2], [1, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], constant target value=3. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).","answer":{"kind":"expression","value":"(4, 4, 3, [3, 3, 3, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [], Yes], [[0, 1, 2], [], Yes], [[3], [0, 1, 2, 3], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 3, [3, 3, 3, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [], Yes], [[0, 1, 2], [], Yes], [[3], [0, 1, 2, 3], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["All Preimages of a Qualified Constant Finite Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0063","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63eabd79cc2de57d2d464aa991e9b373b6f07d4e4e1aece06faada699c22e40b","family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255063,"semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,3],[1,2]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":0}"},"fields":{"title":"All Preimages of a Qualified Constant Finite Map","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [1, 2], [0, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], constant target value=0. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).","answer":{"kind":"expression","value":"(4, 4, 0, [0, 0, 0, 0], [[[], [], Yes], [[0], [0, 1, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 0, [0, 0, 0, 0], [[[], [], Yes], [[0], [0, 1, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["All Preimages of a Qualified Constant Finite Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0064","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a65841aabe69a236a14dc8cb89dc807080f8468cfd628c18560957aa57136e53","family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255064,"semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1,2],[3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":1}"},"fields":{"title":"All Preimages of a Qualified Constant Finite Map","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [0, 1, 2], [3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], constant target value=1. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).","answer":{"kind":"expression","value":"(4, 4, 1, [1, 1, 1, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 1, 2, 3], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 1, [1, 1, 1, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 1, 2, 3], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["All Preimages of a Qualified Constant Finite Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0065","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8b5f3fa51f393240733c07175651cfefe7fbe0cf1e694faeca7a2cc6cbb0fca","family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255065,"semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0],[1,2,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":2}"},"fields":{"title":"All Preimages of a Qualified Constant Finite Map","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [0], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], constant target value=2. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).","answer":{"kind":"expression","value":"(4, 4, 2, [2, 2, 2, 2], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [0, 1, 2, 3], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [], Yes], [[0, 1, 3], [], Yes], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 2, [2, 2, 2, 2], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [0, 1, 2, 3], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [], Yes], [[0, 1, 3], [], Yes], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["All Preimages of a Qualified Constant Finite Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0066","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed9a3a59d80ca58296a0c60ec290f2fe43c4673f243df6f37679d48d95a44f0a","family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255066,"semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,2,3],[1]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":3}"},"fields":{"title":"All Preimages of a Qualified Constant Finite Map","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [1], [0, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], constant target value=3. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).","answer":{"kind":"expression","value":"(4, 4, 3, [3, 3, 3, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [], Yes], [[0, 1, 2], [], Yes], [[3], [0, 1, 2, 3], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 3, [3, 3, 3, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [], Yes], [[0, 1, 2], [], Yes], [[3], [0, 1, 2, 3], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["All Preimages of a Qualified Constant Finite Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0067","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb1b3284498de7d9b372b77d8b3889c0ea7d1181454e0300d21a7bbefbd6a34c","family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255067,"semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1],[2,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":0}"},"fields":{"title":"All Preimages of a Qualified Constant Finite Map","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [0, 1], [2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], constant target value=0. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).","answer":{"kind":"expression","value":"(4, 4, 0, [0, 0, 0, 0], [[[], [], Yes], [[0], [0, 1, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 0, [0, 0, 0, 0], [[[], [], Yes], [[0], [0, 1, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["All Preimages of a Qualified Constant Finite Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0068","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7fc9891b20bb04506dec4128f93f64d4abd5645b95f3cfc374fe5a684568a6fb","family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255068,"semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1,3],[2]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":1}"},"fields":{"title":"All Preimages of a Qualified Constant Finite Map","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [2], [0, 1, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], constant target value=1. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).","answer":{"kind":"expression","value":"(4, 4, 1, [1, 1, 1, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 1, 2, 3], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 1, [1, 1, 1, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 1, 2, 3], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["All Preimages of a Qualified Constant Finite Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0069","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e54862faf3667bc5ea6f6ac7b750d90e8238657634ec5a106a1e9b77a30b245","family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255069,"semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,2],[1,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":2}"},"fields":{"title":"All Preimages of a Qualified Constant Finite Map","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [0, 2], [1, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], constant target value=2. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).","answer":{"kind":"expression","value":"(4, 4, 2, [2, 2, 2, 2], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [0, 1, 2, 3], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [], Yes], [[0, 1, 3], [], Yes], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 2, [2, 2, 2, 2], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [0, 1, 2, 3], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [], Yes], [[0, 1, 3], [], Yes], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["All Preimages of a Qualified Constant Finite Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0070","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"80d2e3b852d1033564935eb57c0921a3283730738394f007b4aa777530094eee","family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255070,"semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,3],[1,2]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":3}"},"fields":{"title":"All Preimages of a Qualified Constant Finite Map","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [1, 2], [0, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], constant target value=3. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).","answer":{"kind":"expression","value":"(4, 4, 3, [3, 3, 3, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [], Yes], [[0, 1, 2], [], Yes], [[3], [0, 1, 2, 3], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 3, [3, 3, 3, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [], Yes], [[0, 1, 2], [], Yes], [[3], [0, 1, 2, 3], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["All Preimages of a Qualified Constant Finite Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0071","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aaa0d07f367ce421d875522305fcca9123e8232aaf02bc685d2a1bd10b1db3e4","family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255071,"semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1,2],[3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":0}"},"fields":{"title":"All Preimages of a Qualified Constant Finite Map","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [0, 1, 2], [3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], constant target value=0. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).","answer":{"kind":"expression","value":"(4, 4, 0, [0, 0, 0, 0], [[[], [], Yes], [[0], [0, 1, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 0, [0, 0, 0, 0], [[[], [], Yes], [[0], [0, 1, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["All Preimages of a Qualified Constant Finite Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0072","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12e1f759a5bc812e0dfa7f00dcded195ce1caaef65eb1f9bdaa499656b229d6a","family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255072,"semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0],[1,2,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":1}"},"fields":{"title":"All Preimages of a Qualified Constant Finite Map","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [0], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], constant target value=1. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).","answer":{"kind":"expression","value":"(4, 4, 1, [1, 1, 1, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 1, 2, 3], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 1, [1, 1, 1, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 1, 2, 3], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["All Preimages of a Qualified Constant Finite Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0073","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aef78ce5ddf5d689aa575a03432c4fddb99a41188e2d06f6201b6c7b33f021a9","family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255073,"semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,2,3],[1]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":2}"},"fields":{"title":"All Preimages of a Qualified Constant Finite Map","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [1], [0, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], constant target value=2. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).","answer":{"kind":"expression","value":"(4, 4, 2, [2, 2, 2, 2], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [0, 1, 2, 3], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [], Yes], [[0, 1, 3], [], Yes], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 2, [2, 2, 2, 2], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [0, 1, 2, 3], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [], Yes], [[0, 1, 3], [], Yes], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["All Preimages of a Qualified Constant Finite Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0074","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d1b61a84b464bc6ece55c33362d0bf136aae5dbdd53d1ee1063583da199bddb","family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255074,"semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1],[2,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":3}"},"fields":{"title":"All Preimages of a Qualified Constant Finite Map","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [0, 1], [2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], constant target value=3. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).","answer":{"kind":"expression","value":"(4, 4, 3, [3, 3, 3, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [], Yes], [[0, 1, 2], [], Yes], [[3], [0, 1, 2, 3], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 3, [3, 3, 3, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [], Yes], [[0, 1, 2], [], Yes], [[3], [0, 1, 2, 3], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["All Preimages of a Qualified Constant Finite Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0075","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"930c63aa2b6ebfa770c26ef9745c9945d989fa44a6ed94ccf862a485e8f4ad4c","family_id":"qf-mod25-measdeep-constant-map-audit","seed":202610255075,"semantic_template":"qf-mod25-measdeep-constant-map-audit/{\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1,3],[2]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"value\":0}"},"fields":{"title":"All Preimages of a Qualified Constant Finite Map","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [2], [0, 1, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], constant target value=0. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).","answer":{"kind":"expression","value":"(4, 4, 0, [0, 0, 0, 0], [[[], [], Yes], [[0], [0, 1, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 0, [0, 0, 0, 0], [[[], [], Yes], [[0], [0, 1, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["All Preimages of a Qualified Constant Finite Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0076","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99c8edb31b285e04d9a3c0539e1cad55ae61a97e9836269788a2a49034e1f624","family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255076,"semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[0,0,0,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0],[1,2,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"},"fields":{"title":"Verify Refinement Rather than Assuming the Proposed Family Is Larger","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [0], [1, 2, 3], [0, 1, 2, 3]], proposed refined sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[0, 0, 0, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.","answer":{"kind":"expression","value":"([0, 0, 0, 0], Yes, [[[], [], Yes], [[0], [0, 1, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, [[[], [], Yes], [[0], [0, 1, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 0, 0], Yes, [[[], [], Yes], [[0], [0, 1, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, [[[], [], Yes], [[0], [0, 1, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["Verify Refinement Rather than Assuming the Proposed Family Is Larger"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0077","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0be40e444356196f88a5a6b92b25dc63d0caa5f585d72f372037dbf8b24d49a","family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255077,"semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[1,0,0,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,2,3],[1]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"},"fields":{"title":"Verify Refinement Rather than Assuming the Proposed Family Is Larger","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [1], [0, 2, 3], [0, 1, 2, 3]], proposed refined sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[1, 0, 0, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.","answer":{"kind":"expression","value":"([1, 0, 0, 0], Yes, [[[], [], Yes], [[0], [1, 2, 3], No], [[1], [0], No], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [1, 2, 3], No], [[1, 2], [0], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [1, 2, 3], No], [[1, 3], [0], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [1, 2, 3], No], [[1, 2, 3], [0], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [1, 2, 3], Yes], [[1], [0], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [1, 2, 3], Yes], [[1, 2], [0], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [1, 2, 3], Yes], [[1, 3], [0], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [1, 2, 3], Yes], [[1, 2, 3], [0], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 0, 0], Yes, [[[], [], Yes], [[0], [1, 2, 3], No], [[1], [0], No], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [1, 2, 3], No], [[1, 2], [0], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [1, 2, 3], No], [[1, 3], [0], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [1, 2, 3], No], [[1, 2, 3], [0], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [1, 2, 3], Yes], [[1], [0], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [1, 2, 3], Yes], [[1, 2], [0], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [1, 2, 3], Yes], [[1, 3], [0], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [1, 2, 3], Yes], [[1, 2, 3], [0], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["Verify Refinement Rather than Assuming the Proposed Family Is Larger"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0078","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1b1cffc129f4aab390032113564796243631551e5dc8c3de1d6c51f5a61a8e7","family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255078,"semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[2,0,0,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1],[2,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"},"fields":{"title":"Verify Refinement Rather than Assuming the Proposed Family Is Larger","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [0, 1], [2, 3], [0, 1, 2, 3]], proposed refined sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[2, 0, 0, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.","answer":{"kind":"expression","value":"([2, 0, 0, 0], Yes, [[[], [], Yes], [[0], [1, 2, 3], No], [[1], [], Yes], [[0, 1], [1, 2, 3], No], [[2], [0], No], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [0], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [1, 2, 3], No], [[1, 3], [], Yes], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0], No], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [1, 2, 3], Yes], [[1], [], Yes], [[0, 1], [1, 2, 3], Yes], [[2], [0], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [0], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [1, 2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [1, 2, 3], Yes], [[2, 3], [0], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 0, 0, 0], Yes, [[[], [], Yes], [[0], [1, 2, 3], No], [[1], [], Yes], [[0, 1], [1, 2, 3], No], [[2], [0], No], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [0], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [1, 2, 3], No], [[1, 3], [], Yes], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0], No], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [1, 2, 3], Yes], [[1], [], Yes], [[0, 1], [1, 2, 3], Yes], [[2], [0], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [0], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [1, 2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [1, 2, 3], Yes], [[2, 3], [0], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["Verify Refinement Rather than Assuming the Proposed Family Is Larger"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0079","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef2a019cac263e3f96910a380ae5492060bdfbd096ec991f047362c0ab004f54","family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255079,"semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[3,0,0,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1,3],[2]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"},"fields":{"title":"Verify Refinement Rather than Assuming the Proposed Family Is Larger","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [2], [0, 1, 3], [0, 1, 2, 3]], proposed refined sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[3, 0, 0, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.","answer":{"kind":"expression","value":"([3, 0, 0, 0], Yes, [[[], [], Yes], [[0], [1, 2, 3], No], [[1], [], Yes], [[0, 1], [1, 2, 3], No], [[2], [], Yes], [[0, 2], [1, 2, 3], No], [[1, 2], [], Yes], [[0, 1, 2], [1, 2, 3], No], [[3], [0], No], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0], No], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [1, 2, 3], Yes], [[1], [], Yes], [[0, 1], [1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [1, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [1, 2, 3], Yes], [[3], [0], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 0, 0, 0], Yes, [[[], [], Yes], [[0], [1, 2, 3], No], [[1], [], Yes], [[0, 1], [1, 2, 3], No], [[2], [], Yes], [[0, 2], [1, 2, 3], No], [[1, 2], [], Yes], [[0, 1, 2], [1, 2, 3], No], [[3], [0], No], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0], No], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [1, 2, 3], Yes], [[1], [], Yes], [[0, 1], [1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [1, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [1, 2, 3], Yes], [[3], [0], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["Verify Refinement Rather than Assuming the Proposed Family Is Larger"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0080","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5951e9feaa7b7492791b03802115217de248ca8497b62779267a996d394bd60","family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255080,"semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[0,1,0,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,2],[1,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"},"fields":{"title":"Verify Refinement Rather than Assuming the Proposed Family Is Larger","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [0, 2], [1, 3], [0, 1, 2, 3]], proposed refined sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[0, 1, 0, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.","answer":{"kind":"expression","value":"([0, 1, 0, 0], Yes, [[[], [], Yes], [[0], [0, 2, 3], No], [[1], [1], No], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 2, 3], No], [[1, 2], [1], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 2, 3], No], [[1, 3], [1], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [0, 2, 3], Yes], [[1], [1], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 2, 3], Yes], [[1, 2], [1], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 2, 3], Yes], [[1, 3], [1], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 2, 3], Yes], [[1, 2, 3], [1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0, 0], Yes, [[[], [], Yes], [[0], [0, 2, 3], No], [[1], [1], No], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 2, 3], No], [[1, 2], [1], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 2, 3], No], [[1, 3], [1], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [0, 2, 3], Yes], [[1], [1], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 2, 3], Yes], [[1, 2], [1], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 2, 3], Yes], [[1, 3], [1], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 2, 3], Yes], [[1, 2, 3], [1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["Verify Refinement Rather than Assuming the Proposed Family Is Larger"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0081","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f9c48519bc1ff36a4ecebb07a4495bc2d90c14e249c6af591485c6978e58033","family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255081,"semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[1,1,0,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,3],[1,2]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"},"fields":{"title":"Verify Refinement Rather than Assuming the Proposed Family Is Larger","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [1, 2], [0, 3], [0, 1, 2, 3]], proposed refined sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[1, 1, 0, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.","answer":{"kind":"expression","value":"([1, 1, 0, 0], Yes, [[[], [], Yes], [[0], [2, 3], No], [[1], [0, 1], No], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [2, 3], No], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [2, 3], No], [[1, 3], [0, 1], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [2, 3], No], [[1, 2, 3], [0, 1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [2, 3], Yes], [[1], [0, 1], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [2, 3], Yes], [[1, 2], [0, 1], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [2, 3], Yes], [[1, 3], [0, 1], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 0, 0], Yes, [[[], [], Yes], [[0], [2, 3], No], [[1], [0, 1], No], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [2, 3], No], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [2, 3], No], [[1, 3], [0, 1], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [2, 3], No], [[1, 2, 3], [0, 1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [2, 3], Yes], [[1], [0, 1], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [2, 3], Yes], [[1, 2], [0, 1], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [2, 3], Yes], [[1, 3], [0, 1], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["Verify Refinement Rather than Assuming the Proposed Family Is Larger"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0082","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"763cd1c0e84a5a4190f350ba4db3db46dc88044c61e415c50898fe473b78e290","family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255082,"semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[2,1,0,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1,2],[3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"},"fields":{"title":"Verify Refinement Rather than Assuming the Proposed Family Is Larger","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [0, 1, 2], [3], [0, 1, 2, 3]], proposed refined sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[2, 1, 0, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.","answer":{"kind":"expression","value":"([2, 1, 0, 0], Yes, [[[], [], Yes], [[0], [2, 3], No], [[1], [1], No], [[0, 1], [1, 2, 3], No], [[2], [0], No], [[0, 2], [0, 2, 3], No], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [2, 3], No], [[1, 3], [1], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0], No], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [0, 1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [2, 3], Yes], [[1], [1], Yes], [[0, 1], [1, 2, 3], Yes], [[2], [0], Yes], [[0, 2], [0, 2, 3], Yes], [[1, 2], [0, 1], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [2, 3], Yes], [[1, 3], [1], Yes], [[0, 1, 3], [1, 2, 3], Yes], [[2, 3], [0], Yes], [[0, 2, 3], [0, 2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 1, 0, 0], Yes, [[[], [], Yes], [[0], [2, 3], No], [[1], [1], No], [[0, 1], [1, 2, 3], No], [[2], [0], No], [[0, 2], [0, 2, 3], No], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [2, 3], No], [[1, 3], [1], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0], No], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [0, 1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [2, 3], Yes], [[1], [1], Yes], [[0, 1], [1, 2, 3], Yes], [[2], [0], Yes], [[0, 2], [0, 2, 3], Yes], [[1, 2], [0, 1], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [2, 3], Yes], [[1, 3], [1], Yes], [[0, 1, 3], [1, 2, 3], Yes], [[2, 3], [0], Yes], [[0, 2, 3], [0, 2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["Verify Refinement Rather than Assuming the Proposed Family Is Larger"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0083","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b440c24810f12b72a518b8d900b0a73b373982d044dbff53e7f2a058160b3ce8","family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255083,"semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[3,1,0,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0],[1,2,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"},"fields":{"title":"Verify Refinement Rather than Assuming the Proposed Family Is Larger","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [0], [1, 2, 3], [0, 1, 2, 3]], proposed refined sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[3, 1, 0, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.","answer":{"kind":"expression","value":"([3, 1, 0, 0], Yes, [[[], [], Yes], [[0], [2, 3], No], [[1], [1], No], [[0, 1], [1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [2, 3], No], [[1, 2], [1], No], [[0, 1, 2], [1, 2, 3], Yes], [[3], [0], Yes], [[0, 3], [0, 2, 3], No], [[1, 3], [0, 1], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0], Yes], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [0, 1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [2, 3], Yes], [[1], [1], Yes], [[0, 1], [1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [2, 3], Yes], [[1, 2], [1], Yes], [[0, 1, 2], [1, 2, 3], Yes], [[3], [0], Yes], [[0, 3], [0, 2, 3], Yes], [[1, 3], [0, 1], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0], Yes], [[0, 2, 3], [0, 2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 1, 0, 0], Yes, [[[], [], Yes], [[0], [2, 3], No], [[1], [1], No], [[0, 1], [1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [2, 3], No], [[1, 2], [1], No], [[0, 1, 2], [1, 2, 3], Yes], [[3], [0], Yes], [[0, 3], [0, 2, 3], No], [[1, 3], [0, 1], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0], Yes], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [0, 1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [2, 3], Yes], [[1], [1], Yes], [[0, 1], [1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [2, 3], Yes], [[1, 2], [1], Yes], [[0, 1, 2], [1, 2, 3], Yes], [[3], [0], Yes], [[0, 3], [0, 2, 3], Yes], [[1, 3], [0, 1], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0], Yes], [[0, 2, 3], [0, 2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["Verify Refinement Rather than Assuming the Proposed Family Is Larger"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0084","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9bf12508abeecd84af0866bd70c57f21ad7d1cd2644fd229f2299fc10d0b2a3c","family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255084,"semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[0,2,0,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,2,3],[1]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"},"fields":{"title":"Verify Refinement Rather than Assuming the Proposed Family Is Larger","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [1], [0, 2, 3], [0, 1, 2, 3]], proposed refined sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[0, 2, 0, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.","answer":{"kind":"expression","value":"([0, 2, 0, 0], Yes, [[[], [], Yes], [[0], [0, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 2, 3], Yes], [[2], [1], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [1], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 2, 3], Yes], [[2, 3], [1], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, [[[], [], Yes], [[0], [0, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 2, 3], Yes], [[2], [1], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [1], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 2, 3], Yes], [[2, 3], [1], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 2, 0, 0], Yes, [[[], [], Yes], [[0], [0, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 2, 3], Yes], [[2], [1], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [1], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 2, 3], Yes], [[2, 3], [1], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, [[[], [], Yes], [[0], [0, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 2, 3], Yes], [[2], [1], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [1], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 2, 3], Yes], [[2, 3], [1], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["Verify Refinement Rather than Assuming the Proposed Family Is Larger"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0085","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0149b9f7505b64de2c65e2bca44010edaed02759e054355ab2c8ba89d68109c","family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255085,"semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[1,2,0,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1],[2,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"},"fields":{"title":"Verify Refinement Rather than Assuming the Proposed Family Is Larger","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [0, 1], [2, 3], [0, 1, 2, 3]], proposed refined sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[1, 2, 0, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.","answer":{"kind":"expression","value":"([1, 2, 0, 0], Yes, [[[], [], Yes], [[0], [2, 3], Yes], [[1], [0], No], [[0, 1], [0, 2, 3], No], [[2], [1], No], [[0, 2], [1, 2, 3], No], [[1, 2], [0, 1], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [2, 3], Yes], [[1, 3], [0], No], [[0, 1, 3], [0, 2, 3], No], [[2, 3], [1], No], [[0, 2, 3], [1, 2, 3], No], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [2, 3], Yes], [[1], [0], Yes], [[0, 1], [0, 2, 3], Yes], [[2], [1], Yes], [[0, 2], [1, 2, 3], Yes], [[1, 2], [0, 1], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [2, 3], Yes], [[1, 3], [0], Yes], [[0, 1, 3], [0, 2, 3], Yes], [[2, 3], [1], Yes], [[0, 2, 3], [1, 2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 2, 0, 0], Yes, [[[], [], Yes], [[0], [2, 3], Yes], [[1], [0], No], [[0, 1], [0, 2, 3], No], [[2], [1], No], [[0, 2], [1, 2, 3], No], [[1, 2], [0, 1], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [2, 3], Yes], [[1, 3], [0], No], [[0, 1, 3], [0, 2, 3], No], [[2, 3], [1], No], [[0, 2, 3], [1, 2, 3], No], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [2, 3], Yes], [[1], [0], Yes], [[0, 1], [0, 2, 3], Yes], [[2], [1], Yes], [[0, 2], [1, 2, 3], Yes], [[1, 2], [0, 1], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [2, 3], Yes], [[1, 3], [0], Yes], [[0, 1, 3], [0, 2, 3], Yes], [[2, 3], [1], Yes], [[0, 2, 3], [1, 2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["Verify Refinement Rather than Assuming the Proposed Family Is Larger"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0086","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"303a353a0c088c6956d8649eff020b6bd656abdd5b8927df54526cab720bde48","family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255086,"semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[2,2,0,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1,3],[2]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"},"fields":{"title":"Verify Refinement Rather than Assuming the Proposed Family Is Larger","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [2], [0, 1, 3], [0, 1, 2, 3]], proposed refined sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[2, 2, 0, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.","answer":{"kind":"expression","value":"([2, 2, 0, 0], Yes, [[[], [], Yes], [[0], [2, 3], No], [[1], [], Yes], [[0, 1], [2, 3], No], [[2], [0, 1], No], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [2, 3], No], [[1, 3], [], Yes], [[0, 1, 3], [2, 3], No], [[2, 3], [0, 1], No], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [2, 3], Yes], [[1], [], Yes], [[0, 1], [2, 3], Yes], [[2], [0, 1], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [0, 1], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [2, 3], Yes], [[2, 3], [0, 1], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, 0, 0], Yes, [[[], [], Yes], [[0], [2, 3], No], [[1], [], Yes], [[0, 1], [2, 3], No], [[2], [0, 1], No], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [2, 3], No], [[1, 3], [], Yes], [[0, 1, 3], [2, 3], No], [[2, 3], [0, 1], No], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [2, 3], Yes], [[1], [], Yes], [[0, 1], [2, 3], Yes], [[2], [0, 1], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [0, 1], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [2, 3], Yes], [[2, 3], [0, 1], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["Verify Refinement Rather than Assuming the Proposed Family Is Larger"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0087","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"452fb5c67b7a63b203c07f8267ce0298b0e0fb5e70cd3d15310efde983d86bc8","family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255087,"semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[3,2,0,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,2],[1,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"},"fields":{"title":"Verify Refinement Rather than Assuming the Proposed Family Is Larger","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [0, 2], [1, 3], [0, 1, 2, 3]], proposed refined sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[3, 2, 0, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.","answer":{"kind":"expression","value":"([3, 2, 0, 0], Yes, [[[], [], Yes], [[0], [2, 3], No], [[1], [], Yes], [[0, 1], [2, 3], No], [[2], [1], No], [[0, 2], [1, 2, 3], No], [[1, 2], [1], No], [[0, 1, 2], [1, 2, 3], No], [[3], [0], No], [[0, 3], [0, 2, 3], No], [[1, 3], [0], No], [[0, 1, 3], [0, 2, 3], No], [[2, 3], [0, 1], No], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [2, 3], Yes], [[1], [], Yes], [[0, 1], [2, 3], Yes], [[2], [1], Yes], [[0, 2], [1, 2, 3], Yes], [[1, 2], [1], Yes], [[0, 1, 2], [1, 2, 3], Yes], [[3], [0], Yes], [[0, 3], [0, 2, 3], Yes], [[1, 3], [0], Yes], [[0, 1, 3], [0, 2, 3], Yes], [[2, 3], [0, 1], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 0, 0], Yes, [[[], [], Yes], [[0], [2, 3], No], [[1], [], Yes], [[0, 1], [2, 3], No], [[2], [1], No], [[0, 2], [1, 2, 3], No], [[1, 2], [1], No], [[0, 1, 2], [1, 2, 3], No], [[3], [0], No], [[0, 3], [0, 2, 3], No], [[1, 3], [0], No], [[0, 1, 3], [0, 2, 3], No], [[2, 3], [0, 1], No], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [2, 3], Yes], [[1], [], Yes], [[0, 1], [2, 3], Yes], [[2], [1], Yes], [[0, 2], [1, 2, 3], Yes], [[1, 2], [1], Yes], [[0, 1, 2], [1, 2, 3], Yes], [[3], [0], Yes], [[0, 3], [0, 2, 3], Yes], [[1, 3], [0], Yes], [[0, 1, 3], [0, 2, 3], Yes], [[2, 3], [0, 1], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["Verify Refinement Rather than Assuming the Proposed Family Is Larger"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0088","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72e8906a303590601e2b527dea21821ffa2b361dd0e63d64fa6b478273d027ca","family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255088,"semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[0,3,0,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,3],[1,2]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"},"fields":{"title":"Verify Refinement Rather than Assuming the Proposed Family Is Larger","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [1, 2], [0, 3], [0, 1, 2, 3]], proposed refined sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[0, 3, 0, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.","answer":{"kind":"expression","value":"([0, 3, 0, 0], Yes, [[[], [], Yes], [[0], [0, 2, 3], No], [[1], [], Yes], [[0, 1], [0, 2, 3], No], [[2], [], Yes], [[0, 2], [0, 2, 3], No], [[1, 2], [], Yes], [[0, 1, 2], [0, 2, 3], No], [[3], [1], No], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [1], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [1], No], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [0, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 2, 3], Yes], [[3], [1], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [1], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [1], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 3, 0, 0], Yes, [[[], [], Yes], [[0], [0, 2, 3], No], [[1], [], Yes], [[0, 1], [0, 2, 3], No], [[2], [], Yes], [[0, 2], [0, 2, 3], No], [[1, 2], [], Yes], [[0, 1, 2], [0, 2, 3], No], [[3], [1], No], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [1], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [1], No], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [0, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 2, 3], Yes], [[3], [1], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [1], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [1], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["Verify Refinement Rather than Assuming the Proposed Family Is Larger"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0089","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7aa4ba9ea9d54618e8e68118763a857f6b5d0739d93784f8c29fc8fc581c4b51","family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255089,"semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[1,3,0,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1,2],[3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"},"fields":{"title":"Verify Refinement Rather than Assuming the Proposed Family Is Larger","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [0, 1, 2], [3], [0, 1, 2, 3]], proposed refined sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[1, 3, 0, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.","answer":{"kind":"expression","value":"([1, 3, 0, 0], Yes, [[[], [], Yes], [[0], [2, 3], No], [[1], [0], No], [[0, 1], [0, 2, 3], No], [[2], [], Yes], [[0, 2], [2, 3], No], [[1, 2], [0], No], [[0, 1, 2], [0, 2, 3], No], [[3], [1], No], [[0, 3], [1, 2, 3], No], [[1, 3], [0, 1], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [1], No], [[0, 2, 3], [1, 2, 3], No], [[1, 2, 3], [0, 1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [2, 3], Yes], [[1], [0], Yes], [[0, 1], [0, 2, 3], Yes], [[2], [], Yes], [[0, 2], [2, 3], Yes], [[1, 2], [0], Yes], [[0, 1, 2], [0, 2, 3], Yes], [[3], [1], Yes], [[0, 3], [1, 2, 3], Yes], [[1, 3], [0, 1], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [1], Yes], [[0, 2, 3], [1, 2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 3, 0, 0], Yes, [[[], [], Yes], [[0], [2, 3], No], [[1], [0], No], [[0, 1], [0, 2, 3], No], [[2], [], Yes], [[0, 2], [2, 3], No], [[1, 2], [0], No], [[0, 1, 2], [0, 2, 3], No], [[3], [1], No], [[0, 3], [1, 2, 3], No], [[1, 3], [0, 1], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [1], No], [[0, 2, 3], [1, 2, 3], No], [[1, 2, 3], [0, 1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [2, 3], Yes], [[1], [0], Yes], [[0, 1], [0, 2, 3], Yes], [[2], [], Yes], [[0, 2], [2, 3], Yes], [[1, 2], [0], Yes], [[0, 1, 2], [0, 2, 3], Yes], [[3], [1], Yes], [[0, 3], [1, 2, 3], Yes], [[1, 3], [0, 1], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [1], Yes], [[0, 2, 3], [1, 2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["Verify Refinement Rather than Assuming the Proposed Family Is Larger"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0090","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee5ca9dc1479593087eebc11df54646da7debdea063803673e2007c026aec6e1","family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255090,"semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[2,3,0,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0],[1,2,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"},"fields":{"title":"Verify Refinement Rather than Assuming the Proposed Family Is Larger","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [0], [1, 2, 3], [0, 1, 2, 3]], proposed refined sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[2, 3, 0, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.","answer":{"kind":"expression","value":"([2, 3, 0, 0], Yes, [[[], [], Yes], [[0], [2, 3], No], [[1], [], Yes], [[0, 1], [2, 3], No], [[2], [0], Yes], [[0, 2], [0, 2, 3], No], [[1, 2], [0], Yes], [[0, 1, 2], [0, 2, 3], No], [[3], [1], No], [[0, 3], [1, 2, 3], Yes], [[1, 3], [1], No], [[0, 1, 3], [1, 2, 3], Yes], [[2, 3], [0, 1], No], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [2, 3], Yes], [[1], [], Yes], [[0, 1], [2, 3], Yes], [[2], [0], Yes], [[0, 2], [0, 2, 3], Yes], [[1, 2], [0], Yes], [[0, 1, 2], [0, 2, 3], Yes], [[3], [1], Yes], [[0, 3], [1, 2, 3], Yes], [[1, 3], [1], Yes], [[0, 1, 3], [1, 2, 3], Yes], [[2, 3], [0, 1], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 3, 0, 0], Yes, [[[], [], Yes], [[0], [2, 3], No], [[1], [], Yes], [[0, 1], [2, 3], No], [[2], [0], Yes], [[0, 2], [0, 2, 3], No], [[1, 2], [0], Yes], [[0, 1, 2], [0, 2, 3], No], [[3], [1], No], [[0, 3], [1, 2, 3], Yes], [[1, 3], [1], No], [[0, 1, 3], [1, 2, 3], Yes], [[2, 3], [0, 1], No], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [2, 3], Yes], [[1], [], Yes], [[0, 1], [2, 3], Yes], [[2], [0], Yes], [[0, 2], [0, 2, 3], Yes], [[1, 2], [0], Yes], [[0, 1, 2], [0, 2, 3], Yes], [[3], [1], Yes], [[0, 3], [1, 2, 3], Yes], [[1, 3], [1], Yes], [[0, 1, 3], [1, 2, 3], Yes], [[2, 3], [0, 1], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["Verify Refinement Rather than Assuming the Proposed Family Is Larger"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0091","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12e40d6849de538e81c5446c5b34d820434e53a1fc983d6f68b8ab333070f8ee","family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255091,"semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[3,3,0,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,2,3],[1]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"},"fields":{"title":"Verify Refinement Rather than Assuming the Proposed Family Is Larger","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [1], [0, 2, 3], [0, 1, 2, 3]], proposed refined sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[3, 3, 0, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.","answer":{"kind":"expression","value":"([3, 3, 0, 0], Yes, [[[], [], Yes], [[0], [2, 3], No], [[1], [], Yes], [[0, 1], [2, 3], No], [[2], [], Yes], [[0, 2], [2, 3], No], [[1, 2], [], Yes], [[0, 1, 2], [2, 3], No], [[3], [0, 1], No], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0, 1], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0, 1], No], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [2, 3], Yes], [[1], [], Yes], [[0, 1], [2, 3], Yes], [[2], [], Yes], [[0, 2], [2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [2, 3], Yes], [[3], [0, 1], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0, 1], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0, 1], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 3, 0, 0], Yes, [[[], [], Yes], [[0], [2, 3], No], [[1], [], Yes], [[0, 1], [2, 3], No], [[2], [], Yes], [[0, 2], [2, 3], No], [[1, 2], [], Yes], [[0, 1, 2], [2, 3], No], [[3], [0, 1], No], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0, 1], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0, 1], No], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [2, 3], Yes], [[1], [], Yes], [[0, 1], [2, 3], Yes], [[2], [], Yes], [[0, 2], [2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [2, 3], Yes], [[3], [0, 1], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0, 1], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0, 1], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["Verify Refinement Rather than Assuming the Proposed Family Is Larger"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0092","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"753bfd07d6eff0e8e9b281f40da9b2969218fb8461f972d97ae43ead9b6d8396","family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255092,"semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[0,0,1,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1],[2,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"},"fields":{"title":"Verify Refinement Rather than Assuming the Proposed Family Is Larger","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [0, 1], [2, 3], [0, 1, 2, 3]], proposed refined sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[0, 0, 1, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.","answer":{"kind":"expression","value":"([0, 0, 1, 0], Yes, [[[], [], Yes], [[0], [0, 1, 3], No], [[1], [2], No], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 1, 3], No], [[1, 2], [2], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 3], No], [[1, 3], [2], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [0, 1, 3], Yes], [[1], [2], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 1, 3], Yes], [[1, 2], [2], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 3], Yes], [[1, 3], [2], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 1, 3], Yes], [[1, 2, 3], [2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 1, 0], Yes, [[[], [], Yes], [[0], [0, 1, 3], No], [[1], [2], No], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 1, 3], No], [[1, 2], [2], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 3], No], [[1, 3], [2], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [0, 1, 3], Yes], [[1], [2], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 1, 3], Yes], [[1, 2], [2], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 3], Yes], [[1, 3], [2], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 1, 3], Yes], [[1, 2, 3], [2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["Verify Refinement Rather than Assuming the Proposed Family Is Larger"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0093","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3dc425347ce4502f7c8e0ccc4508cbb7f27de4adcd0c470d56b4aff0de5836f7","family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255093,"semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[1,0,1,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1,3],[2]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"},"fields":{"title":"Verify Refinement Rather than Assuming the Proposed Family Is Larger","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [2], [0, 1, 3], [0, 1, 2, 3]], proposed refined sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[1, 0, 1, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.","answer":{"kind":"expression","value":"([1, 0, 1, 0], Yes, [[[], [], Yes], [[0], [1, 3], No], [[1], [0, 2], No], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [1, 3], No], [[1, 2], [0, 2], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [1, 3], No], [[1, 3], [0, 2], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [1, 3], No], [[1, 2, 3], [0, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [1, 3], Yes], [[1], [0, 2], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [1, 3], Yes], [[1, 2], [0, 2], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [1, 3], Yes], [[1, 3], [0, 2], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [1, 3], Yes], [[1, 2, 3], [0, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1, 0], Yes, [[[], [], Yes], [[0], [1, 3], No], [[1], [0, 2], No], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [1, 3], No], [[1, 2], [0, 2], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [1, 3], No], [[1, 3], [0, 2], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [1, 3], No], [[1, 2, 3], [0, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [1, 3], Yes], [[1], [0, 2], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [1, 3], Yes], [[1, 2], [0, 2], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [1, 3], Yes], [[1, 3], [0, 2], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [1, 3], Yes], [[1, 2, 3], [0, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["Verify Refinement Rather than Assuming the Proposed Family Is Larger"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0094","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78d793a2d440647c99d9625b866892b9eeb1e2b7e2b8fd9a58d48dc43108756e","family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255094,"semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[2,0,1,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,2],[1,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"},"fields":{"title":"Verify Refinement Rather than Assuming the Proposed Family Is Larger","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [0, 2], [1, 3], [0, 1, 2, 3]], proposed refined sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[2, 0, 1, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.","answer":{"kind":"expression","value":"([2, 0, 1, 0], Yes, [[[], [], Yes], [[0], [1, 3], Yes], [[1], [2], No], [[0, 1], [1, 2, 3], No], [[2], [0], No], [[0, 2], [0, 1, 3], No], [[1, 2], [0, 2], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [1, 3], Yes], [[1, 3], [2], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0], No], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [0, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [1, 3], Yes], [[1], [2], Yes], [[0, 1], [1, 2, 3], Yes], [[2], [0], Yes], [[0, 2], [0, 1, 3], Yes], [[1, 2], [0, 2], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [1, 3], Yes], [[1, 3], [2], Yes], [[0, 1, 3], [1, 2, 3], Yes], [[2, 3], [0], Yes], [[0, 2, 3], [0, 1, 3], Yes], [[1, 2, 3], [0, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 0, 1, 0], Yes, [[[], [], Yes], [[0], [1, 3], Yes], [[1], [2], No], [[0, 1], [1, 2, 3], No], [[2], [0], No], [[0, 2], [0, 1, 3], No], [[1, 2], [0, 2], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [1, 3], Yes], [[1, 3], [2], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0], No], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [0, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [1, 3], Yes], [[1], [2], Yes], [[0, 1], [1, 2, 3], Yes], [[2], [0], Yes], [[0, 2], [0, 1, 3], Yes], [[1, 2], [0, 2], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [1, 3], Yes], [[1, 3], [2], Yes], [[0, 1, 3], [1, 2, 3], Yes], [[2, 3], [0], Yes], [[0, 2, 3], [0, 1, 3], Yes], [[1, 2, 3], [0, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["Verify Refinement Rather than Assuming the Proposed Family Is Larger"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0095","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1a5f935ee64e90f3af4875ce1ba43290d503610914c74fd23f39f5c3e9cb0f1","family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255095,"semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[3,0,1,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,3],[1,2]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"},"fields":{"title":"Verify Refinement Rather than Assuming the Proposed Family Is Larger","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [1, 2], [0, 3], [0, 1, 2, 3]], proposed refined sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[3, 0, 1, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.","answer":{"kind":"expression","value":"([3, 0, 1, 0], Yes, [[[], [], Yes], [[0], [1, 3], No], [[1], [2], No], [[0, 1], [1, 2, 3], No], [[2], [], Yes], [[0, 2], [1, 3], No], [[1, 2], [2], No], [[0, 1, 2], [1, 2, 3], No], [[3], [0], No], [[0, 3], [0, 1, 3], No], [[1, 3], [0, 2], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0], No], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [0, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [1, 3], Yes], [[1], [2], Yes], [[0, 1], [1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [1, 3], Yes], [[1, 2], [2], Yes], [[0, 1, 2], [1, 2, 3], Yes], [[3], [0], Yes], [[0, 3], [0, 1, 3], Yes], [[1, 3], [0, 2], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0], Yes], [[0, 2, 3], [0, 1, 3], Yes], [[1, 2, 3], [0, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 0, 1, 0], Yes, [[[], [], Yes], [[0], [1, 3], No], [[1], [2], No], [[0, 1], [1, 2, 3], No], [[2], [], Yes], [[0, 2], [1, 3], No], [[1, 2], [2], No], [[0, 1, 2], [1, 2, 3], No], [[3], [0], No], [[0, 3], [0, 1, 3], No], [[1, 3], [0, 2], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0], No], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [0, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [1, 3], Yes], [[1], [2], Yes], [[0, 1], [1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [1, 3], Yes], [[1, 2], [2], Yes], [[0, 1, 2], [1, 2, 3], Yes], [[3], [0], Yes], [[0, 3], [0, 1, 3], Yes], [[1, 3], [0, 2], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0], Yes], [[0, 2, 3], [0, 1, 3], Yes], [[1, 2, 3], [0, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["Verify Refinement Rather than Assuming the Proposed Family Is Larger"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0096","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f2073eae5975bb91f3e2cca8f457c5fe6c03e78abfc416305f54f2525dd0d6c","family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255096,"semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[0,1,1,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1,2],[3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"},"fields":{"title":"Verify Refinement Rather than Assuming the Proposed Family Is Larger","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [0, 1, 2], [3], [0, 1, 2, 3]], proposed refined sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[0, 1, 1, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.","answer":{"kind":"expression","value":"([0, 1, 1, 0], Yes, [[[], [], Yes], [[0], [0, 3], No], [[1], [1, 2], No], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 3], No], [[1, 2], [1, 2], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 3], No], [[1, 3], [1, 2], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 3], No], [[1, 2, 3], [1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [0, 3], Yes], [[1], [1, 2], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 3], Yes], [[1, 2], [1, 2], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 3], Yes], [[1, 3], [1, 2], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 3], Yes], [[1, 2, 3], [1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 1, 0], Yes, [[[], [], Yes], [[0], [0, 3], No], [[1], [1, 2], No], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 3], No], [[1, 2], [1, 2], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 3], No], [[1, 3], [1, 2], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 3], No], [[1, 2, 3], [1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [0, 3], Yes], [[1], [1, 2], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 3], Yes], [[1, 2], [1, 2], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 3], Yes], [[1, 3], [1, 2], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 3], Yes], [[1, 2, 3], [1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["Verify Refinement Rather than Assuming the Proposed Family Is Larger"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0097","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c63da7917c42c4a4e5a9fe6a51a68710c1b327d61365a1fe2787f71245b4582c","family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255097,"semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[1,1,1,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0],[1,2,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"},"fields":{"title":"Verify Refinement Rather than Assuming the Proposed Family Is Larger","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [0], [1, 2, 3], [0, 1, 2, 3]], proposed refined sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[1, 1, 1, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.","answer":{"kind":"expression","value":"([1, 1, 1, 0], Yes, [[[], [], Yes], [[0], [3], No], [[1], [0, 1, 2], No], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [3], No], [[1, 2], [0, 1, 2], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [3], No], [[1, 3], [0, 1, 2], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [3], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [3], Yes], [[1], [0, 1, 2], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [3], Yes], [[1, 2], [0, 1, 2], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [3], Yes], [[1, 3], [0, 1, 2], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [3], Yes], [[1, 2, 3], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 1, 0], Yes, [[[], [], Yes], [[0], [3], No], [[1], [0, 1, 2], No], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [3], No], [[1, 2], [0, 1, 2], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [3], No], [[1, 3], [0, 1, 2], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [3], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [3], Yes], [[1], [0, 1, 2], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [3], Yes], [[1, 2], [0, 1, 2], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [3], Yes], [[1, 3], [0, 1, 2], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [3], Yes], [[1, 2, 3], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["Verify Refinement Rather than Assuming the Proposed Family Is Larger"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0098","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aab82c68c5606fa7dfab76e7846da511fb11908a2c6c503bc1105ce4adc04376","family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255098,"semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[2,1,1,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,2,3],[1]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"},"fields":{"title":"Verify Refinement Rather than Assuming the Proposed Family Is Larger","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [1], [0, 2, 3], [0, 1, 2, 3]], proposed refined sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[2, 1, 1, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.","answer":{"kind":"expression","value":"([2, 1, 1, 0], Yes, [[[], [], Yes], [[0], [3], No], [[1], [1, 2], No], [[0, 1], [1, 2, 3], No], [[2], [0], No], [[0, 2], [0, 3], No], [[1, 2], [0, 1, 2], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [3], No], [[1, 3], [1, 2], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0], No], [[0, 2, 3], [0, 3], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [3], Yes], [[1], [1, 2], Yes], [[0, 1], [1, 2, 3], Yes], [[2], [0], Yes], [[0, 2], [0, 3], Yes], [[1, 2], [0, 1, 2], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [3], Yes], [[1, 3], [1, 2], Yes], [[0, 1, 3], [1, 2, 3], Yes], [[2, 3], [0], Yes], [[0, 2, 3], [0, 3], Yes], [[1, 2, 3], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 1, 1, 0], Yes, [[[], [], Yes], [[0], [3], No], [[1], [1, 2], No], [[0, 1], [1, 2, 3], No], [[2], [0], No], [[0, 2], [0, 3], No], [[1, 2], [0, 1, 2], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [3], No], [[1, 3], [1, 2], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0], No], [[0, 2, 3], [0, 3], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [3], Yes], [[1], [1, 2], Yes], [[0, 1], [1, 2, 3], Yes], [[2], [0], Yes], [[0, 2], [0, 3], Yes], [[1, 2], [0, 1, 2], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [3], Yes], [[1, 3], [1, 2], Yes], [[0, 1, 3], [1, 2, 3], Yes], [[2, 3], [0], Yes], [[0, 2, 3], [0, 3], Yes], [[1, 2, 3], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["Verify Refinement Rather than Assuming the Proposed Family Is Larger"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0099","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee8212a1de2d9f6177ce0a4176e7005df1c6d39176fe27eebc6881a78479b3aa","family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255099,"semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[3,1,1,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1],[2,3]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"},"fields":{"title":"Verify Refinement Rather than Assuming the Proposed Family Is Larger","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [0, 1], [2, 3], [0, 1, 2, 3]], proposed refined sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[3, 1, 1, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.","answer":{"kind":"expression","value":"([3, 1, 1, 0], Yes, [[[], [], Yes], [[0], [3], No], [[1], [1, 2], No], [[0, 1], [1, 2, 3], No], [[2], [], Yes], [[0, 2], [3], No], [[1, 2], [1, 2], No], [[0, 1, 2], [1, 2, 3], No], [[3], [0], No], [[0, 3], [0, 3], No], [[1, 3], [0, 1, 2], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0], No], [[0, 2, 3], [0, 3], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [3], Yes], [[1], [1, 2], Yes], [[0, 1], [1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [3], Yes], [[1, 2], [1, 2], Yes], [[0, 1, 2], [1, 2, 3], Yes], [[3], [0], Yes], [[0, 3], [0, 3], Yes], [[1, 3], [0, 1, 2], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0], Yes], [[0, 2, 3], [0, 3], Yes], [[1, 2, 3], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 1, 1, 0], Yes, [[[], [], Yes], [[0], [3], No], [[1], [1, 2], No], [[0, 1], [1, 2, 3], No], [[2], [], Yes], [[0, 2], [3], No], [[1, 2], [1, 2], No], [[0, 1, 2], [1, 2, 3], No], [[3], [0], No], [[0, 3], [0, 3], No], [[1, 3], [0, 1, 2], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0], No], [[0, 2, 3], [0, 3], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [3], Yes], [[1], [1, 2], Yes], [[0, 1], [1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [3], Yes], [[1, 2], [1, 2], Yes], [[0, 1, 2], [1, 2, 3], Yes], [[3], [0], Yes], [[0, 3], [0, 3], Yes], [[1, 3], [0, 1, 2], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0], Yes], [[0, 2, 3], [0, 3], Yes], [[1, 2, 3], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["Verify Refinement Rather than Assuming the Proposed Family Is Larger"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261025-r05-measurable-function-0100","batch_id":"batch-20261013-advance-20261025-r05-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49690581c2977c0787b87635ddfc0c583bde3b6b9701e3dae356cded27e0f59d","family_id":"qf-mod25-measdeep-source-sigma-refinement-audit","seed":202610255100,"semantic_template":"qf-mod25-measdeep-source-sigma-refinement-audit/{\"mapping\":[0,2,1,0],\"refined_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]],\"source_n\":4,\"source_sigma\":[[],[0,1,2,3],[0,1,3],[2]],\"target_n\":4,\"target_sigma\":[[],[0,1,2,3],[0,1,2],[0,1,3],[0,1],[0,2,3],[0,2],[0,3],[0],[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3]]}"},"fields":{"title":"Verify Refinement Rather than Assuming the Proposed Family Is Larger","stem":"Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [2], [0, 1, 3], [0, 1, 2, 3]], proposed refined sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[0, 2, 1, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.","answer":{"kind":"expression","value":"([0, 2, 1, 0], Yes, [[[], [], Yes], [[0], [0, 3], No], [[1], [2], Yes], [[0, 1], [0, 2, 3], No], [[2], [1], No], [[0, 2], [0, 1, 3], Yes], [[1, 2], [1, 2], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 3], No], [[1, 3], [2], Yes], [[0, 1, 3], [0, 2, 3], No], [[2, 3], [1], No], [[0, 2, 3], [0, 1, 3], Yes], [[1, 2, 3], [1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [0, 3], Yes], [[1], [2], Yes], [[0, 1], [0, 2, 3], Yes], [[2], [1], Yes], [[0, 2], [0, 1, 3], Yes], [[1, 2], [1, 2], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 3], Yes], [[1, 3], [2], Yes], [[0, 1, 3], [0, 2, 3], Yes], [[2, 3], [1], Yes], [[0, 2, 3], [0, 1, 3], Yes], [[1, 2, 3], [1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)","explanation":"First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 2, 1, 0], Yes, [[[], [], Yes], [[0], [0, 3], No], [[1], [2], Yes], [[0, 1], [0, 2, 3], No], [[2], [1], No], [[0, 2], [0, 1, 3], Yes], [[1, 2], [1, 2], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 3], No], [[1, 3], [2], Yes], [[0, 1, 3], [0, 2, 3], No], [[2, 3], [1], No], [[0, 2, 3], [0, 1, 3], Yes], [[1, 2, 3], [1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], Yes], [[0], [0, 3], Yes], [[1], [2], Yes], [[0, 1], [0, 2, 3], Yes], [[2], [1], Yes], [[0, 2], [0, 1, 3], Yes], [[1, 2], [1, 2], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 3], Yes], [[1, 3], [2], Yes], [[0, 1, 3], [0, 2, 3], Yes], [[2, 3], [1], Yes], [[0, 2, 3], [0, 1, 3], Yes], [[1, 2, 3], [1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)."},"skills":["Verify Refinement Rather than Assuming the Proposed Family Is Larger"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}