{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0001","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76a6607f3fa493e2655af980a25d920dc69f7f2e8799d79e52885662b02035c9","family_id":"qf-mod24-measurable-preimage","seed":202610242001,"semantic_template":"qf-mod24-measurable-preimage/{\"mapping\":[0,0,0,0],\"source_n\":4,\"target_event\":[0,1],\"target_n\":4}"},"fields":{"title":"The Inverse Image under a Complete Finite Total Map","stem":"Use carriers Fin(4) and Fin(4), complete total mapping=[0, 0, 0, 0] indexed by source labels, and target event=[0, 1] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.","answer":{"kind":"expression","value":"(4, 4, [0, 0, 0, 0], [0, 1], [0, 1, 2, 3])","explanation":"Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 0, 0, 0], [0, 1], [0, 1, 2, 3])."},"skills":["The Inverse Image under a Complete Finite Total 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-20261024-r02-measurable-function-0002","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb5fd24c5656f2bce9fddf69be07d7cf2a6b3cfe73e796804e7978a2c83560d2","family_id":"qf-mod24-measurable-preimage","seed":202610242002,"semantic_template":"qf-mod24-measurable-preimage/{\"mapping\":[1,0,0,0],\"source_n\":4,\"target_event\":[1,2],\"target_n\":4}"},"fields":{"title":"The Inverse Image under a Complete Finite Total Map","stem":"Use carriers Fin(4) and Fin(4), complete total mapping=[1, 0, 0, 0] indexed by source labels, and target event=[1, 2] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.","answer":{"kind":"expression","value":"(4, 4, [1, 0, 0, 0], [1, 2], [0])","explanation":"Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [1, 0, 0, 0], [1, 2], [0])."},"skills":["The Inverse Image under a Complete Finite Total 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-20261024-r02-measurable-function-0003","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a135ecfc6600f4b62dce37c9b353d29ce7007626fa48a94ee52c01ddb8c78407","family_id":"qf-mod24-measurable-preimage","seed":202610242003,"semantic_template":"qf-mod24-measurable-preimage/{\"mapping\":[2,0,0,0],\"source_n\":4,\"target_event\":[2,3],\"target_n\":4}"},"fields":{"title":"The Inverse Image under a Complete Finite Total Map","stem":"Use carriers Fin(4) and Fin(4), complete total mapping=[2, 0, 0, 0] indexed by source labels, and target event=[2, 3] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.","answer":{"kind":"expression","value":"(4, 4, [2, 0, 0, 0], [2, 3], [0])","explanation":"Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [2, 0, 0, 0], [2, 3], [0])."},"skills":["The Inverse Image under a Complete Finite Total 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-20261024-r02-measurable-function-0004","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bee0ff87e6268650936ab56d999c46d37d70483f333eb656bc82d404f882ef79","family_id":"qf-mod24-measurable-preimage","seed":202610242004,"semantic_template":"qf-mod24-measurable-preimage/{\"mapping\":[3,0,0,0],\"source_n\":4,\"target_event\":[0,3],\"target_n\":4}"},"fields":{"title":"The Inverse Image under a Complete Finite Total Map","stem":"Use carriers Fin(4) and Fin(4), complete total mapping=[3, 0, 0, 0] indexed by source labels, and target event=[0, 3] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.","answer":{"kind":"expression","value":"(4, 4, [3, 0, 0, 0], [0, 3], [0, 1, 2, 3])","explanation":"Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [3, 0, 0, 0], [0, 3], [0, 1, 2, 3])."},"skills":["The Inverse Image under a Complete Finite Total 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-20261024-r02-measurable-function-0005","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ac84e48ecb74efdc8d5af1aea223cca02f68cd65a9a4a7e858c7dc196c5cb1d","family_id":"qf-mod24-measurable-preimage","seed":202610242005,"semantic_template":"qf-mod24-measurable-preimage/{\"mapping\":[0,1,0,0],\"source_n\":4,\"target_event\":[0,1],\"target_n\":4}"},"fields":{"title":"The Inverse Image under a Complete Finite Total Map","stem":"Use carriers Fin(4) and Fin(4), complete total mapping=[0, 1, 0, 0] indexed by source labels, and target event=[0, 1] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.","answer":{"kind":"expression","value":"(4, 4, [0, 1, 0, 0], [0, 1], [0, 1, 2, 3])","explanation":"Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 1, 0, 0], [0, 1], [0, 1, 2, 3])."},"skills":["The Inverse Image under a Complete Finite Total 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-20261024-r02-measurable-function-0006","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ba13a869693f7145bf4b0f03bf8ed7ab1850fa2284104ccb81ea888c9cd215d","family_id":"qf-mod24-measurable-preimage","seed":202610242006,"semantic_template":"qf-mod24-measurable-preimage/{\"mapping\":[1,1,0,0],\"source_n\":4,\"target_event\":[1,2],\"target_n\":4}"},"fields":{"title":"The Inverse Image under a Complete Finite Total Map","stem":"Use carriers Fin(4) and Fin(4), complete total mapping=[1, 1, 0, 0] indexed by source labels, and target event=[1, 2] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.","answer":{"kind":"expression","value":"(4, 4, [1, 1, 0, 0], [1, 2], [0, 1])","explanation":"Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [1, 1, 0, 0], [1, 2], [0, 1])."},"skills":["The Inverse Image under a Complete Finite Total 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-20261024-r02-measurable-function-0007","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c655f6a6cb3906e7aa67a4ea8d8fa2b01bac7eb4f28836a0b0a15421339e6a3","family_id":"qf-mod24-measurable-preimage","seed":202610242007,"semantic_template":"qf-mod24-measurable-preimage/{\"mapping\":[2,1,0,0],\"source_n\":4,\"target_event\":[2,3],\"target_n\":4}"},"fields":{"title":"The Inverse Image under a Complete Finite Total Map","stem":"Use carriers Fin(4) and Fin(4), complete total mapping=[2, 1, 0, 0] indexed by source labels, and target event=[2, 3] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.","answer":{"kind":"expression","value":"(4, 4, [2, 1, 0, 0], [2, 3], [0])","explanation":"Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [2, 1, 0, 0], [2, 3], [0])."},"skills":["The Inverse Image under a Complete Finite Total 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-20261024-r02-measurable-function-0008","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9cf32a953a4d0d5bb6b8f8143da38b6264bfc896aae5d79d6079fa194f833335","family_id":"qf-mod24-measurable-preimage","seed":202610242008,"semantic_template":"qf-mod24-measurable-preimage/{\"mapping\":[3,1,0,0],\"source_n\":4,\"target_event\":[0,3],\"target_n\":4}"},"fields":{"title":"The Inverse Image under a Complete Finite Total Map","stem":"Use carriers Fin(4) and Fin(4), complete total mapping=[3, 1, 0, 0] indexed by source labels, and target event=[0, 3] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.","answer":{"kind":"expression","value":"(4, 4, [3, 1, 0, 0], [0, 3], [0, 2, 3])","explanation":"Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [3, 1, 0, 0], [0, 3], [0, 2, 3])."},"skills":["The Inverse Image under a Complete Finite Total 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-20261024-r02-measurable-function-0009","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4fc7f4575964d06c69bc19afc637aa30ad78ff17a1b399f3a92a9ee019842307","family_id":"qf-mod24-measurable-preimage","seed":202610242009,"semantic_template":"qf-mod24-measurable-preimage/{\"mapping\":[0,2,0,0],\"source_n\":4,\"target_event\":[0,1],\"target_n\":4}"},"fields":{"title":"The Inverse Image under a Complete Finite Total Map","stem":"Use carriers Fin(4) and Fin(4), complete total mapping=[0, 2, 0, 0] indexed by source labels, and target event=[0, 1] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.","answer":{"kind":"expression","value":"(4, 4, [0, 2, 0, 0], [0, 1], [0, 2, 3])","explanation":"Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 2, 0, 0], [0, 1], [0, 2, 3])."},"skills":["The Inverse Image under a Complete Finite Total 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-20261024-r02-measurable-function-0010","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cce495a3261a7400f3d58b00249c3b40599349581d9d3c5ec204db124d9f5042","family_id":"qf-mod24-measurable-preimage","seed":202610242010,"semantic_template":"qf-mod24-measurable-preimage/{\"mapping\":[1,2,0,0],\"source_n\":4,\"target_event\":[1,2],\"target_n\":4}"},"fields":{"title":"The Inverse Image under a Complete Finite Total Map","stem":"Use carriers Fin(4) and Fin(4), complete total mapping=[1, 2, 0, 0] indexed by source labels, and target event=[1, 2] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.","answer":{"kind":"expression","value":"(4, 4, [1, 2, 0, 0], [1, 2], [0, 1])","explanation":"Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [1, 2, 0, 0], [1, 2], [0, 1])."},"skills":["The Inverse Image under a Complete Finite Total 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-20261024-r02-measurable-function-0011","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad966e94a3ccd0f4cf3b6a7e34e1d9fbd18a5b96f352f14374b07a7f7480c015","family_id":"qf-mod24-measurable-preimage","seed":202610242011,"semantic_template":"qf-mod24-measurable-preimage/{\"mapping\":[2,2,0,0],\"source_n\":4,\"target_event\":[2,3],\"target_n\":4}"},"fields":{"title":"The Inverse Image under a Complete Finite Total Map","stem":"Use carriers Fin(4) and Fin(4), complete total mapping=[2, 2, 0, 0] indexed by source labels, and target event=[2, 3] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.","answer":{"kind":"expression","value":"(4, 4, [2, 2, 0, 0], [2, 3], [0, 1])","explanation":"Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [2, 2, 0, 0], [2, 3], [0, 1])."},"skills":["The Inverse Image under a Complete Finite Total 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-20261024-r02-measurable-function-0012","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc004d32957b4f08bf7f7e4b18f93eb077f918bee669efedadbbed55e37f42f6","family_id":"qf-mod24-measurable-preimage","seed":202610242012,"semantic_template":"qf-mod24-measurable-preimage/{\"mapping\":[3,2,0,0],\"source_n\":4,\"target_event\":[0,3],\"target_n\":4}"},"fields":{"title":"The Inverse Image under a Complete Finite Total Map","stem":"Use carriers Fin(4) and Fin(4), complete total mapping=[3, 2, 0, 0] indexed by source labels, and target event=[0, 3] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.","answer":{"kind":"expression","value":"(4, 4, [3, 2, 0, 0], [0, 3], [0, 2, 3])","explanation":"Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [3, 2, 0, 0], [0, 3], [0, 2, 3])."},"skills":["The Inverse Image under a Complete Finite Total 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-20261024-r02-measurable-function-0013","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3614f29ad647a94a918424fb0fd1a87ec09bc19991402ee318e777be00205f11","family_id":"qf-mod24-measurable-preimage","seed":202610242013,"semantic_template":"qf-mod24-measurable-preimage/{\"mapping\":[0,3,0,0],\"source_n\":4,\"target_event\":[0,1],\"target_n\":4}"},"fields":{"title":"The Inverse Image under a Complete Finite Total Map","stem":"Use carriers Fin(4) and Fin(4), complete total mapping=[0, 3, 0, 0] indexed by source labels, and target event=[0, 1] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.","answer":{"kind":"expression","value":"(4, 4, [0, 3, 0, 0], [0, 1], [0, 2, 3])","explanation":"Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 3, 0, 0], [0, 1], [0, 2, 3])."},"skills":["The Inverse Image under a Complete Finite Total 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-20261024-r02-measurable-function-0014","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2b0b21f8b9eb7956a80d71d3edd336608c6c6d0ad7a5156178dcd38ff8adc4f","family_id":"qf-mod24-measurable-preimage","seed":202610242014,"semantic_template":"qf-mod24-measurable-preimage/{\"mapping\":[1,3,0,0],\"source_n\":4,\"target_event\":[1,2],\"target_n\":4}"},"fields":{"title":"The Inverse Image under a Complete Finite Total Map","stem":"Use carriers Fin(4) and Fin(4), complete total mapping=[1, 3, 0, 0] indexed by source labels, and target event=[1, 2] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.","answer":{"kind":"expression","value":"(4, 4, [1, 3, 0, 0], [1, 2], [0])","explanation":"Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [1, 3, 0, 0], [1, 2], [0])."},"skills":["The Inverse Image under a Complete Finite Total 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-20261024-r02-measurable-function-0015","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4bb7fdbfc8cab9c8a525e96d5fb19db714c498b5a9ba262f149fecab19f426d7","family_id":"qf-mod24-measurable-preimage","seed":202610242015,"semantic_template":"qf-mod24-measurable-preimage/{\"mapping\":[2,3,0,0],\"source_n\":4,\"target_event\":[2,3],\"target_n\":4}"},"fields":{"title":"The Inverse Image under a Complete Finite Total Map","stem":"Use carriers Fin(4) and Fin(4), complete total mapping=[2, 3, 0, 0] indexed by source labels, and target event=[2, 3] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.","answer":{"kind":"expression","value":"(4, 4, [2, 3, 0, 0], [2, 3], [0, 1])","explanation":"Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [2, 3, 0, 0], [2, 3], [0, 1])."},"skills":["The Inverse Image under a Complete Finite Total 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-20261024-r02-measurable-function-0016","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5e1edc26d403b49542d073d193595657b4a8035d953d796ed1be96c182b2a02","family_id":"qf-mod24-measurable-preimage","seed":202610242016,"semantic_template":"qf-mod24-measurable-preimage/{\"mapping\":[3,3,0,0],\"source_n\":4,\"target_event\":[0,3],\"target_n\":4}"},"fields":{"title":"The Inverse Image under a Complete Finite Total Map","stem":"Use carriers Fin(4) and Fin(4), complete total mapping=[3, 3, 0, 0] indexed by source labels, and target event=[0, 3] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.","answer":{"kind":"expression","value":"(4, 4, [3, 3, 0, 0], [0, 3], [0, 1, 2, 3])","explanation":"Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [3, 3, 0, 0], [0, 3], [0, 1, 2, 3])."},"skills":["The Inverse Image under a Complete Finite Total 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-20261024-r02-measurable-function-0017","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d77ca4974bfd0e0ccf30fbd8fef818ba7289bed51d258ab1c5805fc1826e71bd","family_id":"qf-mod24-measurable-preimage","seed":202610242017,"semantic_template":"qf-mod24-measurable-preimage/{\"mapping\":[0,0,1,0],\"source_n\":4,\"target_event\":[0,1],\"target_n\":4}"},"fields":{"title":"The Inverse Image under a Complete Finite Total Map","stem":"Use carriers Fin(4) and Fin(4), complete total mapping=[0, 0, 1, 0] indexed by source labels, and target event=[0, 1] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.","answer":{"kind":"expression","value":"(4, 4, [0, 0, 1, 0], [0, 1], [0, 1, 2, 3])","explanation":"Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 0, 1, 0], [0, 1], [0, 1, 2, 3])."},"skills":["The Inverse Image under a Complete Finite Total 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-20261024-r02-measurable-function-0018","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65bdfea042bcf8b0f68014e9baae5a915ac0b635759c7b4dafba8603f97eb1cd","family_id":"qf-mod24-measurable-preimage","seed":202610242018,"semantic_template":"qf-mod24-measurable-preimage/{\"mapping\":[1,0,1,0],\"source_n\":4,\"target_event\":[1,2],\"target_n\":4}"},"fields":{"title":"The Inverse Image under a Complete Finite Total Map","stem":"Use carriers Fin(4) and Fin(4), complete total mapping=[1, 0, 1, 0] indexed by source labels, and target event=[1, 2] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.","answer":{"kind":"expression","value":"(4, 4, [1, 0, 1, 0], [1, 2], [0, 2])","explanation":"Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [1, 0, 1, 0], [1, 2], [0, 2])."},"skills":["The Inverse Image under a Complete Finite Total 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-20261024-r02-measurable-function-0019","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f87fa7e86c30e5a6fcb49213fe4f78158ac70e8d4c96b817b24558857b11a8fd","family_id":"qf-mod24-measurable-preimage","seed":202610242019,"semantic_template":"qf-mod24-measurable-preimage/{\"mapping\":[2,0,1,0],\"source_n\":4,\"target_event\":[2,3],\"target_n\":4}"},"fields":{"title":"The Inverse Image under a Complete Finite Total Map","stem":"Use carriers Fin(4) and Fin(4), complete total mapping=[2, 0, 1, 0] indexed by source labels, and target event=[2, 3] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.","answer":{"kind":"expression","value":"(4, 4, [2, 0, 1, 0], [2, 3], [0])","explanation":"Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [2, 0, 1, 0], [2, 3], [0])."},"skills":["The Inverse Image under a Complete Finite Total 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-20261024-r02-measurable-function-0020","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fda50702fb285d088abf1926470dac8ebb6567425d029b8dbf0b108a2659096b","family_id":"qf-mod24-measurable-preimage","seed":202610242020,"semantic_template":"qf-mod24-measurable-preimage/{\"mapping\":[3,0,1,0],\"source_n\":4,\"target_event\":[0,3],\"target_n\":4}"},"fields":{"title":"The Inverse Image under a Complete Finite Total Map","stem":"Use carriers Fin(4) and Fin(4), complete total mapping=[3, 0, 1, 0] indexed by source labels, and target event=[0, 3] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.","answer":{"kind":"expression","value":"(4, 4, [3, 0, 1, 0], [0, 3], [0, 1, 3])","explanation":"Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [3, 0, 1, 0], [0, 3], [0, 1, 3])."},"skills":["The Inverse Image under a Complete Finite Total 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-20261024-r02-measurable-function-0021","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3251e82d4a95f90d0a2b0856cd9fa9cf3cf4de7e875f9e5dc358e189acc4fcdc","family_id":"qf-mod24-measurable-preimage","seed":202610242021,"semantic_template":"qf-mod24-measurable-preimage/{\"mapping\":[0,1,1,0],\"source_n\":4,\"target_event\":[0,1],\"target_n\":4}"},"fields":{"title":"The Inverse Image under a Complete Finite Total Map","stem":"Use carriers Fin(4) and Fin(4), complete total mapping=[0, 1, 1, 0] indexed by source labels, and target event=[0, 1] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.","answer":{"kind":"expression","value":"(4, 4, [0, 1, 1, 0], [0, 1], [0, 1, 2, 3])","explanation":"Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 1, 1, 0], [0, 1], [0, 1, 2, 3])."},"skills":["The Inverse Image under a Complete Finite Total 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-20261024-r02-measurable-function-0022","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2adf52f1ab9403f0695ef2771bad566faf5cdb8004b0d5886b79694febc78dd5","family_id":"qf-mod24-measurable-preimage","seed":202610242022,"semantic_template":"qf-mod24-measurable-preimage/{\"mapping\":[1,1,1,0],\"source_n\":4,\"target_event\":[1,2],\"target_n\":4}"},"fields":{"title":"The Inverse Image under a Complete Finite Total Map","stem":"Use carriers Fin(4) and Fin(4), complete total mapping=[1, 1, 1, 0] indexed by source labels, and target event=[1, 2] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.","answer":{"kind":"expression","value":"(4, 4, [1, 1, 1, 0], [1, 2], [0, 1, 2])","explanation":"Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [1, 1, 1, 0], [1, 2], [0, 1, 2])."},"skills":["The Inverse Image under a Complete Finite Total 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-20261024-r02-measurable-function-0023","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16163bf20d8c714eca113f212892a02d57730dee4bb85140692981d46f4093f4","family_id":"qf-mod24-measurable-preimage","seed":202610242023,"semantic_template":"qf-mod24-measurable-preimage/{\"mapping\":[2,1,1,0],\"source_n\":4,\"target_event\":[2,3],\"target_n\":4}"},"fields":{"title":"The Inverse Image under a Complete Finite Total Map","stem":"Use carriers Fin(4) and Fin(4), complete total mapping=[2, 1, 1, 0] indexed by source labels, and target event=[2, 3] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.","answer":{"kind":"expression","value":"(4, 4, [2, 1, 1, 0], [2, 3], [0])","explanation":"Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [2, 1, 1, 0], [2, 3], [0])."},"skills":["The Inverse Image under a Complete Finite Total 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-20261024-r02-measurable-function-0024","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21419072ee1bb1890aae26dbf2e2c992059c4943168e1119735a9d58fa69fc6a","family_id":"qf-mod24-measurable-preimage","seed":202610242024,"semantic_template":"qf-mod24-measurable-preimage/{\"mapping\":[3,1,1,0],\"source_n\":4,\"target_event\":[0,3],\"target_n\":4}"},"fields":{"title":"The Inverse Image under a Complete Finite Total Map","stem":"Use carriers Fin(4) and Fin(4), complete total mapping=[3, 1, 1, 0] indexed by source labels, and target event=[0, 3] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.","answer":{"kind":"expression","value":"(4, 4, [3, 1, 1, 0], [0, 3], [0, 3])","explanation":"Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [3, 1, 1, 0], [0, 3], [0, 3])."},"skills":["The Inverse Image under a Complete Finite Total 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-20261024-r02-measurable-function-0025","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"853eb3ed645a32af89621dfc81208d04154bd66ac6bc952e18bafc5b1467057a","family_id":"qf-mod24-measurable-preimage","seed":202610242025,"semantic_template":"qf-mod24-measurable-preimage/{\"mapping\":[0,2,1,0],\"source_n\":4,\"target_event\":[0,1],\"target_n\":4}"},"fields":{"title":"The Inverse Image under a Complete Finite Total Map","stem":"Use carriers Fin(4) and Fin(4), complete total mapping=[0, 2, 1, 0] indexed by source labels, and target event=[0, 1] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.","answer":{"kind":"expression","value":"(4, 4, [0, 2, 1, 0], [0, 1], [0, 2, 3])","explanation":"Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 2, 1, 0], [0, 1], [0, 2, 3])."},"skills":["The Inverse Image under a Complete Finite Total 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-20261024-r02-measurable-function-0026","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b277ff6db5806f8085a3b710fdf9d047d11be2dda186a65e865d51ff2e85fa45","family_id":"qf-mod24-measurable-map-audit","seed":202610242026,"semantic_template":"qf-mod24-measurable-map-audit/{\"mapping\":[0,0,0,0],\"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":"Audit Every Target Measurable Set by Preimage","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0], [1, 2, 3], [0, 1, 2, 3]], target size=4, 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 (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).","answer":{"kind":"expression","value":"(4, 4, [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":"Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [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":["Audit Every Target Measurable Set by 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-20261024-r02-measurable-function-0027","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b15c2f52762e447ebe3e5298b1c253197156a5ad1056efa71321f2ebb9851537","family_id":"qf-mod24-measurable-map-audit","seed":202610242027,"semantic_template":"qf-mod24-measurable-map-audit/{\"mapping\":[1,0,0,0],\"source_n\":4,\"source_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]],\"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":"Audit Every Target Measurable Set by Preimage","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source 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 size=4, 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 (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).","answer":{"kind":"expression","value":"(4, 4, [1, 0, 0, 0], [[[], [], 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":"Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [1, 0, 0, 0], [[[], [], 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":["Audit Every Target Measurable Set by 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-20261024-r02-measurable-function-0028","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e7e297603ba9b52702d7f0dcb6ffb27de78acf12ec0244aca78de921193c250","family_id":"qf-mod24-measurable-map-audit","seed":202610242028,"semantic_template":"qf-mod24-measurable-map-audit/{\"mapping\":[2,0,0,0],\"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":"Audit Every Target Measurable Set by Preimage","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0, 1], [2, 3], [0, 1, 2, 3]], target size=4, 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 (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).","answer":{"kind":"expression","value":"(4, 4, [2, 0, 0, 0], [[[], [], 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)","explanation":"Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [2, 0, 0, 0], [[[], [], 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)."},"skills":["Audit Every Target Measurable Set by 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-20261024-r02-measurable-function-0029","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1272bcf4c35ad67f5513c8b44181b86d968b88d808b71d0bcc64e64a4ccfc35b","family_id":"qf-mod24-measurable-map-audit","seed":202610242029,"semantic_template":"qf-mod24-measurable-map-audit/{\"mapping\":[3,0,0,0],\"source_n\":4,\"source_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]],\"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":"Audit Every Target Measurable Set by Preimage","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source 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 size=4, 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 (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).","answer":{"kind":"expression","value":"(4, 4, [3, 0, 0, 0], [[[], [], 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":"Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [3, 0, 0, 0], [[[], [], 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":["Audit Every Target Measurable Set by 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-20261024-r02-measurable-function-0030","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6690e8157d2be5d4b1683f86707be0e354543bbcb95cd38a12d5150beab6811c","family_id":"qf-mod24-measurable-map-audit","seed":202610242030,"semantic_template":"qf-mod24-measurable-map-audit/{\"mapping\":[0,1,0,0],\"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":"Audit Every Target Measurable Set by Preimage","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0, 2], [1, 3], [0, 1, 2, 3]], target size=4, 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 (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).","answer":{"kind":"expression","value":"(4, 4, [0, 1, 0, 0], [[[], [], 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)","explanation":"Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 1, 0, 0], [[[], [], 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)."},"skills":["Audit Every Target Measurable Set by 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-20261024-r02-measurable-function-0031","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03f882b427cd24a8bfdc2480f77aa65e6c517fbf3efbbc314faffe6517280d84","family_id":"qf-mod24-measurable-map-audit","seed":202610242031,"semantic_template":"qf-mod24-measurable-map-audit/{\"mapping\":[1,1,0,0],\"source_n\":4,\"source_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]],\"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":"Audit Every Target Measurable Set by Preimage","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source 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 size=4, 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 (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).","answer":{"kind":"expression","value":"(4, 4, [1, 1, 0, 0], [[[], [], 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":"Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [1, 1, 0, 0], [[[], [], 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":["Audit Every Target Measurable Set by 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-20261024-r02-measurable-function-0032","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d37cdbfed3e99271d5c8d5924111cc54b9a31e3fa094c889630c6088b5d41c09","family_id":"qf-mod24-measurable-map-audit","seed":202610242032,"semantic_template":"qf-mod24-measurable-map-audit/{\"mapping\":[2,1,0,0],\"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":"Audit Every Target Measurable Set by Preimage","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0, 1, 2], [3], [0, 1, 2, 3]], target size=4, 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 (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).","answer":{"kind":"expression","value":"(4, 4, [2, 1, 0, 0], [[[], [], 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)","explanation":"Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [2, 1, 0, 0], [[[], [], 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)."},"skills":["Audit Every Target Measurable Set by 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-20261024-r02-measurable-function-0033","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fbdccedac7519010888e959600ace8d9ff4ce4d0b8727aea96e977f536112335","family_id":"qf-mod24-measurable-map-audit","seed":202610242033,"semantic_template":"qf-mod24-measurable-map-audit/{\"mapping\":[3,1,0,0],\"source_n\":4,\"source_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]],\"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":"Audit Every Target Measurable Set by Preimage","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source 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 size=4, 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 (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).","answer":{"kind":"expression","value":"(4, 4, [3, 1, 0, 0], [[[], [], 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":"Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [3, 1, 0, 0], [[[], [], 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":["Audit Every Target Measurable Set by 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-20261024-r02-measurable-function-0034","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b489eeff1abc695349090a232c252dc9c401d519ff5ffd11f7d4496e1499e08d","family_id":"qf-mod24-measurable-map-audit","seed":202610242034,"semantic_template":"qf-mod24-measurable-map-audit/{\"mapping\":[0,2,0,0],\"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":"Audit Every Target Measurable Set by Preimage","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [1], [0, 2, 3], [0, 1, 2, 3]], target size=4, 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 (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).","answer":{"kind":"expression","value":"(4, 4, [0, 2, 0, 0], [[[], [], 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":"Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 2, 0, 0], [[[], [], 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":["Audit Every Target Measurable Set by 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-20261024-r02-measurable-function-0035","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"305ed7be2bcfc93cc3256cd06af1cf7fb5a74df8ca45ae8f70e563cbdc9d6a06","family_id":"qf-mod24-measurable-map-audit","seed":202610242035,"semantic_template":"qf-mod24-measurable-map-audit/{\"mapping\":[1,2,0,0],\"source_n\":4,\"source_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]],\"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":"Audit Every Target Measurable Set by Preimage","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source 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 size=4, 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 (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).","answer":{"kind":"expression","value":"(4, 4, [1, 2, 0, 0], [[[], [], 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":"Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [1, 2, 0, 0], [[[], [], 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":["Audit Every Target Measurable Set by 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-20261024-r02-measurable-function-0036","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"047b999dd6d38b0cc26f366d4a5fb711a3b8402e8526876c6bdd3ddcc55e636e","family_id":"qf-mod24-measurable-map-audit","seed":202610242036,"semantic_template":"qf-mod24-measurable-map-audit/{\"mapping\":[2,2,0,0],\"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":"Audit Every Target Measurable Set by Preimage","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [2], [0, 1, 3], [0, 1, 2, 3]], target size=4, 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 (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).","answer":{"kind":"expression","value":"(4, 4, [2, 2, 0, 0], [[[], [], 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)","explanation":"Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [2, 2, 0, 0], [[[], [], 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)."},"skills":["Audit Every Target Measurable Set by 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-20261024-r02-measurable-function-0037","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2504973d75d70ce93a999462f59e4f63209789fbac76c13133042af959ea3854","family_id":"qf-mod24-measurable-map-audit","seed":202610242037,"semantic_template":"qf-mod24-measurable-map-audit/{\"mapping\":[3,2,0,0],\"source_n\":4,\"source_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]],\"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":"Audit Every Target Measurable Set by Preimage","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source 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 size=4, 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 (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).","answer":{"kind":"expression","value":"(4, 4, [3, 2, 0, 0], [[[], [], 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":"Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [3, 2, 0, 0], [[[], [], 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":["Audit Every Target Measurable Set by 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-20261024-r02-measurable-function-0038","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18b7e951fbb52b0d45d4995998b7223e27d5f501dcd6a4209f6e20da4c0edf79","family_id":"qf-mod24-measurable-map-audit","seed":202610242038,"semantic_template":"qf-mod24-measurable-map-audit/{\"mapping\":[0,3,0,0],\"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":"Audit Every Target Measurable Set by Preimage","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [1, 2], [0, 3], [0, 1, 2, 3]], target size=4, 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 (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).","answer":{"kind":"expression","value":"(4, 4, [0, 3, 0, 0], [[[], [], 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)","explanation":"Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 3, 0, 0], [[[], [], 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)."},"skills":["Audit Every Target Measurable Set by 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-20261024-r02-measurable-function-0039","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"691a880d83476ad24785e6bc91953bfbd6f7e66256b0cf9b4a2bf610b7fc6da3","family_id":"qf-mod24-measurable-map-audit","seed":202610242039,"semantic_template":"qf-mod24-measurable-map-audit/{\"mapping\":[1,3,0,0],\"source_n\":4,\"source_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]],\"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":"Audit Every Target Measurable Set by Preimage","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source 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 size=4, 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 (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).","answer":{"kind":"expression","value":"(4, 4, [1, 3, 0, 0], [[[], [], 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":"Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [1, 3, 0, 0], [[[], [], 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":["Audit Every Target Measurable Set by 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-20261024-r02-measurable-function-0040","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"acb696b3b0bd75e183550644ace157399a06c6090d7a270ba242064dc4a59ddc","family_id":"qf-mod24-measurable-map-audit","seed":202610242040,"semantic_template":"qf-mod24-measurable-map-audit/{\"mapping\":[2,3,0,0],\"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":"Audit Every Target Measurable Set by Preimage","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0], [1, 2, 3], [0, 1, 2, 3]], target size=4, 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 (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).","answer":{"kind":"expression","value":"(4, 4, [2, 3, 0, 0], [[[], [], 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)","explanation":"Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [2, 3, 0, 0], [[[], [], 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)."},"skills":["Audit Every Target Measurable Set by 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-20261024-r02-measurable-function-0041","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07f80dceee4778d01462bf76a3d30cb4ffb77c0da78a9224b4cad55be39cac15","family_id":"qf-mod24-measurable-map-audit","seed":202610242041,"semantic_template":"qf-mod24-measurable-map-audit/{\"mapping\":[3,3,0,0],\"source_n\":4,\"source_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]],\"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":"Audit Every Target Measurable Set by Preimage","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source 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 size=4, 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 (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).","answer":{"kind":"expression","value":"(4, 4, [3, 3, 0, 0], [[[], [], 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":"Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [3, 3, 0, 0], [[[], [], 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":["Audit Every Target Measurable Set by 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-20261024-r02-measurable-function-0042","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"403a6f81e507a1e6dbe105e3df5bfa40002b1617fe793b72e036b6caedcd6c9b","family_id":"qf-mod24-measurable-map-audit","seed":202610242042,"semantic_template":"qf-mod24-measurable-map-audit/{\"mapping\":[0,0,1,0],\"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":"Audit Every Target Measurable Set by Preimage","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0, 1], [2, 3], [0, 1, 2, 3]], target size=4, 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 (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).","answer":{"kind":"expression","value":"(4, 4, [0, 0, 1, 0], [[[], [], 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)","explanation":"Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 0, 1, 0], [[[], [], 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)."},"skills":["Audit Every Target Measurable Set by 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-20261024-r02-measurable-function-0043","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dba8f45c48c8aaed723cd5aa5aa74b5482385bdbeb6808d26abe86f45c06dac5","family_id":"qf-mod24-measurable-map-audit","seed":202610242043,"semantic_template":"qf-mod24-measurable-map-audit/{\"mapping\":[1,0,1,0],\"source_n\":4,\"source_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]],\"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":"Audit Every Target Measurable Set by Preimage","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source 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 size=4, 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 (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).","answer":{"kind":"expression","value":"(4, 4, [1, 0, 1, 0], [[[], [], 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":"Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [1, 0, 1, 0], [[[], [], 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":["Audit Every Target Measurable Set by 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-20261024-r02-measurable-function-0044","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5389279706bf636157546734827da2d0ec87130d7c5079071ca280f56b199e26","family_id":"qf-mod24-measurable-map-audit","seed":202610242044,"semantic_template":"qf-mod24-measurable-map-audit/{\"mapping\":[2,0,1,0],\"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":"Audit Every Target Measurable Set by Preimage","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0, 2], [1, 3], [0, 1, 2, 3]], target size=4, 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 (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).","answer":{"kind":"expression","value":"(4, 4, [2, 0, 1, 0], [[[], [], 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)","explanation":"Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [2, 0, 1, 0], [[[], [], 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)."},"skills":["Audit Every Target Measurable Set by 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-20261024-r02-measurable-function-0045","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db19669c80d9ce0636737e3448b885391a1441e766f96ed12072a5f945722127","family_id":"qf-mod24-measurable-map-audit","seed":202610242045,"semantic_template":"qf-mod24-measurable-map-audit/{\"mapping\":[3,0,1,0],\"source_n\":4,\"source_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]],\"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":"Audit Every Target Measurable Set by Preimage","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source 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 size=4, 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 (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).","answer":{"kind":"expression","value":"(4, 4, [3, 0, 1, 0], [[[], [], 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":"Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [3, 0, 1, 0], [[[], [], 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":["Audit Every Target Measurable Set by 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-20261024-r02-measurable-function-0046","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"762faa2fb3c7d8edcdfc5d8cf4e1c6e5a8b8236ed527f7b2fa8681d11a1b6566","family_id":"qf-mod24-measurable-map-audit","seed":202610242046,"semantic_template":"qf-mod24-measurable-map-audit/{\"mapping\":[0,1,1,0],\"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":"Audit Every Target Measurable Set by Preimage","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0, 1, 2], [3], [0, 1, 2, 3]], target size=4, 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 (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).","answer":{"kind":"expression","value":"(4, 4, [0, 1, 1, 0], [[[], [], 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)","explanation":"Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 1, 1, 0], [[[], [], 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)."},"skills":["Audit Every Target Measurable Set by 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-20261024-r02-measurable-function-0047","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb1b15fb5aaca3bfdf33059646358f56d375533e3079fed0b28266ff4141afc3","family_id":"qf-mod24-measurable-map-audit","seed":202610242047,"semantic_template":"qf-mod24-measurable-map-audit/{\"mapping\":[1,1,1,0],\"source_n\":4,\"source_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]],\"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":"Audit Every Target Measurable Set by Preimage","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source 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 size=4, 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 (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).","answer":{"kind":"expression","value":"(4, 4, [1, 1, 1, 0], [[[], [], 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":"Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [1, 1, 1, 0], [[[], [], 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":["Audit Every Target Measurable Set by 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-20261024-r02-measurable-function-0048","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5fe351da30720f3434fa94e3a4fc54dcf1e1ab137b5b28a8b3c2421b971fc4c3","family_id":"qf-mod24-measurable-map-audit","seed":202610242048,"semantic_template":"qf-mod24-measurable-map-audit/{\"mapping\":[2,1,1,0],\"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":"Audit Every Target Measurable Set by Preimage","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [1], [0, 2, 3], [0, 1, 2, 3]], target size=4, 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 (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).","answer":{"kind":"expression","value":"(4, 4, [2, 1, 1, 0], [[[], [], 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)","explanation":"Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [2, 1, 1, 0], [[[], [], 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)."},"skills":["Audit Every Target Measurable Set by 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-20261024-r02-measurable-function-0049","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"522ee9ac6ec4a6748fcd04779d074f544777afb8a8842d27ef0967c7ba5b0968","family_id":"qf-mod24-measurable-map-audit","seed":202610242049,"semantic_template":"qf-mod24-measurable-map-audit/{\"mapping\":[3,1,1,0],\"source_n\":4,\"source_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]],\"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":"Audit Every Target Measurable Set by Preimage","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source 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 size=4, 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 (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).","answer":{"kind":"expression","value":"(4, 4, [3, 1, 1, 0], [[[], [], 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":"Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [3, 1, 1, 0], [[[], [], 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":["Audit Every Target Measurable Set by 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-20261024-r02-measurable-function-0050","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28026aa8ecc7fe9f95d71599bb5bd42d95e49456743d86327d6e9e6ef14454f4","family_id":"qf-mod24-measurable-map-audit","seed":202610242050,"semantic_template":"qf-mod24-measurable-map-audit/{\"mapping\":[0,2,1,0],\"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":"Audit Every Target Measurable Set by Preimage","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [2], [0, 1, 3], [0, 1, 2, 3]], target size=4, 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 (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).","answer":{"kind":"expression","value":"(4, 4, [0, 2, 1, 0], [[[], [], 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)","explanation":"Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 2, 1, 0], [[[], [], 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)."},"skills":["Audit Every Target Measurable Set by 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-20261024-r02-measurable-function-0051","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"518d5da4942cafbcfb5a48069c57c671c73a9fba59a45c71739dba7e27de018b","family_id":"qf-mod24-measurable-indicator-audit","seed":202610242051,"semantic_template":"qf-mod24-measurable-indicator-audit/{\"event\":[0],\"source_n\":6,\"source_sigma\":[[],[0,1,2,3,4,5],[0],[1,2,3,4,5]]}"},"fields":{"title":"An Indicator’s Measurability Depends on Its Event","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0], [1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5]], event=[0] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).","answer":{"kind":"expression","value":"(6, [0], [1, 0, 0, 0, 0, 0], Yes)","explanation":"The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0], [1, 0, 0, 0, 0, 0], Yes)."},"skills":["An Indicator’s Measurability Depends on Its Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0052","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bfe26678ded5131ab83f27071397462f463b6dbc8152de12f7ee0bcc8d5e53bb","family_id":"qf-mod24-measurable-indicator-audit","seed":202610242052,"semantic_template":"qf-mod24-measurable-indicator-audit/{\"event\":[1],\"source_n\":6,\"source_sigma\":[[],[0,1,2,3,4,5],[0,2,3,4,5],[1]]}"},"fields":{"title":"An Indicator’s Measurability Depends on Its Event","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [1], [0, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5]], event=[1] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).","answer":{"kind":"expression","value":"(6, [1], [0, 1, 0, 0, 0, 0], Yes)","explanation":"The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [1], [0, 1, 0, 0, 0, 0], Yes)."},"skills":["An Indicator’s Measurability Depends on Its Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0053","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b597f839c879e074ec790da8fd9756cbf54835ae8db50942a1ec12e844040fa","family_id":"qf-mod24-measurable-indicator-audit","seed":202610242053,"semantic_template":"qf-mod24-measurable-indicator-audit/{\"event\":[0,1],\"source_n\":6,\"source_sigma\":[[],[0,1,2,3,4,5],[0,1],[2,3,4,5]]}"},"fields":{"title":"An Indicator’s Measurability Depends on Its Event","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0, 1], [2, 3, 4, 5], [0, 1, 2, 3, 4, 5]], event=[0, 1] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).","answer":{"kind":"expression","value":"(6, [0, 1], [1, 1, 0, 0, 0, 0], Yes)","explanation":"The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0, 1], [1, 1, 0, 0, 0, 0], Yes)."},"skills":["An Indicator’s Measurability Depends on Its Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0054","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06d46d5dc918d2b0ff46258c452914594b160ceea1db8509da110b00ea2616b5","family_id":"qf-mod24-measurable-indicator-audit","seed":202610242054,"semantic_template":"qf-mod24-measurable-indicator-audit/{\"event\":[0],\"source_n\":6,\"source_sigma\":[[],[0,1,2,3,4,5],[0,1,3,4,5],[2]]}"},"fields":{"title":"An Indicator’s Measurability Depends on Its Event","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [2], [0, 1, 3, 4, 5], [0, 1, 2, 3, 4, 5]], event=[0] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).","answer":{"kind":"expression","value":"(6, [0], [1, 0, 0, 0, 0, 0], No)","explanation":"The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0], [1, 0, 0, 0, 0, 0], No)."},"skills":["An Indicator’s Measurability Depends on Its Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0055","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b2dd45299ff2db64647a32bcbc7d623001b69d173a5dd07e645ae45c08d646d","family_id":"qf-mod24-measurable-indicator-audit","seed":202610242055,"semantic_template":"qf-mod24-measurable-indicator-audit/{\"event\":[0,2],\"source_n\":6,\"source_sigma\":[[],[0,1,2,3,4,5],[0,2],[1,3,4,5]]}"},"fields":{"title":"An Indicator’s Measurability Depends on Its Event","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0, 2], [1, 3, 4, 5], [0, 1, 2, 3, 4, 5]], event=[0, 2] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).","answer":{"kind":"expression","value":"(6, [0, 2], [1, 0, 1, 0, 0, 0], Yes)","explanation":"The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0, 2], [1, 0, 1, 0, 0, 0], Yes)."},"skills":["An Indicator’s Measurability Depends on Its Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0056","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2215662a2bddd48d4d04bd3ecb1fe5998e9d656d2a081f01b5e3373fb33efa5c","family_id":"qf-mod24-measurable-indicator-audit","seed":202610242056,"semantic_template":"qf-mod24-measurable-indicator-audit/{\"event\":[1,2],\"source_n\":6,\"source_sigma\":[[],[0,1,2,3,4,5],[0,3,4,5],[1,2]]}"},"fields":{"title":"An Indicator’s Measurability Depends on Its Event","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [1, 2], [0, 3, 4, 5], [0, 1, 2, 3, 4, 5]], event=[1, 2] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).","answer":{"kind":"expression","value":"(6, [1, 2], [0, 1, 1, 0, 0, 0], Yes)","explanation":"The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [1, 2], [0, 1, 1, 0, 0, 0], Yes)."},"skills":["An Indicator’s Measurability Depends on Its Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0057","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7de1ab21d3b8070967093b037cdafb9ea032b4957044d45058deffa5174c407b","family_id":"qf-mod24-measurable-indicator-audit","seed":202610242057,"semantic_template":"qf-mod24-measurable-indicator-audit/{\"event\":[0],\"source_n\":6,\"source_sigma\":[[],[0,1,2,3,4,5],[0,1,2],[3,4,5]]}"},"fields":{"title":"An Indicator’s Measurability Depends on Its Event","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0, 1, 2], [3, 4, 5], [0, 1, 2, 3, 4, 5]], event=[0] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).","answer":{"kind":"expression","value":"(6, [0], [1, 0, 0, 0, 0, 0], No)","explanation":"The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0], [1, 0, 0, 0, 0, 0], No)."},"skills":["An Indicator’s Measurability Depends on Its Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0058","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e6ea84fafa0a48b46e6f37a373b216d1294c36595d0a14351a00f7f8edb66b0","family_id":"qf-mod24-measurable-indicator-audit","seed":202610242058,"semantic_template":"qf-mod24-measurable-indicator-audit/{\"event\":[3],\"source_n\":6,\"source_sigma\":[[],[0,1,2,3,4,5],[0,1,2,4,5],[3]]}"},"fields":{"title":"An Indicator’s Measurability Depends on Its Event","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [3], [0, 1, 2, 4, 5], [0, 1, 2, 3, 4, 5]], event=[3] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).","answer":{"kind":"expression","value":"(6, [3], [0, 0, 0, 1, 0, 0], Yes)","explanation":"The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [3], [0, 0, 0, 1, 0, 0], Yes)."},"skills":["An Indicator’s Measurability Depends on Its Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0059","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02e59b57b56f080c1a8e2ac623ee947fc5174fc505def9c5c060b4eca1e0fbfe","family_id":"qf-mod24-measurable-indicator-audit","seed":202610242059,"semantic_template":"qf-mod24-measurable-indicator-audit/{\"event\":[0,3],\"source_n\":6,\"source_sigma\":[[],[0,1,2,3,4,5],[0,3],[1,2,4,5]]}"},"fields":{"title":"An Indicator’s Measurability Depends on Its Event","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0, 3], [1, 2, 4, 5], [0, 1, 2, 3, 4, 5]], event=[0, 3] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).","answer":{"kind":"expression","value":"(6, [0, 3], [1, 0, 0, 1, 0, 0], Yes)","explanation":"The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0, 3], [1, 0, 0, 1, 0, 0], Yes)."},"skills":["An Indicator’s Measurability Depends on Its Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0060","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5704e6193a827bff5357584870f2d8d0052d24a9b255cd616de834d6ec513c5","family_id":"qf-mod24-measurable-indicator-audit","seed":202610242060,"semantic_template":"qf-mod24-measurable-indicator-audit/{\"event\":[0],\"source_n\":6,\"source_sigma\":[[],[0,1,2,3,4,5],[0,2,4,5],[1,3]]}"},"fields":{"title":"An Indicator’s Measurability Depends on Its Event","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [1, 3], [0, 2, 4, 5], [0, 1, 2, 3, 4, 5]], event=[0] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).","answer":{"kind":"expression","value":"(6, [0], [1, 0, 0, 0, 0, 0], No)","explanation":"The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0], [1, 0, 0, 0, 0, 0], No)."},"skills":["An Indicator’s Measurability Depends on Its Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0061","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd126dd3b9b734fe53b813d30572e6631ff3fd3a49202d47d66c271fd23f454b","family_id":"qf-mod24-measurable-indicator-audit","seed":202610242061,"semantic_template":"qf-mod24-measurable-indicator-audit/{\"event\":[0,1,3],\"source_n\":6,\"source_sigma\":[[],[0,1,2,3,4,5],[0,1,3],[2,4,5]]}"},"fields":{"title":"An Indicator’s Measurability Depends on Its Event","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0, 1, 3], [2, 4, 5], [0, 1, 2, 3, 4, 5]], event=[0, 1, 3] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).","answer":{"kind":"expression","value":"(6, [0, 1, 3], [1, 1, 0, 1, 0, 0], Yes)","explanation":"The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0, 1, 3], [1, 1, 0, 1, 0, 0], Yes)."},"skills":["An Indicator’s Measurability Depends on Its Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0062","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e95ff3c68b9f572dd95721a413aaab152ff8cb14a3a219ad64b9bca2e9938000","family_id":"qf-mod24-measurable-indicator-audit","seed":202610242062,"semantic_template":"qf-mod24-measurable-indicator-audit/{\"event\":[2,3],\"source_n\":6,\"source_sigma\":[[],[0,1,2,3,4,5],[0,1,4,5],[2,3]]}"},"fields":{"title":"An Indicator’s Measurability Depends on Its Event","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [2, 3], [0, 1, 4, 5], [0, 1, 2, 3, 4, 5]], event=[2, 3] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).","answer":{"kind":"expression","value":"(6, [2, 3], [0, 0, 1, 1, 0, 0], Yes)","explanation":"The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [2, 3], [0, 0, 1, 1, 0, 0], Yes)."},"skills":["An Indicator’s Measurability Depends on Its Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0063","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ddfe34f2335327d2b34f641bf976cd98f8ae0339022bad5e8fef99956cba828","family_id":"qf-mod24-measurable-indicator-audit","seed":202610242063,"semantic_template":"qf-mod24-measurable-indicator-audit/{\"event\":[0],\"source_n\":6,\"source_sigma\":[[],[0,1,2,3,4,5],[0,2,3],[1,4,5]]}"},"fields":{"title":"An Indicator’s Measurability Depends on Its Event","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0, 2, 3], [1, 4, 5], [0, 1, 2, 3, 4, 5]], event=[0] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).","answer":{"kind":"expression","value":"(6, [0], [1, 0, 0, 0, 0, 0], No)","explanation":"The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0], [1, 0, 0, 0, 0, 0], No)."},"skills":["An Indicator’s Measurability Depends on Its Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0064","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4df3caa78ddb6963b1d207b853596f02df916894ca6b95c44c0a7a5f38b660b","family_id":"qf-mod24-measurable-indicator-audit","seed":202610242064,"semantic_template":"qf-mod24-measurable-indicator-audit/{\"event\":[1,2,3],\"source_n\":6,\"source_sigma\":[[],[0,1,2,3,4,5],[0,4,5],[1,2,3]]}"},"fields":{"title":"An Indicator’s Measurability Depends on Its Event","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [1, 2, 3], [0, 4, 5], [0, 1, 2, 3, 4, 5]], event=[1, 2, 3] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).","answer":{"kind":"expression","value":"(6, [1, 2, 3], [0, 1, 1, 1, 0, 0], Yes)","explanation":"The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [1, 2, 3], [0, 1, 1, 1, 0, 0], Yes)."},"skills":["An Indicator’s Measurability Depends on Its Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0065","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6cbe0520ae95f877df644aa5b452d6bc64bacc0f62e1b6fcedd982dabd2610d8","family_id":"qf-mod24-measurable-indicator-audit","seed":202610242065,"semantic_template":"qf-mod24-measurable-indicator-audit/{\"event\":[0,1,2,3],\"source_n\":6,\"source_sigma\":[[],[0,1,2,3,4,5],[0,1,2,3],[4,5]]}"},"fields":{"title":"An Indicator’s Measurability Depends on Its Event","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0, 1, 2, 3], [4, 5], [0, 1, 2, 3, 4, 5]], event=[0, 1, 2, 3] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).","answer":{"kind":"expression","value":"(6, [0, 1, 2, 3], [1, 1, 1, 1, 0, 0], Yes)","explanation":"The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0, 1, 2, 3], [1, 1, 1, 1, 0, 0], Yes)."},"skills":["An Indicator’s Measurability Depends on Its Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0066","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aeb9cc972c440fe0bfc9be189ce25178dea0bbe0cd1148eedf4c6de246048ca1","family_id":"qf-mod24-measurable-indicator-audit","seed":202610242066,"semantic_template":"qf-mod24-measurable-indicator-audit/{\"event\":[0],\"source_n\":6,\"source_sigma\":[[],[0,1,2,3,4,5],[0,1,2,3,5],[4]]}"},"fields":{"title":"An Indicator’s Measurability Depends on Its Event","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [4], [0, 1, 2, 3, 5], [0, 1, 2, 3, 4, 5]], event=[0] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).","answer":{"kind":"expression","value":"(6, [0], [1, 0, 0, 0, 0, 0], No)","explanation":"The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0], [1, 0, 0, 0, 0, 0], No)."},"skills":["An Indicator’s Measurability Depends on Its Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0067","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b2735b87ae528f2e4a6b49c6aa944d2af7a2f9ce2d74192e4536e80246adf06","family_id":"qf-mod24-measurable-indicator-audit","seed":202610242067,"semantic_template":"qf-mod24-measurable-indicator-audit/{\"event\":[0,4],\"source_n\":6,\"source_sigma\":[[],[0,1,2,3,4,5],[0,4],[1,2,3,5]]}"},"fields":{"title":"An Indicator’s Measurability Depends on Its Event","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0, 4], [1, 2, 3, 5], [0, 1, 2, 3, 4, 5]], event=[0, 4] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).","answer":{"kind":"expression","value":"(6, [0, 4], [1, 0, 0, 0, 1, 0], Yes)","explanation":"The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0, 4], [1, 0, 0, 0, 1, 0], Yes)."},"skills":["An Indicator’s Measurability Depends on Its Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0068","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ce12a42d55433e5701d8e2d62d4ad9e5d950eb011a712f199f1ce6927a6314f","family_id":"qf-mod24-measurable-indicator-audit","seed":202610242068,"semantic_template":"qf-mod24-measurable-indicator-audit/{\"event\":[1,4],\"source_n\":6,\"source_sigma\":[[],[0,1,2,3,4,5],[0,2,3,5],[1,4]]}"},"fields":{"title":"An Indicator’s Measurability Depends on Its Event","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [1, 4], [0, 2, 3, 5], [0, 1, 2, 3, 4, 5]], event=[1, 4] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).","answer":{"kind":"expression","value":"(6, [1, 4], [0, 1, 0, 0, 1, 0], Yes)","explanation":"The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [1, 4], [0, 1, 0, 0, 1, 0], Yes)."},"skills":["An Indicator’s Measurability Depends on Its Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0069","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f089d9a5130cf0d8fffc588ef3119106ada0516049a032f7b47893ce3fe0107","family_id":"qf-mod24-measurable-indicator-audit","seed":202610242069,"semantic_template":"qf-mod24-measurable-indicator-audit/{\"event\":[0],\"source_n\":6,\"source_sigma\":[[],[0,1,2,3,4,5],[0,1,4],[2,3,5]]}"},"fields":{"title":"An Indicator’s Measurability Depends on Its Event","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0, 1, 4], [2, 3, 5], [0, 1, 2, 3, 4, 5]], event=[0] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).","answer":{"kind":"expression","value":"(6, [0], [1, 0, 0, 0, 0, 0], No)","explanation":"The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0], [1, 0, 0, 0, 0, 0], No)."},"skills":["An Indicator’s Measurability Depends on Its Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0070","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d5740f5372d002b3ff9ccb66733dc1c392c13daf19268fa2a527b63026cb9e5","family_id":"qf-mod24-measurable-indicator-audit","seed":202610242070,"semantic_template":"qf-mod24-measurable-indicator-audit/{\"event\":[2,4],\"source_n\":6,\"source_sigma\":[[],[0,1,2,3,4,5],[0,1,3,5],[2,4]]}"},"fields":{"title":"An Indicator’s Measurability Depends on Its Event","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [2, 4], [0, 1, 3, 5], [0, 1, 2, 3, 4, 5]], event=[2, 4] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).","answer":{"kind":"expression","value":"(6, [2, 4], [0, 0, 1, 0, 1, 0], Yes)","explanation":"The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [2, 4], [0, 0, 1, 0, 1, 0], Yes)."},"skills":["An Indicator’s Measurability Depends on Its Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0071","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2d2f4b7cf65e73af040316681373b68d8c53ed88f3392662d468f7c4720dc9a","family_id":"qf-mod24-measurable-indicator-audit","seed":202610242071,"semantic_template":"qf-mod24-measurable-indicator-audit/{\"event\":[0,2,4],\"source_n\":6,\"source_sigma\":[[],[0,1,2,3,4,5],[0,2,4],[1,3,5]]}"},"fields":{"title":"An Indicator’s Measurability Depends on Its Event","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0, 2, 4], [1, 3, 5], [0, 1, 2, 3, 4, 5]], event=[0, 2, 4] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).","answer":{"kind":"expression","value":"(6, [0, 2, 4], [1, 0, 1, 0, 1, 0], Yes)","explanation":"The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0, 2, 4], [1, 0, 1, 0, 1, 0], Yes)."},"skills":["An Indicator’s Measurability Depends on Its Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0072","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c17b2a1cac8a3504a2e95d05528925886780519cf22b2fbd157e66e90605acfb","family_id":"qf-mod24-measurable-indicator-audit","seed":202610242072,"semantic_template":"qf-mod24-measurable-indicator-audit/{\"event\":[0],\"source_n\":6,\"source_sigma\":[[],[0,1,2,3,4,5],[0,3,5],[1,2,4]]}"},"fields":{"title":"An Indicator’s Measurability Depends on Its Event","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [1, 2, 4], [0, 3, 5], [0, 1, 2, 3, 4, 5]], event=[0] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).","answer":{"kind":"expression","value":"(6, [0], [1, 0, 0, 0, 0, 0], No)","explanation":"The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0], [1, 0, 0, 0, 0, 0], No)."},"skills":["An Indicator’s Measurability Depends on Its Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0073","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f172e00959b60f99cb7a78986581772a226c315108a373c7ba1421886ea5fe0","family_id":"qf-mod24-measurable-indicator-audit","seed":202610242073,"semantic_template":"qf-mod24-measurable-indicator-audit/{\"event\":[0,1,2,4],\"source_n\":6,\"source_sigma\":[[],[0,1,2,3,4,5],[0,1,2,4],[3,5]]}"},"fields":{"title":"An Indicator’s Measurability Depends on Its Event","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0, 1, 2, 4], [3, 5], [0, 1, 2, 3, 4, 5]], event=[0, 1, 2, 4] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).","answer":{"kind":"expression","value":"(6, [0, 1, 2, 4], [1, 1, 1, 0, 1, 0], Yes)","explanation":"The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0, 1, 2, 4], [1, 1, 1, 0, 1, 0], Yes)."},"skills":["An Indicator’s Measurability Depends on Its Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0074","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d6f4add2d0ff219b30e5e86111f783930d438542eaa0b9f41624f306ace8cf9","family_id":"qf-mod24-measurable-indicator-audit","seed":202610242074,"semantic_template":"qf-mod24-measurable-indicator-audit/{\"event\":[3,4],\"source_n\":6,\"source_sigma\":[[],[0,1,2,3,4,5],[0,1,2,5],[3,4]]}"},"fields":{"title":"An Indicator’s Measurability Depends on Its Event","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [3, 4], [0, 1, 2, 5], [0, 1, 2, 3, 4, 5]], event=[3, 4] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).","answer":{"kind":"expression","value":"(6, [3, 4], [0, 0, 0, 1, 1, 0], Yes)","explanation":"The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [3, 4], [0, 0, 0, 1, 1, 0], Yes)."},"skills":["An Indicator’s Measurability Depends on Its Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0075","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c28bd12776b09825cc441cbd370ef35f57fb7b2359c761492f08419854b52caf","family_id":"qf-mod24-measurable-indicator-audit","seed":202610242075,"semantic_template":"qf-mod24-measurable-indicator-audit/{\"event\":[0],\"source_n\":6,\"source_sigma\":[[],[0,1,2,3,4,5],[0,3,4],[1,2,5]]}"},"fields":{"title":"An Indicator’s Measurability Depends on Its Event","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0, 3, 4], [1, 2, 5], [0, 1, 2, 3, 4, 5]], event=[0] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).","answer":{"kind":"expression","value":"(6, [0], [1, 0, 0, 0, 0, 0], No)","explanation":"The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0], [1, 0, 0, 0, 0, 0], No)."},"skills":["An Indicator’s Measurability Depends on Its Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0076","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"353ce3fb245bfb2672439160da9d8f20e380e2c177c813779fdcc2dc46799477","family_id":"qf-mod24-measurable-composition-audit","seed":202610242076,"semantic_template":"qf-mod24-measurable-composition-audit/{\"first\":[0,0,0,0],\"middle_n\":4,\"middle_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]],\"second\":[1,1,2,0],\"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":"Composition: Individual Checks, Composite Check and Their Conjunction","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0], [1, 2, 3], [0, 1, 2, 3]], middle 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]]. First map=[0, 0, 0, 0], second=[1, 1, 2, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.","answer":{"kind":"expression","value":"(Yes, Yes, [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, Yes)","explanation":"Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [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, Yes)."},"skills":["Composition: Individual Checks, Composite Check and Their Conjunction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0077","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0877a6cb1560a6f5b01e7c75cb8fca50f718e96b2dabf9a8abbcb9c5c22fa4b","family_id":"qf-mod24-measurable-composition-audit","seed":202610242077,"semantic_template":"qf-mod24-measurable-composition-audit/{\"first\":[1,0,0,0],\"middle_n\":4,\"middle_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]],\"second\":[2,1,2,0],\"source_n\":4,\"source_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]],\"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":"Composition: Individual Checks, Composite Check and Their Conjunction","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source 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]], middle 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]]. First map=[1, 0, 0, 0], second=[2, 1, 2, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.","answer":{"kind":"expression","value":"(Yes, Yes, [1, 2, 2, 2], [[[], [], Yes], [[0], [], Yes], [[1], [0], Yes], [[0, 1], [0], Yes], [[2], [1, 2, 3], Yes], [[0, 2], [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], [0], Yes], [[0, 1, 3], [0], Yes], [[2, 3], [1, 2, 3], Yes], [[0, 2, 3], [1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes)","explanation":"Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [1, 2, 2, 2], [[[], [], Yes], [[0], [], Yes], [[1], [0], Yes], [[0, 1], [0], Yes], [[2], [1, 2, 3], Yes], [[0, 2], [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], [0], Yes], [[0, 1, 3], [0], Yes], [[2, 3], [1, 2, 3], Yes], [[0, 2, 3], [1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes)."},"skills":["Composition: Individual Checks, Composite Check and Their Conjunction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0078","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe9952b7f2afe846eed6e42d60c04dfd12a29310f8d4919ea6b708a5d69cc084","family_id":"qf-mod24-measurable-composition-audit","seed":202610242078,"semantic_template":"qf-mod24-measurable-composition-audit/{\"first\":[2,0,0,0],\"middle_n\":4,\"middle_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]],\"second\":[3,1,2,0],\"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":"Composition: Individual Checks, Composite Check and Their Conjunction","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0, 1], [2, 3], [0, 1, 2, 3]], middle 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]]. First map=[2, 0, 0, 0], second=[3, 1, 2, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.","answer":{"kind":"expression","value":"(No, Yes, [2, 3, 3, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [0], No], [[0, 2], [0], No], [[1, 2], [0], No], [[0, 1, 2], [0], No], [[3], [1, 2, 3], No], [[0, 3], [1, 2, 3], No], [[1, 3], [1, 2, 3], No], [[0, 1, 3], [1, 2, 3], No], [[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]], No, No)","explanation":"Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (No, Yes, [2, 3, 3, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [0], No], [[0, 2], [0], No], [[1, 2], [0], No], [[0, 1, 2], [0], No], [[3], [1, 2, 3], No], [[0, 3], [1, 2, 3], No], [[1, 3], [1, 2, 3], No], [[0, 1, 3], [1, 2, 3], No], [[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]], No, No)."},"skills":["Composition: Individual Checks, Composite Check and Their Conjunction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0079","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c37c94bd4f549b55357adf1615ec7c94f0f90c32a5657fd2fde520cca7e684a8","family_id":"qf-mod24-measurable-composition-audit","seed":202610242079,"semantic_template":"qf-mod24-measurable-composition-audit/{\"first\":[3,0,0,0],\"middle_n\":4,\"middle_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]],\"second\":[0,2,2,0],\"source_n\":4,\"source_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]],\"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":"Composition: Individual Checks, Composite Check and Their Conjunction","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source 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]], middle 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]]. First map=[3, 0, 0, 0], second=[0, 2, 2, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.","answer":{"kind":"expression","value":"(Yes, Yes, [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, Yes)","explanation":"Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [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, Yes)."},"skills":["Composition: Individual Checks, Composite Check and Their Conjunction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0080","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c61a0ff181b8eaefbadb6ff393208b0c478ca231a1597a5db3dc27aff808beb8","family_id":"qf-mod24-measurable-composition-audit","seed":202610242080,"semantic_template":"qf-mod24-measurable-composition-audit/{\"first\":[0,1,0,0],\"middle_n\":4,\"middle_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]],\"second\":[1,2,2,0],\"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":"Composition: Individual Checks, Composite Check and Their Conjunction","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0, 2], [1, 3], [0, 1, 2, 3]], middle 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]]. First map=[0, 1, 0, 0], second=[1, 2, 2, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.","answer":{"kind":"expression","value":"(No, Yes, [1, 2, 1, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 2, 3], No], [[0, 1], [0, 2, 3], No], [[2], [1], No], [[0, 2], [1], No], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 2, 3], No], [[0, 1, 3], [0, 2, 3], No], [[2, 3], [1], No], [[0, 2, 3], [1], No], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No)","explanation":"Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (No, Yes, [1, 2, 1, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 2, 3], No], [[0, 1], [0, 2, 3], No], [[2], [1], No], [[0, 2], [1], No], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 2, 3], No], [[0, 1, 3], [0, 2, 3], No], [[2, 3], [1], No], [[0, 2, 3], [1], No], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No)."},"skills":["Composition: Individual Checks, Composite Check and Their Conjunction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0081","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c318171b5d685d87c59d4df897e94d615e3ef608b137ee6a3c667cb77e191a5","family_id":"qf-mod24-measurable-composition-audit","seed":202610242081,"semantic_template":"qf-mod24-measurable-composition-audit/{\"first\":[1,1,0,0],\"middle_n\":4,\"middle_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]],\"second\":[2,2,2,0],\"source_n\":4,\"source_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]],\"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":"Composition: Individual Checks, Composite Check and Their Conjunction","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source 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]], middle 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]]. First map=[1, 1, 0, 0], second=[2, 2, 2, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.","answer":{"kind":"expression","value":"(Yes, Yes, [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, Yes)","explanation":"Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [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, Yes)."},"skills":["Composition: Individual Checks, Composite Check and Their Conjunction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0082","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77815e2d7efac0e23f7e5869e3a71e2487f71c04c0770a51c894bbfb4d2d4952","family_id":"qf-mod24-measurable-composition-audit","seed":202610242082,"semantic_template":"qf-mod24-measurable-composition-audit/{\"first\":[2,1,0,0],\"middle_n\":4,\"middle_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]],\"second\":[3,2,2,0],\"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":"Composition: Individual Checks, Composite Check and Their Conjunction","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0, 1, 2], [3], [0, 1, 2, 3]], middle 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]]. First map=[2, 1, 0, 0], second=[3, 2, 2, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.","answer":{"kind":"expression","value":"(No, Yes, [2, 2, 3, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [0, 1], No], [[0, 2], [0, 1], No], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1], No], [[3], [2, 3], No], [[0, 3], [2, 3], No], [[1, 3], [2, 3], No], [[0, 1, 3], [2, 3], No], [[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]], No, No)","explanation":"Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (No, Yes, [2, 2, 3, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [0, 1], No], [[0, 2], [0, 1], No], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1], No], [[3], [2, 3], No], [[0, 3], [2, 3], No], [[1, 3], [2, 3], No], [[0, 1, 3], [2, 3], No], [[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]], No, No)."},"skills":["Composition: Individual Checks, Composite Check and Their Conjunction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0083","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ffb27f11985b0162f2108740d3b9b15fb132c64e0d341f184a86a7dcdc9a15f","family_id":"qf-mod24-measurable-composition-audit","seed":202610242083,"semantic_template":"qf-mod24-measurable-composition-audit/{\"first\":[3,1,0,0],\"middle_n\":4,\"middle_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]],\"second\":[0,3,2,0],\"source_n\":4,\"source_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]],\"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":"Composition: Individual Checks, Composite Check and Their Conjunction","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source 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]], middle 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]]. First map=[3, 1, 0, 0], second=[0, 3, 2, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.","answer":{"kind":"expression","value":"(Yes, Yes, [0, 3, 0, 0], [[[], [], 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, Yes)","explanation":"Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [0, 3, 0, 0], [[[], [], 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, Yes)."},"skills":["Composition: Individual Checks, Composite Check and Their Conjunction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0084","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a31b39dc9377ef7412cb4377f5f06ab541a7543ff13858c5e8b0b0e71b06e3b","family_id":"qf-mod24-measurable-composition-audit","seed":202610242084,"semantic_template":"qf-mod24-measurable-composition-audit/{\"first\":[0,2,0,0],\"middle_n\":4,\"middle_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]],\"second\":[1,3,2,0],\"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":"Composition: Individual Checks, Composite Check and Their Conjunction","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [1], [0, 2, 3], [0, 1, 2, 3]], middle 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]]. First map=[0, 2, 0, 0], second=[1, 3, 2, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.","answer":{"kind":"expression","value":"(Yes, Yes, [1, 2, 1, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 2, 3], Yes], [[0, 1], [0, 2, 3], Yes], [[2], [1], Yes], [[0, 2], [1], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 2, 3], Yes], [[0, 1, 3], [0, 2, 3], Yes], [[2, 3], [1], Yes], [[0, 2, 3], [1], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes)","explanation":"Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [1, 2, 1, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 2, 3], Yes], [[0, 1], [0, 2, 3], Yes], [[2], [1], Yes], [[0, 2], [1], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 2, 3], Yes], [[0, 1, 3], [0, 2, 3], Yes], [[2, 3], [1], Yes], [[0, 2, 3], [1], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes)."},"skills":["Composition: Individual Checks, Composite Check and Their Conjunction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0085","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22f6b9d0a0ca193a7a32b338967c221e34753f14b85690c70efbf37b42e1562f","family_id":"qf-mod24-measurable-composition-audit","seed":202610242085,"semantic_template":"qf-mod24-measurable-composition-audit/{\"first\":[1,2,0,0],\"middle_n\":4,\"middle_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]],\"second\":[2,3,2,0],\"source_n\":4,\"source_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]],\"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":"Composition: Individual Checks, Composite Check and Their Conjunction","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source 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]], middle 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]]. First map=[1, 2, 0, 0], second=[2, 3, 2, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.","answer":{"kind":"expression","value":"(Yes, Yes, [3, 2, 2, 2], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [1, 2, 3], Yes], [[0, 2], [1, 2, 3], Yes], [[1, 2], [1, 2, 3], Yes], [[0, 1, 2], [1, 2, 3], Yes], [[3], [0], Yes], [[0, 3], [0], Yes], [[1, 3], [0], Yes], [[0, 1, 3], [0], 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, Yes)","explanation":"Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [3, 2, 2, 2], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [1, 2, 3], Yes], [[0, 2], [1, 2, 3], Yes], [[1, 2], [1, 2, 3], Yes], [[0, 1, 2], [1, 2, 3], Yes], [[3], [0], Yes], [[0, 3], [0], Yes], [[1, 3], [0], Yes], [[0, 1, 3], [0], 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, Yes)."},"skills":["Composition: Individual Checks, Composite Check and Their Conjunction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0086","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce01a9f43c8008ba3d3dd6e6981bea862ef7b5a159bc8e98fe5396e73add793a","family_id":"qf-mod24-measurable-composition-audit","seed":202610242086,"semantic_template":"qf-mod24-measurable-composition-audit/{\"first\":[2,2,0,0],\"middle_n\":4,\"middle_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]],\"second\":[3,3,2,0],\"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":"Composition: Individual Checks, Composite Check and Their Conjunction","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [2], [0, 1, 3], [0, 1, 2, 3]], middle 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]]. First map=[2, 2, 0, 0], second=[3, 3, 2, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.","answer":{"kind":"expression","value":"(No, Yes, [2, 2, 3, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [0, 1], No], [[0, 2], [0, 1], No], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1], No], [[3], [2, 3], No], [[0, 3], [2, 3], No], [[1, 3], [2, 3], No], [[0, 1, 3], [2, 3], No], [[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]], No, No)","explanation":"Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (No, Yes, [2, 2, 3, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [0, 1], No], [[0, 2], [0, 1], No], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1], No], [[3], [2, 3], No], [[0, 3], [2, 3], No], [[1, 3], [2, 3], No], [[0, 1, 3], [2, 3], No], [[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]], No, No)."},"skills":["Composition: Individual Checks, Composite Check and Their Conjunction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0087","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"152ed6e9a60befef0b6b85d2af405ceca843bef9b2bd6fb930a9ef781a72930a","family_id":"qf-mod24-measurable-composition-audit","seed":202610242087,"semantic_template":"qf-mod24-measurable-composition-audit/{\"first\":[3,2,0,0],\"middle_n\":4,\"middle_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]],\"second\":[0,0,3,0],\"source_n\":4,\"source_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]],\"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":"Composition: Individual Checks, Composite Check and Their Conjunction","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source 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]], middle 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]]. First map=[3, 2, 0, 0], second=[0, 0, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.","answer":{"kind":"expression","value":"(Yes, Yes, [0, 3, 0, 0], [[[], [], 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, Yes)","explanation":"Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [0, 3, 0, 0], [[[], [], 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, Yes)."},"skills":["Composition: Individual Checks, Composite Check and Their Conjunction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0088","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1258d340c443d73f3818117d75bd049b738477d27f19697b7d643549d74e7ac9","family_id":"qf-mod24-measurable-composition-audit","seed":202610242088,"semantic_template":"qf-mod24-measurable-composition-audit/{\"first\":[0,3,0,0],\"middle_n\":4,\"middle_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]],\"second\":[1,0,3,0],\"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":"Composition: Individual Checks, Composite Check and Their Conjunction","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [1, 2], [0, 3], [0, 1, 2, 3]], middle 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]]. First map=[0, 3, 0, 0], second=[1, 0, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.","answer":{"kind":"expression","value":"(No, Yes, [1, 0, 1, 1], [[[], [], Yes], [[0], [1], No], [[1], [0, 2, 3], No], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [1], No], [[1, 2], [0, 2, 3], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [1], No], [[1, 3], [0, 2, 3], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [1], No], [[1, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No)","explanation":"Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (No, Yes, [1, 0, 1, 1], [[[], [], Yes], [[0], [1], No], [[1], [0, 2, 3], No], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [1], No], [[1, 2], [0, 2, 3], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [1], No], [[1, 3], [0, 2, 3], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [1], No], [[1, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No)."},"skills":["Composition: Individual Checks, Composite Check and Their Conjunction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0089","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cfa36514967ecf4d0f94c35e2ab80c10a7778b3929737f32d653f954af4a8db4","family_id":"qf-mod24-measurable-composition-audit","seed":202610242089,"semantic_template":"qf-mod24-measurable-composition-audit/{\"first\":[1,3,0,0],\"middle_n\":4,\"middle_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]],\"second\":[2,0,3,0],\"source_n\":4,\"source_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]],\"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":"Composition: Individual Checks, Composite Check and Their Conjunction","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source 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]], middle 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]]. First map=[1, 3, 0, 0], second=[2, 0, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.","answer":{"kind":"expression","value":"(Yes, Yes, [0, 0, 2, 2], [[[], [], Yes], [[0], [0, 1], Yes], [[1], [], Yes], [[0, 1], [0, 1], Yes], [[2], [2, 3], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1], Yes], [[2, 3], [2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes)","explanation":"Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [0, 0, 2, 2], [[[], [], Yes], [[0], [0, 1], Yes], [[1], [], Yes], [[0, 1], [0, 1], Yes], [[2], [2, 3], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1], Yes], [[2, 3], [2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes)."},"skills":["Composition: Individual Checks, Composite Check and Their Conjunction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0090","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45e82d5dd2054d94d2978fb4d0b2109a794d8c8ac1a54931799768b274311211","family_id":"qf-mod24-measurable-composition-audit","seed":202610242090,"semantic_template":"qf-mod24-measurable-composition-audit/{\"first\":[2,3,0,0],\"middle_n\":4,\"middle_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]],\"second\":[3,0,3,0],\"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":"Composition: Individual Checks, Composite Check and Their Conjunction","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0], [1, 2, 3], [0, 1, 2, 3]], middle 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]]. First map=[2, 3, 0, 0], second=[3, 0, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.","answer":{"kind":"expression","value":"(No, Yes, [3, 0, 3, 3], [[[], [], Yes], [[0], [1], No], [[1], [], Yes], [[0, 1], [1], No], [[2], [], Yes], [[0, 2], [1], No], [[1, 2], [], Yes], [[0, 1, 2], [1], No], [[3], [0, 2, 3], No], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0, 2, 3], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0, 2, 3], No], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No)","explanation":"Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (No, Yes, [3, 0, 3, 3], [[[], [], Yes], [[0], [1], No], [[1], [], Yes], [[0, 1], [1], No], [[2], [], Yes], [[0, 2], [1], No], [[1, 2], [], Yes], [[0, 1, 2], [1], No], [[3], [0, 2, 3], No], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0, 2, 3], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0, 2, 3], No], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No)."},"skills":["Composition: Individual Checks, Composite Check and Their Conjunction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0091","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66bb0fdeab577aefe98741cf532f5b38164f00b296e33ab630def61d5953ccf9","family_id":"qf-mod24-measurable-composition-audit","seed":202610242091,"semantic_template":"qf-mod24-measurable-composition-audit/{\"first\":[3,3,0,0],\"middle_n\":4,\"middle_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]],\"second\":[0,1,3,0],\"source_n\":4,\"source_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]],\"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":"Composition: Individual Checks, Composite Check and Their Conjunction","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source 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]], middle 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]]. First map=[3, 3, 0, 0], second=[0, 1, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.","answer":{"kind":"expression","value":"(Yes, Yes, [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, Yes)","explanation":"Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [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, Yes)."},"skills":["Composition: Individual Checks, Composite Check and Their Conjunction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0092","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"472253395459dc88328f6a49cb9ba54f10e059b16b79340b570537a977956844","family_id":"qf-mod24-measurable-composition-audit","seed":202610242092,"semantic_template":"qf-mod24-measurable-composition-audit/{\"first\":[0,0,1,0],\"middle_n\":4,\"middle_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]],\"second\":[1,1,3,0],\"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":"Composition: Individual Checks, Composite Check and Their Conjunction","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0, 1], [2, 3], [0, 1, 2, 3]], middle 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]]. First map=[0, 0, 1, 0], second=[1, 1, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.","answer":{"kind":"expression","value":"(No, Yes, [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, No)","explanation":"Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (No, Yes, [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, No)."},"skills":["Composition: Individual Checks, Composite Check and Their Conjunction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0093","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"296545563618623807bf08493e89db7438097ac54bc9856fbaae862c293b9ac3","family_id":"qf-mod24-measurable-composition-audit","seed":202610242093,"semantic_template":"qf-mod24-measurable-composition-audit/{\"first\":[1,0,1,0],\"middle_n\":4,\"middle_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]],\"second\":[2,1,3,0],\"source_n\":4,\"source_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]],\"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":"Composition: Individual Checks, Composite Check and Their Conjunction","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source 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]], middle 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]]. First map=[1, 0, 1, 0], second=[2, 1, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.","answer":{"kind":"expression","value":"(Yes, Yes, [1, 2, 1, 2], [[[], [], Yes], [[0], [], Yes], [[1], [0, 2], Yes], [[0, 1], [0, 2], Yes], [[2], [1, 3], Yes], [[0, 2], [1, 3], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 2], Yes], [[0, 1, 3], [0, 2], Yes], [[2, 3], [1, 3], Yes], [[0, 2, 3], [1, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes)","explanation":"Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [1, 2, 1, 2], [[[], [], Yes], [[0], [], Yes], [[1], [0, 2], Yes], [[0, 1], [0, 2], Yes], [[2], [1, 3], Yes], [[0, 2], [1, 3], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 2], Yes], [[0, 1, 3], [0, 2], Yes], [[2, 3], [1, 3], Yes], [[0, 2, 3], [1, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes)."},"skills":["Composition: Individual Checks, Composite Check and Their Conjunction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0094","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0ec3813877c55d2d8a2999c5840691814bc9cd46ea9eced50347f980728990a","family_id":"qf-mod24-measurable-composition-audit","seed":202610242094,"semantic_template":"qf-mod24-measurable-composition-audit/{\"first\":[2,0,1,0],\"middle_n\":4,\"middle_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]],\"second\":[3,1,3,0],\"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":"Composition: Individual Checks, Composite Check and Their Conjunction","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0, 2], [1, 3], [0, 1, 2, 3]], middle 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]]. First map=[2, 0, 1, 0], second=[3, 1, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.","answer":{"kind":"expression","value":"(No, Yes, [3, 3, 1, 3], [[[], [], Yes], [[0], [], Yes], [[1], [2], No], [[0, 1], [2], No], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [2], No], [[0, 1, 2], [2], No], [[3], [0, 1, 3], No], [[0, 3], [0, 1, 3], No], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0, 1, 3], No], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No)","explanation":"Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (No, Yes, [3, 3, 1, 3], [[[], [], Yes], [[0], [], Yes], [[1], [2], No], [[0, 1], [2], No], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [2], No], [[0, 1, 2], [2], No], [[3], [0, 1, 3], No], [[0, 3], [0, 1, 3], No], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0, 1, 3], No], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No)."},"skills":["Composition: Individual Checks, Composite Check and Their Conjunction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0095","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5556ef2d94ca8831a4784f729221b129c11afbb68f761ff114d008c8e66a1d26","family_id":"qf-mod24-measurable-composition-audit","seed":202610242095,"semantic_template":"qf-mod24-measurable-composition-audit/{\"first\":[3,0,1,0],\"middle_n\":4,\"middle_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]],\"second\":[0,2,3,0],\"source_n\":4,\"source_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]],\"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":"Composition: Individual Checks, Composite Check and Their Conjunction","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source 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]], middle 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]]. First map=[3, 0, 1, 0], second=[0, 2, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.","answer":{"kind":"expression","value":"(Yes, Yes, [0, 0, 2, 0], [[[], [], Yes], [[0], [0, 1, 3], Yes], [[1], [], Yes], [[0, 1], [0, 1, 3], Yes], [[2], [2], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [2], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1, 3], Yes], [[2, 3], [2], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes)","explanation":"Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [0, 0, 2, 0], [[[], [], Yes], [[0], [0, 1, 3], Yes], [[1], [], Yes], [[0, 1], [0, 1, 3], Yes], [[2], [2], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [2], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1, 3], Yes], [[2, 3], [2], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes)."},"skills":["Composition: Individual Checks, Composite Check and Their Conjunction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0096","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd9bc79320429e31502c4613c392533afd78dec1cdf4571a8682ebbe61d4e4a0","family_id":"qf-mod24-measurable-composition-audit","seed":202610242096,"semantic_template":"qf-mod24-measurable-composition-audit/{\"first\":[0,1,1,0],\"middle_n\":4,\"middle_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]],\"second\":[1,2,3,0],\"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":"Composition: Individual Checks, Composite Check and Their Conjunction","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0, 1, 2], [3], [0, 1, 2, 3]], middle 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]]. First map=[0, 1, 1, 0], second=[1, 2, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.","answer":{"kind":"expression","value":"(No, Yes, [1, 2, 2, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 3], No], [[0, 1], [0, 3], No], [[2], [1, 2], No], [[0, 2], [1, 2], No], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 3], No], [[0, 1, 3], [0, 3], No], [[2, 3], [1, 2], No], [[0, 2, 3], [1, 2], No], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No)","explanation":"Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (No, Yes, [1, 2, 2, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 3], No], [[0, 1], [0, 3], No], [[2], [1, 2], No], [[0, 2], [1, 2], No], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 3], No], [[0, 1, 3], [0, 3], No], [[2, 3], [1, 2], No], [[0, 2, 3], [1, 2], No], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No)."},"skills":["Composition: Individual Checks, Composite Check and Their Conjunction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0097","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a5fea5887677996d8cfeeaadbf54a3fc7621ba975edc62735d554e7842ed638","family_id":"qf-mod24-measurable-composition-audit","seed":202610242097,"semantic_template":"qf-mod24-measurable-composition-audit/{\"first\":[1,1,1,0],\"middle_n\":4,\"middle_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]],\"second\":[2,2,3,0],\"source_n\":4,\"source_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]],\"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":"Composition: Individual Checks, Composite Check and Their Conjunction","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source 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]], middle 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]]. First map=[1, 1, 1, 0], second=[2, 2, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.","answer":{"kind":"expression","value":"(Yes, Yes, [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, Yes)","explanation":"Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [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, Yes)."},"skills":["Composition: Individual Checks, Composite Check and Their Conjunction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0098","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3929ea52d12ea480ff5f576e97ef376033c3e6ab315f311c736140057f33a8af","family_id":"qf-mod24-measurable-composition-audit","seed":202610242098,"semantic_template":"qf-mod24-measurable-composition-audit/{\"first\":[2,1,1,0],\"middle_n\":4,\"middle_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]],\"second\":[3,2,3,0],\"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":"Composition: Individual Checks, Composite Check and Their Conjunction","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [1], [0, 2, 3], [0, 1, 2, 3]], middle 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]]. First map=[2, 1, 1, 0], second=[3, 2, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.","answer":{"kind":"expression","value":"(No, Yes, [3, 2, 2, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [1, 2], No], [[0, 2], [1, 2], No], [[1, 2], [1, 2], No], [[0, 1, 2], [1, 2], No], [[3], [0, 3], No], [[0, 3], [0, 3], No], [[1, 3], [0, 3], No], [[0, 1, 3], [0, 3], No], [[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]], No, No)","explanation":"Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (No, Yes, [3, 2, 2, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [1, 2], No], [[0, 2], [1, 2], No], [[1, 2], [1, 2], No], [[0, 1, 2], [1, 2], No], [[3], [0, 3], No], [[0, 3], [0, 3], No], [[1, 3], [0, 3], No], [[0, 1, 3], [0, 3], No], [[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]], No, No)."},"skills":["Composition: Individual Checks, Composite Check and Their Conjunction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0099","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"510f074b139681edc4e83a0f9cb22bf6c0ea160728600fa2e10255bcd46a918d","family_id":"qf-mod24-measurable-composition-audit","seed":202610242099,"semantic_template":"qf-mod24-measurable-composition-audit/{\"first\":[3,1,1,0],\"middle_n\":4,\"middle_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]],\"second\":[0,3,3,0],\"source_n\":4,\"source_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]],\"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":"Composition: Individual Checks, Composite Check and Their Conjunction","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source 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]], middle 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]]. First map=[3, 1, 1, 0], second=[0, 3, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.","answer":{"kind":"expression","value":"(Yes, Yes, [0, 3, 3, 0], [[[], [], Yes], [[0], [0, 3], Yes], [[1], [], Yes], [[0, 1], [0, 3], Yes], [[2], [], Yes], [[0, 2], [0, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 3], Yes], [[3], [1, 2], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [1, 2], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [1, 2], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes)","explanation":"Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [0, 3, 3, 0], [[[], [], Yes], [[0], [0, 3], Yes], [[1], [], Yes], [[0, 1], [0, 3], Yes], [[2], [], Yes], [[0, 2], [0, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 3], Yes], [[3], [1, 2], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [1, 2], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [1, 2], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes)."},"skills":["Composition: Individual Checks, Composite Check and Their Conjunction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261024-r02-measurable-function-0100","batch_id":"batch-20261013-advance-20261024-r02-measurable-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf5eee77d122c2892b63358f2ad43ff1ea81f8ed3412aeb4ef0a94970751f9a8","family_id":"qf-mod24-measurable-composition-audit","seed":202610242100,"semantic_template":"qf-mod24-measurable-composition-audit/{\"first\":[0,2,1,0],\"middle_n\":4,\"middle_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]],\"second\":[1,3,3,0],\"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":"Composition: Individual Checks, Composite Check and Their Conjunction","stem":"Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [2], [0, 1, 3], [0, 1, 2, 3]], middle 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]]. First map=[0, 2, 1, 0], second=[1, 3, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.","answer":{"kind":"expression","value":"(No, Yes, [1, 3, 3, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 3], No], [[0, 1], [0, 3], No], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [0, 3], No], [[0, 1, 2], [0, 3], No], [[3], [1, 2], No], [[0, 3], [1, 2], No], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [1, 2], No], [[0, 2, 3], [1, 2], No], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No)","explanation":"Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (No, Yes, [1, 3, 3, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 3], No], [[0, 1], [0, 3], No], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [0, 3], No], [[0, 1, 2], [0, 3], No], [[3], [1, 2], No], [[0, 3], [1, 2], No], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [1, 2], No], [[0, 2, 3], [1, 2], No], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No)."},"skills":["Composition: Individual Checks, Composite Check and Their Conjunction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}