{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0001","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e27b7b707016993a88be78f29a776e8464d113966c3e6a3a68a36cb07aa1cfd5","family_id":"qf-ext19-continuity-single-preimage","seed":202610193001,"semantic_template":"qf-ext19-continuity-single-preimage/{\"images\":[0,1,2,3],\"m\":4,\"n\":4,\"opens_x\":[[],[0],[0,1],[0,1,2],[0,1,2,3]],\"opens_y\":[[],[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_open\":[]}"},"fields":{"title":"One Declared Target-Open Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[0, 1, 2, 3] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.","answer":{"kind":"expression","value":"(4, 4, [], [], Yes)","explanation":"Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [], [], Yes)."},"skills":["One Declared Target-Open 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-20261019-r03-topological-continuity-0002","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11d5a1b7bb7d45defe2e47c1660a5c3e4480454e8b89aabb1c3e700112d387d1","family_id":"qf-ext19-continuity-single-preimage","seed":202610193002,"semantic_template":"qf-ext19-continuity-single-preimage/{\"images\":[1,0,2,3],\"m\":4,\"n\":4,\"opens_x\":[[],[1],[0,1],[0,1,2],[0,1,2,3]],\"opens_y\":[[],[1],[1,2],[0,1,2],[0,1,2,3]],\"target_open\":[1]}"},"fields":{"title":"One Declared Target-Open Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]]. The complete total map has images=[1, 0, 2, 3] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[1] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.","answer":{"kind":"expression","value":"(4, 4, [1], [0], No)","explanation":"Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [1], [0], No)."},"skills":["One Declared Target-Open 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-20261019-r03-topological-continuity-0003","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b2a988989b55b3b766389f3e94a72ef524df5186e1790746a6b30b6592adca7","family_id":"qf-ext19-continuity-single-preimage","seed":202610193003,"semantic_template":"qf-ext19-continuity-single-preimage/{\"images\":[2,0,1,3],\"m\":4,\"n\":4,\"opens_x\":[[],[2],[0,2],[0,1,2],[0,1,2,3]],\"opens_y\":[[],[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_open\":[1]}"},"fields":{"title":"One Declared Target-Open Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [0, 2], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[2, 0, 1, 3] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[1] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.","answer":{"kind":"expression","value":"(4, 4, [1], [2], Yes)","explanation":"Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [1], [2], Yes)."},"skills":["One Declared Target-Open 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-20261019-r03-topological-continuity-0004","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81d8da9dc4ecfaf3fde0589a4da38f0a7efe00afff32ebff9c2e99b2de28167b","family_id":"qf-ext19-continuity-single-preimage","seed":202610193004,"semantic_template":"qf-ext19-continuity-single-preimage/{\"images\":[3,0,1,2],\"m\":4,\"n\":4,\"opens_x\":[[],[3],[0,3],[0,1,3],[0,1,2,3]],\"opens_y\":[[],[3],[0,3],[0,2,3],[0,1,2,3]],\"target_open\":[0,2,3]}"},"fields":{"title":"One Declared Target-Open Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 0, 1, 2] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[0, 2, 3] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.","answer":{"kind":"expression","value":"(4, 4, [0, 2, 3], [0, 1, 3], Yes)","explanation":"Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [0, 2, 3], [0, 1, 3], Yes)."},"skills":["One Declared Target-Open 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-20261019-r03-topological-continuity-0005","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8be875950e6e30cad3e0dcaf1f192792ad0165eb3bfca081fadc8777bc6b27bd","family_id":"qf-ext19-continuity-single-preimage","seed":202610193005,"semantic_template":"qf-ext19-continuity-single-preimage/{\"images\":[0,2,1,3],\"m\":4,\"n\":4,\"opens_x\":[[],[0],[0,2],[0,1,2],[0,1,2,3]],\"opens_y\":[[],[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_open\":[2]}"},"fields":{"title":"One Declared Target-Open Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[0, 2, 1, 3] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[2] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.","answer":{"kind":"expression","value":"(4, 4, [2], [1], No)","explanation":"Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [2], [1], No)."},"skills":["One Declared Target-Open 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-20261019-r03-topological-continuity-0006","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20a9d3a852c782bcfbcfc5ccb6d33583301bfe6c66f27bdc0515838927da19bd","family_id":"qf-ext19-continuity-single-preimage","seed":202610193006,"semantic_template":"qf-ext19-continuity-single-preimage/{\"images\":[1,2,0,3],\"m\":4,\"n\":4,\"opens_x\":[[],[1],[1,2],[0,1,2],[0,1,2,3]],\"opens_y\":[[],[1],[0,1],[0,1,2],[0,1,2,3]],\"target_open\":[]}"},"fields":{"title":"One Declared Target-Open Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]]. The complete total map has images=[1, 2, 0, 3] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.","answer":{"kind":"expression","value":"(4, 4, [], [], Yes)","explanation":"Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [], [], Yes)."},"skills":["One Declared Target-Open 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-20261019-r03-topological-continuity-0007","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d7bf070282f12c6928529fa89dc083ec9ca8792866d51fbe89e2e89e9800bab","family_id":"qf-ext19-continuity-single-preimage","seed":202610193007,"semantic_template":"qf-ext19-continuity-single-preimage/{\"images\":[2,1,0,3],\"m\":4,\"n\":4,\"opens_x\":[[],[2],[1,2],[0,1,2],[0,1,2,3]],\"opens_y\":[[],[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_open\":[1,2]}"},"fields":{"title":"One Declared Target-Open Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [1, 2], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[2, 1, 0, 3] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[1, 2] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.","answer":{"kind":"expression","value":"(4, 4, [1, 2], [0, 1], No)","explanation":"Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [1, 2], [0, 1], No)."},"skills":["One Declared Target-Open 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-20261019-r03-topological-continuity-0008","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95f461d3179745d24a6f5682acdbfea4754450e26f47745e48d77d8327b57f0e","family_id":"qf-ext19-continuity-single-preimage","seed":202610193008,"semantic_template":"qf-ext19-continuity-single-preimage/{\"images\":[3,1,0,2],\"m\":4,\"n\":4,\"opens_x\":[[],[3],[1,3],[0,1,3],[0,1,2,3]],\"opens_y\":[[],[3],[2,3],[0,2,3],[0,1,2,3]],\"target_open\":[2,3]}"},"fields":{"title":"One Declared Target-Open Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 1, 0, 2] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[2, 3] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.","answer":{"kind":"expression","value":"(4, 4, [2, 3], [0, 3], No)","explanation":"Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [2, 3], [0, 3], No)."},"skills":["One Declared Target-Open 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-20261019-r03-topological-continuity-0009","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d4fd4dbc2d821d7f971b74d6bd5bcc4982e727d51e7672a994a2a0026127ff5c","family_id":"qf-ext19-continuity-single-preimage","seed":202610193009,"semantic_template":"qf-ext19-continuity-single-preimage/{\"images\":[0,3,1,2],\"m\":4,\"n\":4,\"opens_x\":[[],[0],[0,3],[0,1,3],[0,1,2,3]],\"opens_y\":[[],[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_open\":[3]}"},"fields":{"title":"One Declared Target-Open Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[0, 3, 1, 2] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[3] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.","answer":{"kind":"expression","value":"(4, 4, [3], [1], No)","explanation":"Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [3], [1], No)."},"skills":["One Declared Target-Open 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-20261019-r03-topological-continuity-0010","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9e23668b3e0581505e489e58f9924c83d7a9210d456abc42e0ee914d9738520","family_id":"qf-ext19-continuity-single-preimage","seed":202610193010,"semantic_template":"qf-ext19-continuity-single-preimage/{\"images\":[1,3,0,2],\"m\":4,\"n\":4,\"opens_x\":[[],[1],[1,3],[0,1,3],[0,1,2,3]],\"opens_y\":[[],[1],[1,3],[1,2,3],[0,1,2,3]],\"target_open\":[0,1,2,3]}"},"fields":{"title":"One Declared Target-Open Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[1, 3, 0, 2] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[0, 1, 2, 3] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.","answer":{"kind":"expression","value":"(4, 4, [0, 1, 2, 3], [0, 1, 2, 3], Yes)","explanation":"Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [0, 1, 2, 3], [0, 1, 2, 3], Yes)."},"skills":["One Declared Target-Open 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-20261019-r03-topological-continuity-0011","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fcd173b2d2e00493f5c5a08f2ce5bcc00c93807fb724a8a1b5865fc85909fdac","family_id":"qf-ext19-continuity-single-preimage","seed":202610193011,"semantic_template":"qf-ext19-continuity-single-preimage/{\"images\":[2,3,0,1],\"m\":4,\"n\":4,\"opens_x\":[[],[2],[2,3],[0,2,3],[0,1,2,3]],\"opens_y\":[[],[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_open\":[1,3]}"},"fields":{"title":"One Declared Target-Open Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[2, 3, 0, 1] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[1, 3] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.","answer":{"kind":"expression","value":"(4, 4, [1, 3], [1, 3], No)","explanation":"Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [1, 3], [1, 3], No)."},"skills":["One Declared Target-Open 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-20261019-r03-topological-continuity-0012","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d41a56e7e269922a4bad74699d6010f50568c9bd2116d9bacf0d93e691868cbf","family_id":"qf-ext19-continuity-single-preimage","seed":202610193012,"semantic_template":"qf-ext19-continuity-single-preimage/{\"images\":[3,2,0,1],\"m\":4,\"n\":4,\"opens_x\":[[],[3],[2,3],[0,2,3],[0,1,2,3]],\"opens_y\":[[],[3],[1,3],[0,1,3],[0,1,2,3]],\"target_open\":[3]}"},"fields":{"title":"One Declared Target-Open Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 2, 0, 1] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[3] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.","answer":{"kind":"expression","value":"(4, 4, [3], [0], No)","explanation":"Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [3], [0], No)."},"skills":["One Declared Target-Open 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-20261019-r03-topological-continuity-0013","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19b4c84352802577de053e30f36fb10a8acd76ae12b8e87cb70d85317512c4e3","family_id":"qf-ext19-continuity-single-preimage","seed":202610193013,"semantic_template":"qf-ext19-continuity-single-preimage/{\"images\":[0,1,3,2],\"m\":4,\"n\":4,\"opens_x\":[[],[0],[0,1],[0,1,3],[0,1,2,3]],\"opens_y\":[[],[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_open\":[2,3]}"},"fields":{"title":"One Declared Target-Open Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[0, 1, 3, 2] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[2, 3] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.","answer":{"kind":"expression","value":"(4, 4, [2, 3], [2, 3], No)","explanation":"Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [2, 3], [2, 3], No)."},"skills":["One Declared Target-Open 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-20261019-r03-topological-continuity-0014","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89e24dcc84b23abc1fb82d11e3b7d4e2beed73b57b65112ae8c19c4749fec38e","family_id":"qf-ext19-continuity-single-preimage","seed":202610193014,"semantic_template":"qf-ext19-continuity-single-preimage/{\"images\":[1,0,3,2],\"m\":4,\"n\":4,\"opens_x\":[[],[1],[0,1],[0,1,3],[0,1,2,3]],\"opens_y\":[[],[1],[1,2],[1,2,3],[0,1,2,3]],\"target_open\":[1,2,3]}"},"fields":{"title":"One Declared Target-Open Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[1, 0, 3, 2] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[1, 2, 3] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.","answer":{"kind":"expression","value":"(4, 4, [1, 2, 3], [0, 2, 3], No)","explanation":"Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [1, 2, 3], [0, 2, 3], No)."},"skills":["One Declared Target-Open 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-20261019-r03-topological-continuity-0015","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9210ea84a597d5aafcab8ad34fcf659db568d6b0ac05f4a1570caeb82b077d29","family_id":"qf-ext19-continuity-single-preimage","seed":202610193015,"semantic_template":"qf-ext19-continuity-single-preimage/{\"images\":[2,0,3,1],\"m\":4,\"n\":4,\"opens_x\":[[],[2],[0,2],[0,2,3],[0,1,2,3]],\"opens_y\":[[],[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_open\":[1,2,3]}"},"fields":{"title":"One Declared Target-Open Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[2, 0, 3, 1] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[1, 2, 3] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.","answer":{"kind":"expression","value":"(4, 4, [1, 2, 3], [0, 2, 3], Yes)","explanation":"Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [1, 2, 3], [0, 2, 3], Yes)."},"skills":["One Declared Target-Open 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-20261019-r03-topological-continuity-0016","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"091e60f1f14e34c608ba2f5170657d22c79f023b9e12503592c17efc38e22d53","family_id":"qf-ext19-continuity-single-preimage","seed":202610193016,"semantic_template":"qf-ext19-continuity-single-preimage/{\"images\":[3,0,2,1],\"m\":4,\"n\":4,\"opens_x\":[[],[3],[0,3],[0,2,3],[0,1,2,3]],\"opens_y\":[[],[3],[0,3],[0,1,3],[0,1,2,3]],\"target_open\":[]}"},"fields":{"title":"One Declared Target-Open Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 0, 2, 1] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.","answer":{"kind":"expression","value":"(4, 4, [], [], Yes)","explanation":"Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [], [], Yes)."},"skills":["One Declared Target-Open 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-20261019-r03-topological-continuity-0017","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31aa5a702360474bf1f886016f085f71d4bc476a6817d794e7a7eb45f7b61cff","family_id":"qf-ext19-continuity-single-preimage","seed":202610193017,"semantic_template":"qf-ext19-continuity-single-preimage/{\"images\":[0,2,3,1],\"m\":4,\"n\":4,\"opens_x\":[[],[0],[0,2],[0,2,3],[0,1,2,3]],\"opens_y\":[[],[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_open\":[]}"},"fields":{"title":"One Declared Target-Open Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[0, 2, 3, 1] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.","answer":{"kind":"expression","value":"(4, 4, [], [], Yes)","explanation":"Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [], [], Yes)."},"skills":["One Declared Target-Open 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-20261019-r03-topological-continuity-0018","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d75fe661bb781fa95358c567c1970426c9aeb0d4ec2242f2c0e9ce6b9786fbf0","family_id":"qf-ext19-continuity-single-preimage","seed":202610193018,"semantic_template":"qf-ext19-continuity-single-preimage/{\"images\":[1,2,3,0],\"m\":4,\"n\":4,\"opens_x\":[[],[1],[1,2],[1,2,3],[0,1,2,3]],\"opens_y\":[[],[1],[0,1],[0,1,3],[0,1,2,3]],\"target_open\":[0,1]}"},"fields":{"title":"One Declared Target-Open Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]]. The complete total map has images=[1, 2, 3, 0] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[0, 1] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.","answer":{"kind":"expression","value":"(4, 4, [0, 1], [0, 3], No)","explanation":"Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [0, 1], [0, 3], No)."},"skills":["One Declared Target-Open 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-20261019-r03-topological-continuity-0019","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e865ab6a218192604ad5329373a863effebe8b44b32900a1dc1cb13d63dfc038","family_id":"qf-ext19-continuity-single-preimage","seed":202610193019,"semantic_template":"qf-ext19-continuity-single-preimage/{\"images\":[2,1,3,0],\"m\":4,\"n\":4,\"opens_x\":[[],[2],[1,2],[1,2,3],[0,1,2,3]],\"opens_y\":[[],[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_open\":[1]}"},"fields":{"title":"One Declared Target-Open Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[2, 1, 3, 0] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[1] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.","answer":{"kind":"expression","value":"(4, 4, [1], [1], No)","explanation":"Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [1], [1], No)."},"skills":["One Declared Target-Open 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-20261019-r03-topological-continuity-0020","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ba36c886aa2ca825c0fe0d56395579d337dd9c1a3eaf0983ca551fe0a627e98","family_id":"qf-ext19-continuity-single-preimage","seed":202610193020,"semantic_template":"qf-ext19-continuity-single-preimage/{\"images\":[3,1,2,0],\"m\":4,\"n\":4,\"opens_x\":[[],[3],[1,3],[1,2,3],[0,1,2,3]],\"opens_y\":[[],[3],[2,3],[1,2,3],[0,1,2,3]],\"target_open\":[0,1,2,3]}"},"fields":{"title":"One Declared Target-Open Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 1, 2, 0] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[0, 1, 2, 3] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.","answer":{"kind":"expression","value":"(4, 4, [0, 1, 2, 3], [0, 1, 2, 3], Yes)","explanation":"Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [0, 1, 2, 3], [0, 1, 2, 3], Yes)."},"skills":["One Declared Target-Open 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-20261019-r03-topological-continuity-0021","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7408631018b1a0edbe1a454e4cfab9f0c5126e327ea3c84f0f3faf87a32224f","family_id":"qf-ext19-continuity-single-preimage","seed":202610193021,"semantic_template":"qf-ext19-continuity-single-preimage/{\"images\":[0,3,2,1],\"m\":4,\"n\":4,\"opens_x\":[[],[0],[0,3],[0,2,3],[0,1,2,3]],\"opens_y\":[[],[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_open\":[2]}"},"fields":{"title":"One Declared Target-Open Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[0, 3, 2, 1] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[2] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.","answer":{"kind":"expression","value":"(4, 4, [2], [2], No)","explanation":"Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [2], [2], No)."},"skills":["One Declared Target-Open 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-20261019-r03-topological-continuity-0022","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bff617af893b6bac8042ce4a5a1b280d6add8d2f051c09ab65ff6f88de60d30d","family_id":"qf-ext19-continuity-single-preimage","seed":202610193022,"semantic_template":"qf-ext19-continuity-single-preimage/{\"images\":[1,3,2,0],\"m\":4,\"n\":4,\"opens_x\":[[],[1],[1,3],[1,2,3],[0,1,2,3]],\"opens_y\":[[],[1],[1,3],[0,1,3],[0,1,2,3]],\"target_open\":[1]}"},"fields":{"title":"One Declared Target-Open Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]]. The complete total map has images=[1, 3, 2, 0] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[1] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.","answer":{"kind":"expression","value":"(4, 4, [1], [0], No)","explanation":"Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [1], [0], No)."},"skills":["One Declared Target-Open 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-20261019-r03-topological-continuity-0023","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b3869e9ff7bb9ef69d359decc1ccbb692f29746177a4381bd41fcab00f6d974","family_id":"qf-ext19-continuity-single-preimage","seed":202610193023,"semantic_template":"qf-ext19-continuity-single-preimage/{\"images\":[2,3,1,0],\"m\":4,\"n\":4,\"opens_x\":[[],[2],[2,3],[1,2,3],[0,1,2,3]],\"opens_y\":[[],[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_open\":[1,2]}"},"fields":{"title":"One Declared Target-Open Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[2, 3, 1, 0] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[1, 2] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.","answer":{"kind":"expression","value":"(4, 4, [1, 2], [0, 2], No)","explanation":"Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [1, 2], [0, 2], No)."},"skills":["One Declared Target-Open 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-20261019-r03-topological-continuity-0024","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d11f38f97c85c520467c5b26b2b607a83a9e3f51e3d220930aad5328108aaf79","family_id":"qf-ext19-continuity-single-preimage","seed":202610193024,"semantic_template":"qf-ext19-continuity-single-preimage/{\"images\":[3,2,1,0],\"m\":4,\"n\":4,\"opens_x\":[[],[3],[2,3],[1,2,3],[0,1,2,3]],\"opens_y\":[[],[3],[1,3],[1,2,3],[0,1,2,3]],\"target_open\":[1,2,3]}"},"fields":{"title":"One Declared Target-Open Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 2, 1, 0] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[1, 2, 3] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.","answer":{"kind":"expression","value":"(4, 4, [1, 2, 3], [0, 1, 2], No)","explanation":"Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 4, [1, 2, 3], [0, 1, 2], No)."},"skills":["One Declared Target-Open 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-20261019-r03-topological-continuity-0025","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"94fe9c56e85bf08474d3fd4d0d440becac1800f9277a862e59af4dce58bc19d7","family_id":"qf-ext19-continuity-single-preimage","seed":202610193025,"semantic_template":"qf-ext19-continuity-single-preimage/{\"images\":[0,1,2],\"m\":3,\"n\":3,\"opens_x\":[[],[0],[0,1],[0,1,2]],\"opens_y\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2]],\"target_open\":[]}"},"fields":{"title":"One Declared Target-Open Preimage","stem":"Use complete finite topologies X=Fin(3)={0,…,2}, opens_X=[[], [0], [0, 1], [0, 1, 2]] and Y=Fin(3)={0,…,2}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2]]. The complete total map has images=[0, 1, 2] in source order. All entries must type-check; set preimages are not inverse functions. Given target_open=[] is required to be open in Y. Give (n, m, target open, its full preimage, preimage open in X?). This one-set check alone is not a complete continuity audit.","answer":{"kind":"expression","value":"(3, 3, [], [], Yes)","explanation":"Collect every source label mapped into the target set before checking source openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 3, [], [], Yes)."},"skills":["One Declared Target-Open 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-20261019-r03-topological-continuity-0026","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22569181fb81f4840ecacae7771929ff171f9a5a084ec476cea4f96411690208","family_id":"qf-ext19-continuity-full-audit","seed":202610193026,"semantic_template":"qf-ext19-continuity-full-audit/{\"images\":[0,1,2,3],\"m\":4,\"n\":4,\"opens_x\":[[],[0],[0,1],[0,1,2],[0,1,2,3]],\"opens_y\":[[],[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]]}"},"fields":{"title":"Every Target Open Set and Its Full Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[0, 1, 2, 3] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.","answer":{"kind":"expression","value":"([0, 1, 2, 3], [[[], [], Yes], [[0], [0], Yes], [[1], [1], No], [[0, 1], [0, 1], Yes], [[2], [2], No], [[0, 2], [0, 2], No], [[1, 2], [1, 2], No], [[0, 1, 2], [0, 1, 2], Yes], [[3], [3], No], [[0, 3], [0, 3], No], [[1, 3], [1, 3], No], [[0, 1, 3], [0, 1, 3], No], [[2, 3], [2, 3], No], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)","explanation":"Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2, 3], [[[], [], Yes], [[0], [0], Yes], [[1], [1], No], [[0, 1], [0, 1], Yes], [[2], [2], No], [[0, 2], [0, 2], No], [[1, 2], [1, 2], No], [[0, 1, 2], [0, 1, 2], Yes], [[3], [3], No], [[0, 3], [0, 3], No], [[1, 3], [1, 3], No], [[0, 1, 3], [0, 1, 3], No], [[2, 3], [2, 3], No], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)."},"skills":["Every Target Open Set and Its Full 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-20261019-r03-topological-continuity-0027","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f462298b244cc19feb2d3cd777015e38fbe021e8031c9accd3e042866f8e54e","family_id":"qf-ext19-continuity-full-audit","seed":202610193027,"semantic_template":"qf-ext19-continuity-full-audit/{\"images\":[1,0,2,3],\"m\":4,\"n\":4,\"opens_x\":[[],[1],[0,1],[0,1,2],[0,1,2,3]],\"opens_y\":[[],[1],[1,2],[0,1,2],[0,1,2,3]]}"},"fields":{"title":"Every Target Open Set and Its Full Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]]. The complete total map has images=[1, 0, 2, 3] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.","answer":{"kind":"expression","value":"([1, 0, 2, 3], [[[], [], Yes], [[1], [0], No], [[1, 2], [0, 2], No], [[0, 1, 2], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)","explanation":"Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 2, 3], [[[], [], Yes], [[1], [0], No], [[1, 2], [0, 2], No], [[0, 1, 2], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)."},"skills":["Every Target Open Set and Its Full 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-20261019-r03-topological-continuity-0028","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1fc84400fb692438c065447ded63146191d62e6d2b8a774e08a5126c1462427","family_id":"qf-ext19-continuity-full-audit","seed":202610193028,"semantic_template":"qf-ext19-continuity-full-audit/{\"images\":[2,0,1,3],\"m\":4,\"n\":4,\"opens_x\":[[],[2],[0,2],[0,1,2],[0,1,2,3]],\"opens_y\":[[],[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]]}"},"fields":{"title":"Every Target Open Set and Its Full Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [0, 2], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[2, 0, 1, 3] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.","answer":{"kind":"expression","value":"([2, 0, 1, 3], [[[], [], Yes], [[0], [1], No], [[1], [2], Yes], [[0, 1], [1, 2], No], [[2], [0], No], [[0, 2], [0, 1], No], [[1, 2], [0, 2], Yes], [[0, 1, 2], [0, 1, 2], Yes], [[3], [3], No], [[0, 3], [1, 3], No], [[1, 3], [2, 3], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0, 3], No], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)","explanation":"Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 1, 3], [[[], [], Yes], [[0], [1], No], [[1], [2], Yes], [[0, 1], [1, 2], No], [[2], [0], No], [[0, 2], [0, 1], No], [[1, 2], [0, 2], Yes], [[0, 1, 2], [0, 1, 2], Yes], [[3], [3], No], [[0, 3], [1, 3], No], [[1, 3], [2, 3], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0, 3], No], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)."},"skills":["Every Target Open Set and Its Full 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-20261019-r03-topological-continuity-0029","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ead0ec1d94593efae1990cdef87d1834505e74b043203dde87228457683403fb","family_id":"qf-ext19-continuity-full-audit","seed":202610193029,"semantic_template":"qf-ext19-continuity-full-audit/{\"images\":[3,0,1,2],\"m\":4,\"n\":4,\"opens_x\":[[],[3],[0,3],[0,1,3],[0,1,2,3]],\"opens_y\":[[],[3],[0,3],[0,2,3],[0,1,2,3]]}"},"fields":{"title":"Every Target Open Set and Its Full Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 0, 1, 2] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.","answer":{"kind":"expression","value":"([3, 0, 1, 2], [[[], [], Yes], [[3], [0], No], [[0, 3], [0, 1], No], [[0, 2, 3], [0, 1, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)","explanation":"Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 0, 1, 2], [[[], [], Yes], [[3], [0], No], [[0, 3], [0, 1], No], [[0, 2, 3], [0, 1, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)."},"skills":["Every Target Open Set and Its Full 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-20261019-r03-topological-continuity-0030","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ffde0315d41ae5016f20ec56685920ae383059b134415c4adb9138ba48e5e012","family_id":"qf-ext19-continuity-full-audit","seed":202610193030,"semantic_template":"qf-ext19-continuity-full-audit/{\"images\":[0,2,1,3],\"m\":4,\"n\":4,\"opens_x\":[[],[0],[0,2],[0,1,2],[0,1,2,3]],\"opens_y\":[[],[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]]}"},"fields":{"title":"Every Target Open Set and Its Full Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[0, 2, 1, 3] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.","answer":{"kind":"expression","value":"([0, 2, 1, 3], [[[], [], Yes], [[0], [0], Yes], [[1], [2], No], [[0, 1], [0, 2], Yes], [[2], [1], No], [[0, 2], [0, 1], No], [[1, 2], [1, 2], No], [[0, 1, 2], [0, 1, 2], Yes], [[3], [3], No], [[0, 3], [0, 3], No], [[1, 3], [2, 3], No], [[0, 1, 3], [0, 2, 3], No], [[2, 3], [1, 3], No], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)","explanation":"Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 1, 3], [[[], [], Yes], [[0], [0], Yes], [[1], [2], No], [[0, 1], [0, 2], Yes], [[2], [1], No], [[0, 2], [0, 1], No], [[1, 2], [1, 2], No], [[0, 1, 2], [0, 1, 2], Yes], [[3], [3], No], [[0, 3], [0, 3], No], [[1, 3], [2, 3], No], [[0, 1, 3], [0, 2, 3], No], [[2, 3], [1, 3], No], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)."},"skills":["Every Target Open Set and Its Full 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-20261019-r03-topological-continuity-0031","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bcf2f64c4cb029ebac77629b11254a7789c4a117d0518cfeb1d007f135ade695","family_id":"qf-ext19-continuity-full-audit","seed":202610193031,"semantic_template":"qf-ext19-continuity-full-audit/{\"images\":[1,2,0,3],\"m\":4,\"n\":4,\"opens_x\":[[],[1],[1,2],[0,1,2],[0,1,2,3]],\"opens_y\":[[],[1],[0,1],[0,1,2],[0,1,2,3]]}"},"fields":{"title":"Every Target Open Set and Its Full Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]]. The complete total map has images=[1, 2, 0, 3] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.","answer":{"kind":"expression","value":"([1, 2, 0, 3], [[[], [], Yes], [[1], [0], No], [[0, 1], [0, 2], No], [[0, 1, 2], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)","explanation":"Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 0, 3], [[[], [], Yes], [[1], [0], No], [[0, 1], [0, 2], No], [[0, 1, 2], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)."},"skills":["Every Target Open Set and Its Full 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-20261019-r03-topological-continuity-0032","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e8687d10a384c1ddad86a97ac1f1f574f380b45e2cac35603b6a68a129d224c","family_id":"qf-ext19-continuity-full-audit","seed":202610193032,"semantic_template":"qf-ext19-continuity-full-audit/{\"images\":[2,1,0,3],\"m\":4,\"n\":4,\"opens_x\":[[],[2],[1,2],[0,1,2],[0,1,2,3]],\"opens_y\":[[],[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]]}"},"fields":{"title":"Every Target Open Set and Its Full Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [1, 2], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[2, 1, 0, 3] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.","answer":{"kind":"expression","value":"([2, 1, 0, 3], [[[], [], Yes], [[0], [2], Yes], [[1], [1], No], [[0, 1], [1, 2], Yes], [[2], [0], No], [[0, 2], [0, 2], No], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1, 2], Yes], [[3], [3], No], [[0, 3], [2, 3], No], [[1, 3], [1, 3], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0, 3], No], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [0, 1, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)","explanation":"Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 0, 3], [[[], [], Yes], [[0], [2], Yes], [[1], [1], No], [[0, 1], [1, 2], Yes], [[2], [0], No], [[0, 2], [0, 2], No], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1, 2], Yes], [[3], [3], No], [[0, 3], [2, 3], No], [[1, 3], [1, 3], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0, 3], No], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [0, 1, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)."},"skills":["Every Target Open Set and Its Full 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-20261019-r03-topological-continuity-0033","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3e2541dc178eb669c1a4c8debbb8e6cff505df4b4f634feadea6921951c7c70","family_id":"qf-ext19-continuity-full-audit","seed":202610193033,"semantic_template":"qf-ext19-continuity-full-audit/{\"images\":[3,1,0,2],\"m\":4,\"n\":4,\"opens_x\":[[],[3],[1,3],[0,1,3],[0,1,2,3]],\"opens_y\":[[],[3],[2,3],[0,2,3],[0,1,2,3]]}"},"fields":{"title":"Every Target Open Set and Its Full Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 1, 0, 2] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.","answer":{"kind":"expression","value":"([3, 1, 0, 2], [[[], [], Yes], [[3], [0], No], [[2, 3], [0, 3], No], [[0, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)","explanation":"Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 1, 0, 2], [[[], [], Yes], [[3], [0], No], [[2, 3], [0, 3], No], [[0, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)."},"skills":["Every Target Open Set and Its Full 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-20261019-r03-topological-continuity-0034","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e28c5629e834d732a67e8febe0d6af3b0a1c3ad0f0aac421ef1c03966ef97a5f","family_id":"qf-ext19-continuity-full-audit","seed":202610193034,"semantic_template":"qf-ext19-continuity-full-audit/{\"images\":[0,3,1,2],\"m\":4,\"n\":4,\"opens_x\":[[],[0],[0,3],[0,1,3],[0,1,2,3]],\"opens_y\":[[],[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]]}"},"fields":{"title":"Every Target Open Set and Its Full Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[0, 3, 1, 2] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.","answer":{"kind":"expression","value":"([0, 3, 1, 2], [[[], [], Yes], [[0], [0], Yes], [[1], [2], No], [[0, 1], [0, 2], No], [[2], [3], No], [[0, 2], [0, 3], Yes], [[1, 2], [2, 3], No], [[0, 1, 2], [0, 2, 3], No], [[3], [1], No], [[0, 3], [0, 1], No], [[1, 3], [1, 2], No], [[0, 1, 3], [0, 1, 2], No], [[2, 3], [1, 3], No], [[0, 2, 3], [0, 1, 3], Yes], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)","explanation":"Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 3, 1, 2], [[[], [], Yes], [[0], [0], Yes], [[1], [2], No], [[0, 1], [0, 2], No], [[2], [3], No], [[0, 2], [0, 3], Yes], [[1, 2], [2, 3], No], [[0, 1, 2], [0, 2, 3], No], [[3], [1], No], [[0, 3], [0, 1], No], [[1, 3], [1, 2], No], [[0, 1, 3], [0, 1, 2], No], [[2, 3], [1, 3], No], [[0, 2, 3], [0, 1, 3], Yes], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)."},"skills":["Every Target Open Set and Its Full 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-20261019-r03-topological-continuity-0035","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"117edc07d4eeba9f37d338a8e1805615990617d6356060da1e3516a6aeb21986","family_id":"qf-ext19-continuity-full-audit","seed":202610193035,"semantic_template":"qf-ext19-continuity-full-audit/{\"images\":[1,3,0,2],\"m\":4,\"n\":4,\"opens_x\":[[],[1],[1,3],[0,1,3],[0,1,2,3]],\"opens_y\":[[],[1],[1,3],[1,2,3],[0,1,2,3]]}"},"fields":{"title":"Every Target Open Set and Its Full Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[1, 3, 0, 2] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.","answer":{"kind":"expression","value":"([1, 3, 0, 2], [[[], [], Yes], [[1], [0], No], [[1, 3], [0, 1], No], [[1, 2, 3], [0, 1, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)","explanation":"Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 3, 0, 2], [[[], [], Yes], [[1], [0], No], [[1, 3], [0, 1], No], [[1, 2, 3], [0, 1, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)."},"skills":["Every Target Open Set and Its Full 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-20261019-r03-topological-continuity-0036","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2cb9fcbabbeac54a5820004363475c6b38a0093135bfe133ec0c85db330f72c","family_id":"qf-ext19-continuity-full-audit","seed":202610193036,"semantic_template":"qf-ext19-continuity-full-audit/{\"images\":[2,3,0,1],\"m\":4,\"n\":4,\"opens_x\":[[],[2],[2,3],[0,2,3],[0,1,2,3]],\"opens_y\":[[],[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]]}"},"fields":{"title":"Every Target Open Set and Its Full Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[2, 3, 0, 1] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.","answer":{"kind":"expression","value":"([2, 3, 0, 1], [[[], [], Yes], [[0], [2], Yes], [[1], [3], No], [[0, 1], [2, 3], Yes], [[2], [0], No], [[0, 2], [0, 2], No], [[1, 2], [0, 3], No], [[0, 1, 2], [0, 2, 3], Yes], [[3], [1], No], [[0, 3], [1, 2], No], [[1, 3], [1, 3], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0, 1], No], [[0, 2, 3], [0, 1, 2], No], [[1, 2, 3], [0, 1, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)","explanation":"Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 3, 0, 1], [[[], [], Yes], [[0], [2], Yes], [[1], [3], No], [[0, 1], [2, 3], Yes], [[2], [0], No], [[0, 2], [0, 2], No], [[1, 2], [0, 3], No], [[0, 1, 2], [0, 2, 3], Yes], [[3], [1], No], [[0, 3], [1, 2], No], [[1, 3], [1, 3], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0, 1], No], [[0, 2, 3], [0, 1, 2], No], [[1, 2, 3], [0, 1, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)."},"skills":["Every Target Open Set and Its Full 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-20261019-r03-topological-continuity-0037","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"733242cd226c8dec1858bf8ca601e7c3e7e80b8961a1869f67624812676c6339","family_id":"qf-ext19-continuity-full-audit","seed":202610193037,"semantic_template":"qf-ext19-continuity-full-audit/{\"images\":[3,2,0,1],\"m\":4,\"n\":4,\"opens_x\":[[],[3],[2,3],[0,2,3],[0,1,2,3]],\"opens_y\":[[],[3],[1,3],[0,1,3],[0,1,2,3]]}"},"fields":{"title":"Every Target Open Set and Its Full Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 2, 0, 1] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.","answer":{"kind":"expression","value":"([3, 2, 0, 1], [[[], [], Yes], [[3], [0], No], [[1, 3], [0, 3], No], [[0, 1, 3], [0, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)","explanation":"Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 2, 0, 1], [[[], [], Yes], [[3], [0], No], [[1, 3], [0, 3], No], [[0, 1, 3], [0, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)."},"skills":["Every Target Open Set and Its Full 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-20261019-r03-topological-continuity-0038","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"475e11c27589c8ff2f8a8d4ac5b99c40e00557e28007c211d171fe79c9d0e04b","family_id":"qf-ext19-continuity-full-audit","seed":202610193038,"semantic_template":"qf-ext19-continuity-full-audit/{\"images\":[0,1,3,2],\"m\":4,\"n\":4,\"opens_x\":[[],[0],[0,1],[0,1,3],[0,1,2,3]],\"opens_y\":[[],[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]]}"},"fields":{"title":"Every Target Open Set and Its Full Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[0, 1, 3, 2] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.","answer":{"kind":"expression","value":"([0, 1, 3, 2], [[[], [], Yes], [[0], [0], Yes], [[1], [1], No], [[0, 1], [0, 1], Yes], [[2], [3], No], [[0, 2], [0, 3], No], [[1, 2], [1, 3], No], [[0, 1, 2], [0, 1, 3], Yes], [[3], [2], No], [[0, 3], [0, 2], No], [[1, 3], [1, 2], No], [[0, 1, 3], [0, 1, 2], No], [[2, 3], [2, 3], No], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)","explanation":"Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 3, 2], [[[], [], Yes], [[0], [0], Yes], [[1], [1], No], [[0, 1], [0, 1], Yes], [[2], [3], No], [[0, 2], [0, 3], No], [[1, 2], [1, 3], No], [[0, 1, 2], [0, 1, 3], Yes], [[3], [2], No], [[0, 3], [0, 2], No], [[1, 3], [1, 2], No], [[0, 1, 3], [0, 1, 2], No], [[2, 3], [2, 3], No], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)."},"skills":["Every Target Open Set and Its Full 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-20261019-r03-topological-continuity-0039","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3800b56ed584f41aca21a846c96b38ae545ce231797443d1e779a2eb09668708","family_id":"qf-ext19-continuity-full-audit","seed":202610193039,"semantic_template":"qf-ext19-continuity-full-audit/{\"images\":[1,0,3,2],\"m\":4,\"n\":4,\"opens_x\":[[],[1],[0,1],[0,1,3],[0,1,2,3]],\"opens_y\":[[],[1],[1,2],[1,2,3],[0,1,2,3]]}"},"fields":{"title":"Every Target Open Set and Its Full Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[1, 0, 3, 2] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.","answer":{"kind":"expression","value":"([1, 0, 3, 2], [[[], [], Yes], [[1], [0], No], [[1, 2], [0, 3], No], [[1, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)","explanation":"Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 3, 2], [[[], [], Yes], [[1], [0], No], [[1, 2], [0, 3], No], [[1, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)."},"skills":["Every Target Open Set and Its Full 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-20261019-r03-topological-continuity-0040","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"561a352973d01ed697af7b73fe2f489340ab99952a6da71339e46a9d66ae35a8","family_id":"qf-ext19-continuity-full-audit","seed":202610193040,"semantic_template":"qf-ext19-continuity-full-audit/{\"images\":[2,0,3,1],\"m\":4,\"n\":4,\"opens_x\":[[],[2],[0,2],[0,2,3],[0,1,2,3]],\"opens_y\":[[],[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]]}"},"fields":{"title":"Every Target Open Set and Its Full Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[2, 0, 3, 1] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.","answer":{"kind":"expression","value":"([2, 0, 3, 1], [[[], [], Yes], [[0], [1], No], [[1], [3], No], [[0, 1], [1, 3], No], [[2], [0], No], [[0, 2], [0, 1], No], [[1, 2], [0, 3], No], [[0, 1, 2], [0, 1, 3], No], [[3], [2], Yes], [[0, 3], [1, 2], No], [[1, 3], [2, 3], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0, 2], Yes], [[0, 2, 3], [0, 1, 2], No], [[1, 2, 3], [0, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)","explanation":"Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 3, 1], [[[], [], Yes], [[0], [1], No], [[1], [3], No], [[0, 1], [1, 3], No], [[2], [0], No], [[0, 2], [0, 1], No], [[1, 2], [0, 3], No], [[0, 1, 2], [0, 1, 3], No], [[3], [2], Yes], [[0, 3], [1, 2], No], [[1, 3], [2, 3], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0, 2], Yes], [[0, 2, 3], [0, 1, 2], No], [[1, 2, 3], [0, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)."},"skills":["Every Target Open Set and Its Full 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-20261019-r03-topological-continuity-0041","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2557bffe261eacf76209de7b42c357d6ede81b61ebe8ee44e8c9d104274b579d","family_id":"qf-ext19-continuity-full-audit","seed":202610193041,"semantic_template":"qf-ext19-continuity-full-audit/{\"images\":[3,0,2,1],\"m\":4,\"n\":4,\"opens_x\":[[],[3],[0,3],[0,2,3],[0,1,2,3]],\"opens_y\":[[],[3],[0,3],[0,1,3],[0,1,2,3]]}"},"fields":{"title":"Every Target Open Set and Its Full Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 0, 2, 1] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.","answer":{"kind":"expression","value":"([3, 0, 2, 1], [[[], [], Yes], [[3], [0], No], [[0, 3], [0, 1], No], [[0, 1, 3], [0, 1, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)","explanation":"Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 0, 2, 1], [[[], [], Yes], [[3], [0], No], [[0, 3], [0, 1], No], [[0, 1, 3], [0, 1, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)."},"skills":["Every Target Open Set and Its Full 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-20261019-r03-topological-continuity-0042","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a0af5af02eca922d5cd04093f34b5ca391748085eb9279520ec406c59aacd97","family_id":"qf-ext19-continuity-full-audit","seed":202610193042,"semantic_template":"qf-ext19-continuity-full-audit/{\"images\":[0,2,3,1],\"m\":4,\"n\":4,\"opens_x\":[[],[0],[0,2],[0,2,3],[0,1,2,3]],\"opens_y\":[[],[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]]}"},"fields":{"title":"Every Target Open Set and Its Full Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[0, 2, 3, 1] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.","answer":{"kind":"expression","value":"([0, 2, 3, 1], [[[], [], Yes], [[0], [0], Yes], [[1], [3], No], [[0, 1], [0, 3], No], [[2], [1], No], [[0, 2], [0, 1], No], [[1, 2], [1, 3], No], [[0, 1, 2], [0, 1, 3], No], [[3], [2], No], [[0, 3], [0, 2], Yes], [[1, 3], [2, 3], No], [[0, 1, 3], [0, 2, 3], Yes], [[2, 3], [1, 2], No], [[0, 2, 3], [0, 1, 2], No], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)","explanation":"Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 3, 1], [[[], [], Yes], [[0], [0], Yes], [[1], [3], No], [[0, 1], [0, 3], No], [[2], [1], No], [[0, 2], [0, 1], No], [[1, 2], [1, 3], No], [[0, 1, 2], [0, 1, 3], No], [[3], [2], No], [[0, 3], [0, 2], Yes], [[1, 3], [2, 3], No], [[0, 1, 3], [0, 2, 3], Yes], [[2, 3], [1, 2], No], [[0, 2, 3], [0, 1, 2], No], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)."},"skills":["Every Target Open Set and Its Full 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-20261019-r03-topological-continuity-0043","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96132341a31900e747bdd6d04fdd3b5a8f91227928cb138ad9e0a29202fd8e21","family_id":"qf-ext19-continuity-full-audit","seed":202610193043,"semantic_template":"qf-ext19-continuity-full-audit/{\"images\":[1,2,3,0],\"m\":4,\"n\":4,\"opens_x\":[[],[1],[1,2],[1,2,3],[0,1,2,3]],\"opens_y\":[[],[1],[0,1],[0,1,3],[0,1,2,3]]}"},"fields":{"title":"Every Target Open Set and Its Full Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]]. The complete total map has images=[1, 2, 3, 0] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.","answer":{"kind":"expression","value":"([1, 2, 3, 0], [[[], [], Yes], [[1], [0], No], [[0, 1], [0, 3], No], [[0, 1, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)","explanation":"Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 3, 0], [[[], [], Yes], [[1], [0], No], [[0, 1], [0, 3], No], [[0, 1, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)."},"skills":["Every Target Open Set and Its Full 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-20261019-r03-topological-continuity-0044","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cdd4f9c6b2db2f2423dbf4933a533e778da0015836e67e57af359517e15a1cca","family_id":"qf-ext19-continuity-full-audit","seed":202610193044,"semantic_template":"qf-ext19-continuity-full-audit/{\"images\":[2,1,3,0],\"m\":4,\"n\":4,\"opens_x\":[[],[2],[1,2],[1,2,3],[0,1,2,3]],\"opens_y\":[[],[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]]}"},"fields":{"title":"Every Target Open Set and Its Full Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[2, 1, 3, 0] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.","answer":{"kind":"expression","value":"([2, 1, 3, 0], [[[], [], Yes], [[0], [3], No], [[1], [1], No], [[0, 1], [1, 3], No], [[2], [0], No], [[0, 2], [0, 3], No], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1, 3], No], [[3], [2], Yes], [[0, 3], [2, 3], No], [[1, 3], [1, 2], Yes], [[0, 1, 3], [1, 2, 3], Yes], [[2, 3], [0, 2], No], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)","explanation":"Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 3, 0], [[[], [], Yes], [[0], [3], No], [[1], [1], No], [[0, 1], [1, 3], No], [[2], [0], No], [[0, 2], [0, 3], No], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1, 3], No], [[3], [2], Yes], [[0, 3], [2, 3], No], [[1, 3], [1, 2], Yes], [[0, 1, 3], [1, 2, 3], Yes], [[2, 3], [0, 2], No], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)."},"skills":["Every Target Open Set and Its Full 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-20261019-r03-topological-continuity-0045","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7bb7baba5eebf5602fd3625dfcc5d803494e29fed269973606f670c6621b3a36","family_id":"qf-ext19-continuity-full-audit","seed":202610193045,"semantic_template":"qf-ext19-continuity-full-audit/{\"images\":[3,1,2,0],\"m\":4,\"n\":4,\"opens_x\":[[],[3],[1,3],[1,2,3],[0,1,2,3]],\"opens_y\":[[],[3],[2,3],[1,2,3],[0,1,2,3]]}"},"fields":{"title":"Every Target Open Set and Its Full Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 1, 2, 0] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.","answer":{"kind":"expression","value":"([3, 1, 2, 0], [[[], [], Yes], [[3], [0], No], [[2, 3], [0, 2], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)","explanation":"Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 1, 2, 0], [[[], [], Yes], [[3], [0], No], [[2, 3], [0, 2], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)."},"skills":["Every Target Open Set and Its Full 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-20261019-r03-topological-continuity-0046","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8dae37b3ffe9202eec29205163d39b001df310cece8608241568b3b266900f76","family_id":"qf-ext19-continuity-full-audit","seed":202610193046,"semantic_template":"qf-ext19-continuity-full-audit/{\"images\":[0,3,2,1],\"m\":4,\"n\":4,\"opens_x\":[[],[0],[0,3],[0,2,3],[0,1,2,3]],\"opens_y\":[[],[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]]}"},"fields":{"title":"Every Target Open Set and Its Full Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[0, 3, 2, 1] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.","answer":{"kind":"expression","value":"([0, 3, 2, 1], [[[], [], Yes], [[0], [0], Yes], [[1], [3], No], [[0, 1], [0, 3], Yes], [[2], [2], No], [[0, 2], [0, 2], No], [[1, 2], [2, 3], No], [[0, 1, 2], [0, 2, 3], Yes], [[3], [1], No], [[0, 3], [0, 1], No], [[1, 3], [1, 3], No], [[0, 1, 3], [0, 1, 3], No], [[2, 3], [1, 2], No], [[0, 2, 3], [0, 1, 2], No], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)","explanation":"Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 3, 2, 1], [[[], [], Yes], [[0], [0], Yes], [[1], [3], No], [[0, 1], [0, 3], Yes], [[2], [2], No], [[0, 2], [0, 2], No], [[1, 2], [2, 3], No], [[0, 1, 2], [0, 2, 3], Yes], [[3], [1], No], [[0, 3], [0, 1], No], [[1, 3], [1, 3], No], [[0, 1, 3], [0, 1, 3], No], [[2, 3], [1, 2], No], [[0, 2, 3], [0, 1, 2], No], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)."},"skills":["Every Target Open Set and Its Full 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-20261019-r03-topological-continuity-0047","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3f987f166601ab985f31fcd5b08ac6830753e768bbda22c9bb842d8913e0d7b","family_id":"qf-ext19-continuity-full-audit","seed":202610193047,"semantic_template":"qf-ext19-continuity-full-audit/{\"images\":[1,3,2,0],\"m\":4,\"n\":4,\"opens_x\":[[],[1],[1,3],[1,2,3],[0,1,2,3]],\"opens_y\":[[],[1],[1,3],[0,1,3],[0,1,2,3]]}"},"fields":{"title":"Every Target Open Set and Its Full Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]]. The complete total map has images=[1, 3, 2, 0] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.","answer":{"kind":"expression","value":"([1, 3, 2, 0], [[[], [], Yes], [[1], [0], No], [[1, 3], [0, 1], No], [[0, 1, 3], [0, 1, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)","explanation":"Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 3, 2, 0], [[[], [], Yes], [[1], [0], No], [[1, 3], [0, 1], No], [[0, 1, 3], [0, 1, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)."},"skills":["Every Target Open Set and Its Full 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-20261019-r03-topological-continuity-0048","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e659be4cee684e7a66c88224c5ec1f3fd3f9c512418c4a21b1e28c0d6eddb39","family_id":"qf-ext19-continuity-full-audit","seed":202610193048,"semantic_template":"qf-ext19-continuity-full-audit/{\"images\":[2,3,1,0],\"m\":4,\"n\":4,\"opens_x\":[[],[2],[2,3],[1,2,3],[0,1,2,3]],\"opens_y\":[[],[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]]}"},"fields":{"title":"Every Target Open Set and Its Full Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[2, 3, 1, 0] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.","answer":{"kind":"expression","value":"([2, 3, 1, 0], [[[], [], Yes], [[0], [3], No], [[1], [2], Yes], [[0, 1], [2, 3], Yes], [[2], [0], No], [[0, 2], [0, 3], No], [[1, 2], [0, 2], No], [[0, 1, 2], [0, 2, 3], No], [[3], [1], No], [[0, 3], [1, 3], No], [[1, 3], [1, 2], No], [[0, 1, 3], [1, 2, 3], Yes], [[2, 3], [0, 1], No], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)","explanation":"Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 3, 1, 0], [[[], [], Yes], [[0], [3], No], [[1], [2], Yes], [[0, 1], [2, 3], Yes], [[2], [0], No], [[0, 2], [0, 3], No], [[1, 2], [0, 2], No], [[0, 1, 2], [0, 2, 3], No], [[3], [1], No], [[0, 3], [1, 3], No], [[1, 3], [1, 2], No], [[0, 1, 3], [1, 2, 3], Yes], [[2, 3], [0, 1], No], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)."},"skills":["Every Target Open Set and Its Full 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-20261019-r03-topological-continuity-0049","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"770557e09db12a8a4ff35a4a76f85d2158333f808ccff906cda5d740bb98a7d8","family_id":"qf-ext19-continuity-full-audit","seed":202610193049,"semantic_template":"qf-ext19-continuity-full-audit/{\"images\":[3,2,1,0],\"m\":4,\"n\":4,\"opens_x\":[[],[3],[2,3],[1,2,3],[0,1,2,3]],\"opens_y\":[[],[3],[1,3],[1,2,3],[0,1,2,3]]}"},"fields":{"title":"Every Target Open Set and Its Full Preimage","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 2, 1, 0] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.","answer":{"kind":"expression","value":"([3, 2, 1, 0], [[[], [], Yes], [[3], [0], No], [[1, 3], [0, 2], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)","explanation":"Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 2, 1, 0], [[[], [], Yes], [[3], [0], No], [[1, 3], [0, 2], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)."},"skills":["Every Target Open Set and Its Full 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-20261019-r03-topological-continuity-0050","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67020d828b30f612ae5a1646bf4b1a939fcb98c3881a934534b3a05bd3caebce","family_id":"qf-ext19-continuity-full-audit","seed":202610193050,"semantic_template":"qf-ext19-continuity-full-audit/{\"images\":[0,1,2],\"m\":3,\"n\":3,\"opens_x\":[[],[0],[0,1],[0,1,2]],\"opens_y\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2]]}"},"fields":{"title":"Every Target Open Set and Its Full Preimage","stem":"Use complete finite topologies X=Fin(3)={0,…,2}, opens_X=[[], [0], [0, 1], [0, 1, 2]] and Y=Fin(3)={0,…,2}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2]]. The complete total map has images=[0, 1, 2] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, every row [target open, full preimage, source-open?], continuity?). Inspect all target opens in subset-mask order.","answer":{"kind":"expression","value":"([0, 1, 2], [[[], [], Yes], [[0], [0], Yes], [[1], [1], No], [[0, 1], [0, 1], Yes], [[2], [2], No], [[0, 2], [0, 2], No], [[1, 2], [1, 2], No], [[0, 1, 2], [0, 1, 2], Yes]], No)","explanation":"Continuity requires all the open-preimage rows; sampling points or one open set is insufficient. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2], [[[], [], Yes], [[0], [0], Yes], [[1], [1], No], [[0, 1], [0, 1], Yes], [[2], [2], No], [[0, 2], [0, 2], No], [[1, 2], [1, 2], No], [[0, 1, 2], [0, 1, 2], Yes]], No)."},"skills":["Every Target Open Set and Its Full 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-20261019-r03-topological-continuity-0051","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1fc8ba028fb7ea116bfdb2f671cdcb78ac65bf372d7781ad411bab7c39482777","family_id":"qf-ext19-continuity-composition","seed":202610193051,"semantic_template":"qf-ext19-continuity-composition/{\"images_f\":[0,1,2,3],\"images_g\":[0,1,2,3],\"k\":4,\"m\":4,\"n\":4,\"opens_x\":[[],[0],[0,1],[0,1,2],[0,1,2,3]],\"opens_y\":[[],[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]],\"opens_z\":[[],[0,1,2,3]]}"},"fields":{"title":"Two Typed Maps and Independently Audited Composition","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[0, 1, 2, 3] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[0, 1, 2, 3]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.","answer":{"kind":"expression","value":"([0, 1, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])","explanation":"Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])."},"skills":["Two Typed Maps and Independently Audited Composition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0052","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eecf72ac2fcbb4a42b5cbaa19285698cc35fbfdb4d5dc2d074c383637b665760","family_id":"qf-ext19-continuity-composition","seed":202610193052,"semantic_template":"qf-ext19-continuity-composition/{\"images_f\":[1,0,2,3],\"images_g\":[3,1,0,2],\"k\":4,\"m\":4,\"n\":4,\"opens_x\":[[],[1],[0,1],[0,1,2],[0,1,2,3]],\"opens_y\":[[],[1],[1,2],[0,1,2],[0,1,2,3]],\"opens_z\":[[],[0,1,2,3]]}"},"fields":{"title":"Two Typed Maps and Independently Audited Composition","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]]. The complete total map has images=[1, 0, 2, 3] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[3, 1, 0, 2]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.","answer":{"kind":"expression","value":"([1, 0, 2, 3], [3, 1, 0, 2], [1, 3, 0, 2], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])","explanation":"Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 2, 3], [3, 1, 0, 2], [1, 3, 0, 2], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])."},"skills":["Two Typed Maps and Independently Audited Composition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0053","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ef92ef1eafbd12dc69de5485bcdba14ad5cf9b3af73247dec6c7cf2668f41b5","family_id":"qf-ext19-continuity-composition","seed":202610193053,"semantic_template":"qf-ext19-continuity-composition/{\"images_f\":[2,0,1,3],\"images_g\":[2,0,3,1],\"k\":4,\"m\":4,\"n\":4,\"opens_x\":[[],[2],[0,2],[0,1,2],[0,1,2,3]],\"opens_y\":[[],[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]],\"opens_z\":[[],[0,1,2,3]]}"},"fields":{"title":"Two Typed Maps and Independently Audited Composition","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [0, 2], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[2, 0, 1, 3] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[2, 0, 3, 1]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.","answer":{"kind":"expression","value":"([2, 0, 1, 3], [2, 0, 3, 1], [3, 2, 0, 1], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])","explanation":"Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 1, 3], [2, 0, 3, 1], [3, 2, 0, 1], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])."},"skills":["Two Typed Maps and Independently Audited Composition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0054","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a40d021eff143d88b7cd259c29874b919961c57031694a27ed9df55ff92fe4e2","family_id":"qf-ext19-continuity-composition","seed":202610193054,"semantic_template":"qf-ext19-continuity-composition/{\"images_f\":[3,0,1,2],\"images_g\":[1,3,2,0],\"k\":4,\"m\":4,\"n\":4,\"opens_x\":[[],[3],[0,3],[0,1,3],[0,1,2,3]],\"opens_y\":[[],[3],[0,3],[0,2,3],[0,1,2,3]],\"opens_z\":[[],[0,1,2,3]]}"},"fields":{"title":"Two Typed Maps and Independently Audited Composition","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 0, 1, 2] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[1, 3, 2, 0]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.","answer":{"kind":"expression","value":"([3, 0, 1, 2], [1, 3, 2, 0], [0, 1, 3, 2], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])","explanation":"Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 0, 1, 2], [1, 3, 2, 0], [0, 1, 3, 2], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])."},"skills":["Two Typed Maps and Independently Audited Composition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0055","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa8e21c2619d24ea69057176b2f8b511d4ef9e87b1de49fc29d21adbb1cfc668","family_id":"qf-ext19-continuity-composition","seed":202610193055,"semantic_template":"qf-ext19-continuity-composition/{\"images_f\":[0,2,1,3],\"images_g\":[0,2,1,3],\"k\":4,\"m\":4,\"n\":4,\"opens_x\":[[],[0],[0,2],[0,1,2],[0,1,2,3]],\"opens_y\":[[],[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]],\"opens_z\":[[],[0,1,2,3]]}"},"fields":{"title":"Two Typed Maps and Independently Audited Composition","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[0, 2, 1, 3] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[0, 2, 1, 3]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.","answer":{"kind":"expression","value":"([0, 2, 1, 3], [0, 2, 1, 3], [0, 1, 2, 3], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])","explanation":"Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 1, 3], [0, 2, 1, 3], [0, 1, 2, 3], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])."},"skills":["Two Typed Maps and Independently Audited Composition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0056","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f553df6bb966dafe02d7dd5e14369c2c4c50c94872247142ec94466ee79dfa30","family_id":"qf-ext19-continuity-composition","seed":202610193056,"semantic_template":"qf-ext19-continuity-composition/{\"images_f\":[1,2,0,3],\"images_g\":[3,2,0,1],\"k\":4,\"m\":4,\"n\":4,\"opens_x\":[[],[1],[1,2],[0,1,2],[0,1,2,3]],\"opens_y\":[[],[1],[0,1],[0,1,2],[0,1,2,3]],\"opens_z\":[[],[0,1,2,3]]}"},"fields":{"title":"Two Typed Maps and Independently Audited Composition","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]]. The complete total map has images=[1, 2, 0, 3] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[3, 2, 0, 1]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.","answer":{"kind":"expression","value":"([1, 2, 0, 3], [3, 2, 0, 1], [2, 0, 3, 1], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])","explanation":"Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 0, 3], [3, 2, 0, 1], [2, 0, 3, 1], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])."},"skills":["Two Typed Maps and Independently Audited Composition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0057","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"402b706110496f53d244347adbad7d28238214c7f7b2d8cecec2253be27201d9","family_id":"qf-ext19-continuity-composition","seed":202610193057,"semantic_template":"qf-ext19-continuity-composition/{\"images_f\":[2,1,0,3],\"images_g\":[2,1,3,0],\"k\":4,\"m\":4,\"n\":4,\"opens_x\":[[],[2],[1,2],[0,1,2],[0,1,2,3]],\"opens_y\":[[],[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]],\"opens_z\":[[],[0,1,2,3]]}"},"fields":{"title":"Two Typed Maps and Independently Audited Composition","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [1, 2], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[2, 1, 0, 3] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[2, 1, 3, 0]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.","answer":{"kind":"expression","value":"([2, 1, 0, 3], [2, 1, 3, 0], [3, 1, 2, 0], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])","explanation":"Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 0, 3], [2, 1, 3, 0], [3, 1, 2, 0], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])."},"skills":["Two Typed Maps and Independently Audited Composition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0058","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b81c970781c18dd4303481ab344b95863cd12710d645379d550d291a0388484","family_id":"qf-ext19-continuity-composition","seed":202610193058,"semantic_template":"qf-ext19-continuity-composition/{\"images_f\":[3,1,0,2],\"images_g\":[1,0,2,3],\"k\":4,\"m\":4,\"n\":4,\"opens_x\":[[],[3],[1,3],[0,1,3],[0,1,2,3]],\"opens_y\":[[],[3],[2,3],[0,2,3],[0,1,2,3]],\"opens_z\":[[],[0,1,2,3]]}"},"fields":{"title":"Two Typed Maps and Independently Audited Composition","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 1, 0, 2] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[1, 0, 2, 3]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.","answer":{"kind":"expression","value":"([3, 1, 0, 2], [1, 0, 2, 3], [3, 0, 1, 2], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])","explanation":"Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 1, 0, 2], [1, 0, 2, 3], [3, 0, 1, 2], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])."},"skills":["Two Typed Maps and Independently Audited Composition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0059","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95f5ec19fffcda7787931566772d8c3adfdf0081c2effa8ec3ae097563801a50","family_id":"qf-ext19-continuity-composition","seed":202610193059,"semantic_template":"qf-ext19-continuity-composition/{\"images_f\":[0,3,1,2],\"images_g\":[0,3,1,2],\"k\":4,\"m\":4,\"n\":4,\"opens_x\":[[],[0],[0,3],[0,1,3],[0,1,2,3]],\"opens_y\":[[],[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]],\"opens_z\":[[],[0,1,2,3]]}"},"fields":{"title":"Two Typed Maps and Independently Audited Composition","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[0, 3, 1, 2] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[0, 3, 1, 2]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.","answer":{"kind":"expression","value":"([0, 3, 1, 2], [0, 3, 1, 2], [0, 2, 3, 1], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])","explanation":"Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 3, 1, 2], [0, 3, 1, 2], [0, 2, 3, 1], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])."},"skills":["Two Typed Maps and Independently Audited Composition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0060","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2018bd4011c484660f45d3e403b739bfdcb8e6f392ea9fc8a4c098c88469bb7c","family_id":"qf-ext19-continuity-composition","seed":202610193060,"semantic_template":"qf-ext19-continuity-composition/{\"images_f\":[1,3,0,2],\"images_g\":[3,0,2,1],\"k\":4,\"m\":4,\"n\":4,\"opens_x\":[[],[1],[1,3],[0,1,3],[0,1,2,3]],\"opens_y\":[[],[1],[1,3],[1,2,3],[0,1,2,3]],\"opens_z\":[[],[0,1,2,3]]}"},"fields":{"title":"Two Typed Maps and Independently Audited Composition","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[1, 3, 0, 2] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[3, 0, 2, 1]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.","answer":{"kind":"expression","value":"([1, 3, 0, 2], [3, 0, 2, 1], [0, 1, 3, 2], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])","explanation":"Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 3, 0, 2], [3, 0, 2, 1], [0, 1, 3, 2], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])."},"skills":["Two Typed Maps and Independently Audited Composition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0061","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aafc57cd0b39b63e7c3ed5434d8c894f9d56900d670137fe6b801cabe55694f0","family_id":"qf-ext19-continuity-composition","seed":202610193061,"semantic_template":"qf-ext19-continuity-composition/{\"images_f\":[2,3,0,1],\"images_g\":[2,3,1,0],\"k\":4,\"m\":4,\"n\":4,\"opens_x\":[[],[2],[2,3],[0,2,3],[0,1,2,3]],\"opens_y\":[[],[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]],\"opens_z\":[[],[0,1,2,3]]}"},"fields":{"title":"Two Typed Maps and Independently Audited Composition","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[2, 3, 0, 1] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[2, 3, 1, 0]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.","answer":{"kind":"expression","value":"([2, 3, 0, 1], [2, 3, 1, 0], [1, 0, 2, 3], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])","explanation":"Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 3, 0, 1], [2, 3, 1, 0], [1, 0, 2, 3], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])."},"skills":["Two Typed Maps and Independently Audited Composition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0062","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9bda1c33558b2fc600849c1a95762240a8700bd7e0b603c90bab8394cfccc7d3","family_id":"qf-ext19-continuity-composition","seed":202610193062,"semantic_template":"qf-ext19-continuity-composition/{\"images_f\":[3,2,0,1],\"images_g\":[1,2,0,3],\"k\":4,\"m\":4,\"n\":4,\"opens_x\":[[],[3],[2,3],[0,2,3],[0,1,2,3]],\"opens_y\":[[],[3],[1,3],[0,1,3],[0,1,2,3]],\"opens_z\":[[],[0,1,2,3]]}"},"fields":{"title":"Two Typed Maps and Independently Audited Composition","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 2, 0, 1] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[1, 2, 0, 3]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.","answer":{"kind":"expression","value":"([3, 2, 0, 1], [1, 2, 0, 3], [3, 0, 1, 2], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])","explanation":"Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 2, 0, 1], [1, 2, 0, 3], [3, 0, 1, 2], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])."},"skills":["Two Typed Maps and Independently Audited Composition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0063","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92fa8fe204449b16cc6e22d51437ded6231cfbb6504eddae368e73c7a4dfa6f2","family_id":"qf-ext19-continuity-composition","seed":202610193063,"semantic_template":"qf-ext19-continuity-composition/{\"images_f\":[0,1,3,2],\"images_g\":[0,1,3,2],\"k\":4,\"m\":4,\"n\":4,\"opens_x\":[[],[0],[0,1],[0,1,3],[0,1,2,3]],\"opens_y\":[[],[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]],\"opens_z\":[[],[0,1,2,3]]}"},"fields":{"title":"Two Typed Maps and Independently Audited Composition","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[0, 1, 3, 2] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[0, 1, 3, 2]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.","answer":{"kind":"expression","value":"([0, 1, 3, 2], [0, 1, 3, 2], [0, 1, 2, 3], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])","explanation":"Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 3, 2], [0, 1, 3, 2], [0, 1, 2, 3], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])."},"skills":["Two Typed Maps and Independently Audited Composition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0064","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c265a263ce63fd61b3d475fd9e63b9b712220aac0f3ddb528219256ae75b539","family_id":"qf-ext19-continuity-composition","seed":202610193064,"semantic_template":"qf-ext19-continuity-composition/{\"images_f\":[1,0,3,2],\"images_g\":[3,1,2,0],\"k\":4,\"m\":4,\"n\":4,\"opens_x\":[[],[1],[0,1],[0,1,3],[0,1,2,3]],\"opens_y\":[[],[1],[1,2],[1,2,3],[0,1,2,3]],\"opens_z\":[[],[0,1,2,3]]}"},"fields":{"title":"Two Typed Maps and Independently Audited Composition","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[1, 0, 3, 2] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[3, 1, 2, 0]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.","answer":{"kind":"expression","value":"([1, 0, 3, 2], [3, 1, 2, 0], [1, 3, 0, 2], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])","explanation":"Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 3, 2], [3, 1, 2, 0], [1, 3, 0, 2], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])."},"skills":["Two Typed Maps and Independently Audited Composition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0065","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b695a99fa64012e23d86c41ab8bdffaa588659a656d84d216afced9ea97c2283","family_id":"qf-ext19-continuity-composition","seed":202610193065,"semantic_template":"qf-ext19-continuity-composition/{\"images_f\":[2,0,3,1],\"images_g\":[2,0,1,3],\"k\":4,\"m\":4,\"n\":4,\"opens_x\":[[],[2],[0,2],[0,2,3],[0,1,2,3]],\"opens_y\":[[],[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]],\"opens_z\":[[],[0,1,2,3]]}"},"fields":{"title":"Two Typed Maps and Independently Audited Composition","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[2, 0, 3, 1] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[2, 0, 1, 3]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.","answer":{"kind":"expression","value":"([2, 0, 3, 1], [2, 0, 1, 3], [1, 2, 3, 0], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])","explanation":"Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 3, 1], [2, 0, 1, 3], [1, 2, 3, 0], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])."},"skills":["Two Typed Maps and Independently Audited Composition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0066","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bea480557b2c39bfbf8aa880d6f701efc91bc10e48c3f85b336f9a6f2f6f8e29","family_id":"qf-ext19-continuity-composition","seed":202610193066,"semantic_template":"qf-ext19-continuity-composition/{\"images_f\":[3,0,2,1],\"images_g\":[1,3,0,2],\"k\":4,\"m\":4,\"n\":4,\"opens_x\":[[],[3],[0,3],[0,2,3],[0,1,2,3]],\"opens_y\":[[],[3],[0,3],[0,1,3],[0,1,2,3]],\"opens_z\":[[],[0,1,2,3]]}"},"fields":{"title":"Two Typed Maps and Independently Audited Composition","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 0, 2, 1] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[1, 3, 0, 2]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.","answer":{"kind":"expression","value":"([3, 0, 2, 1], [1, 3, 0, 2], [2, 1, 0, 3], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])","explanation":"Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 0, 2, 1], [1, 3, 0, 2], [2, 1, 0, 3], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])."},"skills":["Two Typed Maps and Independently Audited Composition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0067","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"82bd3e2c26163236fccc4842ec19c6167ad7140ab24a03fab5d45302e217326a","family_id":"qf-ext19-continuity-composition","seed":202610193067,"semantic_template":"qf-ext19-continuity-composition/{\"images_f\":[0,2,3,1],\"images_g\":[0,2,3,1],\"k\":4,\"m\":4,\"n\":4,\"opens_x\":[[],[0],[0,2],[0,2,3],[0,1,2,3]],\"opens_y\":[[],[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]],\"opens_z\":[[],[0,1,2,3]]}"},"fields":{"title":"Two Typed Maps and Independently Audited Composition","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[0, 2, 3, 1] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[0, 2, 3, 1]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.","answer":{"kind":"expression","value":"([0, 2, 3, 1], [0, 2, 3, 1], [0, 3, 1, 2], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])","explanation":"Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 3, 1], [0, 2, 3, 1], [0, 3, 1, 2], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])."},"skills":["Two Typed Maps and Independently Audited Composition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0068","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0df5d99177666057175c8bba855148912a375e32c6dfe4c4c25759d2eccd388","family_id":"qf-ext19-continuity-composition","seed":202610193068,"semantic_template":"qf-ext19-continuity-composition/{\"images_f\":[1,2,3,0],\"images_g\":[3,2,1,0],\"k\":4,\"m\":4,\"n\":4,\"opens_x\":[[],[1],[1,2],[1,2,3],[0,1,2,3]],\"opens_y\":[[],[1],[0,1],[0,1,3],[0,1,2,3]],\"opens_z\":[[],[0,1,2,3]]}"},"fields":{"title":"Two Typed Maps and Independently Audited Composition","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]]. The complete total map has images=[1, 2, 3, 0] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[3, 2, 1, 0]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.","answer":{"kind":"expression","value":"([1, 2, 3, 0], [3, 2, 1, 0], [2, 1, 0, 3], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])","explanation":"Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 3, 0], [3, 2, 1, 0], [2, 1, 0, 3], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])."},"skills":["Two Typed Maps and Independently Audited Composition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0069","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"abac8b0967ec946acec2b04402f14201184bcda304ee8fccea1f2c11cd517827","family_id":"qf-ext19-continuity-composition","seed":202610193069,"semantic_template":"qf-ext19-continuity-composition/{\"images_f\":[2,1,3,0],\"images_g\":[2,1,0,3],\"k\":4,\"m\":4,\"n\":4,\"opens_x\":[[],[2],[1,2],[1,2,3],[0,1,2,3]],\"opens_y\":[[],[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]],\"opens_z\":[[],[0,1,2,3]]}"},"fields":{"title":"Two Typed Maps and Independently Audited Composition","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[2, 1, 3, 0] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[2, 1, 0, 3]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.","answer":{"kind":"expression","value":"([2, 1, 3, 0], [2, 1, 0, 3], [0, 1, 3, 2], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])","explanation":"Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 3, 0], [2, 1, 0, 3], [0, 1, 3, 2], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])."},"skills":["Two Typed Maps and Independently Audited Composition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0070","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c57d02b1798144f7f5101005b702952c80deaeee5928dc807b82eef3d9a4e77f","family_id":"qf-ext19-continuity-composition","seed":202610193070,"semantic_template":"qf-ext19-continuity-composition/{\"images_f\":[3,1,2,0],\"images_g\":[1,0,3,2],\"k\":4,\"m\":4,\"n\":4,\"opens_x\":[[],[3],[1,3],[1,2,3],[0,1,2,3]],\"opens_y\":[[],[3],[2,3],[1,2,3],[0,1,2,3]],\"opens_z\":[[],[0,1,2,3]]}"},"fields":{"title":"Two Typed Maps and Independently Audited Composition","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 1, 2, 0] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[1, 0, 3, 2]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.","answer":{"kind":"expression","value":"([3, 1, 2, 0], [1, 0, 3, 2], [2, 0, 3, 1], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])","explanation":"Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 1, 2, 0], [1, 0, 3, 2], [2, 0, 3, 1], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])."},"skills":["Two Typed Maps and Independently Audited Composition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0071","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"263a65a68fff7b3d73739859d8394fd17441e24fc5a284e58cf09be0ad8400c2","family_id":"qf-ext19-continuity-composition","seed":202610193071,"semantic_template":"qf-ext19-continuity-composition/{\"images_f\":[0,3,2,1],\"images_g\":[0,3,2,1],\"k\":4,\"m\":4,\"n\":4,\"opens_x\":[[],[0],[0,3],[0,2,3],[0,1,2,3]],\"opens_y\":[[],[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]],\"opens_z\":[[],[0,1,2,3]]}"},"fields":{"title":"Two Typed Maps and Independently Audited Composition","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[0, 3, 2, 1] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[0, 3, 2, 1]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.","answer":{"kind":"expression","value":"([0, 3, 2, 1], [0, 3, 2, 1], [0, 1, 2, 3], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])","explanation":"Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 3, 2, 1], [0, 3, 2, 1], [0, 1, 2, 3], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])."},"skills":["Two Typed Maps and Independently Audited Composition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0072","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b51b0acc1f695bec6f69ce2bc6a5e80e19e1e143a67916cb68d8da37a13c1f3c","family_id":"qf-ext19-continuity-composition","seed":202610193072,"semantic_template":"qf-ext19-continuity-composition/{\"images_f\":[1,3,2,0],\"images_g\":[3,0,1,2],\"k\":4,\"m\":4,\"n\":4,\"opens_x\":[[],[1],[1,3],[1,2,3],[0,1,2,3]],\"opens_y\":[[],[1],[1,3],[0,1,3],[0,1,2,3]],\"opens_z\":[[],[0,1,2,3]]}"},"fields":{"title":"Two Typed Maps and Independently Audited Composition","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]]. The complete total map has images=[1, 3, 2, 0] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[3, 0, 1, 2]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.","answer":{"kind":"expression","value":"([1, 3, 2, 0], [3, 0, 1, 2], [0, 2, 1, 3], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])","explanation":"Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 3, 2, 0], [3, 0, 1, 2], [0, 2, 1, 3], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])."},"skills":["Two Typed Maps and Independently Audited Composition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0073","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2fb963e51a5cba6d59168047076258842e4b6b10c2e4eb6071f017fe0ba0eaff","family_id":"qf-ext19-continuity-composition","seed":202610193073,"semantic_template":"qf-ext19-continuity-composition/{\"images_f\":[2,3,1,0],\"images_g\":[2,3,0,1],\"k\":4,\"m\":4,\"n\":4,\"opens_x\":[[],[2],[2,3],[1,2,3],[0,1,2,3]],\"opens_y\":[[],[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]],\"opens_z\":[[],[0,1,2,3]]}"},"fields":{"title":"Two Typed Maps and Independently Audited Composition","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[2, 3, 1, 0] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[2, 3, 0, 1]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.","answer":{"kind":"expression","value":"([2, 3, 1, 0], [2, 3, 0, 1], [0, 1, 3, 2], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])","explanation":"Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 3, 1, 0], [2, 3, 0, 1], [0, 1, 3, 2], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])."},"skills":["Two Typed Maps and Independently Audited Composition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0074","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38a66256dfe31d43c8525e8dfe1cb1aa492b50bb7540642dd207dc719373607e","family_id":"qf-ext19-continuity-composition","seed":202610193074,"semantic_template":"qf-ext19-continuity-composition/{\"images_f\":[3,2,1,0],\"images_g\":[1,2,3,0],\"k\":4,\"m\":4,\"n\":4,\"opens_x\":[[],[3],[2,3],[1,2,3],[0,1,2,3]],\"opens_y\":[[],[3],[1,3],[1,2,3],[0,1,2,3]],\"opens_z\":[[],[0,1,2,3]]}"},"fields":{"title":"Two Typed Maps and Independently Audited Composition","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 2, 1, 0] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(4)={0,…,3}, opens_Z=[[], [0, 1, 2, 3]], and g:Y→Z with images_g=[1, 2, 3, 0]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.","answer":{"kind":"expression","value":"([3, 2, 1, 0], [1, 2, 3, 0], [0, 3, 2, 1], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])","explanation":"Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 2, 1, 0], [1, 2, 3, 0], [0, 3, 2, 1], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]])."},"skills":["Two Typed Maps and Independently Audited Composition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0075","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5b0023d0d9ec39ab5183d2c9b4028226f5c2a3a97cb8adf5e3ea2f9fe810df1","family_id":"qf-ext19-continuity-composition","seed":202610193075,"semantic_template":"qf-ext19-continuity-composition/{\"images_f\":[0,1,2],\"images_g\":[0,1,2],\"k\":3,\"m\":3,\"n\":3,\"opens_x\":[[],[0],[0,1],[0,1,2]],\"opens_y\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2]],\"opens_z\":[[],[0,1,2]]}"},"fields":{"title":"Two Typed Maps and Independently Audited Composition","stem":"Use complete finite topologies X=Fin(3)={0,…,2}, opens_X=[[], [0], [0, 1], [0, 1, 2]] and Y=Fin(3)={0,…,2}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2]]. The complete total map has images=[0, 1, 2] in source order. All entries must type-check; set preimages are not inverse functions. Add Z=Fin(3)={0,…,2}, opens_Z=[[], [0, 1, 2]], and g:Y→Z with images_g=[0, 1, 2]. Give (images_f, images_g, images of g after f, f continuous?, g continuous?, composition continuous?, all composition preimage rows). Do not assume either factor continuous.","answer":{"kind":"expression","value":"([0, 1, 2], [0, 1, 2], [0, 1, 2], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2], [0, 1, 2], Yes]])","explanation":"Type-check both total tables and compute g(f(x)); composition may be continuous even if one separately audited factor is not. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2], [0, 1, 2], [0, 1, 2], No, Yes, Yes, [[[], [], Yes], [[0, 1, 2], [0, 1, 2], Yes]])."},"skills":["Two Typed Maps and Independently Audited Composition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0076","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"87768c2fcb5819d462ad0b8cb3295d8bffb733738ed5028ec763ef1f19576a73","family_id":"qf-ext19-continuity-open-map-contrast","seed":202610193076,"semantic_template":"qf-ext19-continuity-open-map-contrast/{\"images\":[0,1,2,3],\"m\":4,\"n\":4,\"opens_x\":[[],[0],[0,1],[0,1,2],[0,1,2,3]],\"opens_y\":[[],[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]]}"},"fields":{"title":"Open Images versus Open Preimages","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[0, 1, 2, 3] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).","answer":{"kind":"expression","value":"([0, 1, 2, 3], No, Yes, [[[], [], Yes], [[0], [0], Yes], [[1], [1], No], [[0, 1], [0, 1], Yes], [[2], [2], No], [[0, 2], [0, 2], No], [[1, 2], [1, 2], No], [[0, 1, 2], [0, 1, 2], Yes], [[3], [3], No], [[0, 3], [0, 3], No], [[1, 3], [1, 3], No], [[0, 1, 3], [0, 1, 3], No], [[2, 3], [2, 3], No], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[0], [0], Yes], [[0, 1], [0, 1], Yes], [[0, 1, 2], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ)","explanation":"Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2, 3], No, Yes, [[[], [], Yes], [[0], [0], Yes], [[1], [1], No], [[0, 1], [0, 1], Yes], [[2], [2], No], [[0, 2], [0, 2], No], [[1, 2], [1, 2], No], [[0, 1, 2], [0, 1, 2], Yes], [[3], [3], No], [[0, 3], [0, 3], No], [[1, 3], [1, 3], No], [[0, 1, 3], [0, 1, 3], No], [[2, 3], [2, 3], No], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[0], [0], Yes], [[0, 1], [0, 1], Yes], [[0, 1, 2], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ)."},"skills":["Open Images versus Open Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0077","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b9c4e028d4c0cb299ab305f6af27d40e15f37c7c8856240380414f5e1eef595","family_id":"qf-ext19-continuity-open-map-contrast","seed":202610193077,"semantic_template":"qf-ext19-continuity-open-map-contrast/{\"images\":[1,0,2,3],\"m\":4,\"n\":4,\"opens_x\":[[],[1],[0,1],[0,1,2],[0,1,2,3]],\"opens_y\":[[],[1],[1,2],[0,1,2],[0,1,2,3]]}"},"fields":{"title":"Open Images versus Open Preimages","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]]. The complete total map has images=[1, 0, 2, 3] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).","answer":{"kind":"expression","value":"([1, 0, 2, 3], No, No, [[[], [], Yes], [[1], [0], No], [[1, 2], [0, 2], No], [[0, 1, 2], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[1], [0], No], [[0, 1], [0, 1], No], [[0, 1, 2], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations)","explanation":"Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 2, 3], No, No, [[[], [], Yes], [[1], [0], No], [[1, 2], [0, 2], No], [[0, 1, 2], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[1], [0], No], [[0, 1], [0, 1], No], [[0, 1, 2], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations)."},"skills":["Open Images versus Open Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0078","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d685b8c1bbe0af3ad5907dd52dc11d55e8b7c3aae33051f1c1f0f6fbe7e32c0","family_id":"qf-ext19-continuity-open-map-contrast","seed":202610193078,"semantic_template":"qf-ext19-continuity-open-map-contrast/{\"images\":[2,0,1,3],\"m\":4,\"n\":4,\"opens_x\":[[],[2],[0,2],[0,1,2],[0,1,2,3]],\"opens_y\":[[],[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]]}"},"fields":{"title":"Open Images versus Open Preimages","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [0, 2], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[2, 0, 1, 3] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).","answer":{"kind":"expression","value":"([2, 0, 1, 3], No, Yes, [[[], [], Yes], [[0], [1], No], [[1], [2], Yes], [[0, 1], [1, 2], No], [[2], [0], No], [[0, 2], [0, 1], No], [[1, 2], [0, 2], Yes], [[0, 1, 2], [0, 1, 2], Yes], [[3], [3], No], [[0, 3], [1, 3], No], [[1, 3], [2, 3], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0, 3], No], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[2], [1], Yes], [[0, 2], [1, 2], Yes], [[0, 1, 2], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ)","explanation":"Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 1, 3], No, Yes, [[[], [], Yes], [[0], [1], No], [[1], [2], Yes], [[0, 1], [1, 2], No], [[2], [0], No], [[0, 2], [0, 1], No], [[1, 2], [0, 2], Yes], [[0, 1, 2], [0, 1, 2], Yes], [[3], [3], No], [[0, 3], [1, 3], No], [[1, 3], [2, 3], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0, 3], No], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[2], [1], Yes], [[0, 2], [1, 2], Yes], [[0, 1, 2], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ)."},"skills":["Open Images versus Open Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0079","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aba2a848c9c3f19df6dab194017f9d9a06e9c7e2cb165984feed4eb23f2df72a","family_id":"qf-ext19-continuity-open-map-contrast","seed":202610193079,"semantic_template":"qf-ext19-continuity-open-map-contrast/{\"images\":[3,0,1,2],\"m\":4,\"n\":4,\"opens_x\":[[],[3],[0,3],[0,1,3],[0,1,2,3]],\"opens_y\":[[],[3],[0,3],[0,2,3],[0,1,2,3]]}"},"fields":{"title":"Open Images versus Open Preimages","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 0, 1, 2] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).","answer":{"kind":"expression","value":"([3, 0, 1, 2], No, No, [[[], [], Yes], [[3], [0], No], [[0, 3], [0, 1], No], [[0, 2, 3], [0, 1, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[3], [2], No], [[0, 3], [2, 3], No], [[0, 1, 3], [0, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations)","explanation":"Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 0, 1, 2], No, No, [[[], [], Yes], [[3], [0], No], [[0, 3], [0, 1], No], [[0, 2, 3], [0, 1, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[3], [2], No], [[0, 3], [2, 3], No], [[0, 1, 3], [0, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations)."},"skills":["Open Images versus Open Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0080","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5fe7f64a4d8127917e219ba8ea3a5d82a38d2b21745df3b9d122ba50214ad51","family_id":"qf-ext19-continuity-open-map-contrast","seed":202610193080,"semantic_template":"qf-ext19-continuity-open-map-contrast/{\"images\":[0,2,1,3],\"m\":4,\"n\":4,\"opens_x\":[[],[0],[0,2],[0,1,2],[0,1,2,3]],\"opens_y\":[[],[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]]}"},"fields":{"title":"Open Images versus Open Preimages","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[0, 2, 1, 3] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).","answer":{"kind":"expression","value":"([0, 2, 1, 3], No, Yes, [[[], [], Yes], [[0], [0], Yes], [[1], [2], No], [[0, 1], [0, 2], Yes], [[2], [1], No], [[0, 2], [0, 1], No], [[1, 2], [1, 2], No], [[0, 1, 2], [0, 1, 2], Yes], [[3], [3], No], [[0, 3], [0, 3], No], [[1, 3], [2, 3], No], [[0, 1, 3], [0, 2, 3], No], [[2, 3], [1, 3], No], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[0], [0], Yes], [[0, 2], [0, 1], Yes], [[0, 1, 2], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ)","explanation":"Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 1, 3], No, Yes, [[[], [], Yes], [[0], [0], Yes], [[1], [2], No], [[0, 1], [0, 2], Yes], [[2], [1], No], [[0, 2], [0, 1], No], [[1, 2], [1, 2], No], [[0, 1, 2], [0, 1, 2], Yes], [[3], [3], No], [[0, 3], [0, 3], No], [[1, 3], [2, 3], No], [[0, 1, 3], [0, 2, 3], No], [[2, 3], [1, 3], No], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[0], [0], Yes], [[0, 2], [0, 1], Yes], [[0, 1, 2], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ)."},"skills":["Open Images versus Open Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0081","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7349f80489886cdae3f935b83bdd5c0676de185181dcc14f6ceff96f3e853e3","family_id":"qf-ext19-continuity-open-map-contrast","seed":202610193081,"semantic_template":"qf-ext19-continuity-open-map-contrast/{\"images\":[1,2,0,3],\"m\":4,\"n\":4,\"opens_x\":[[],[1],[1,2],[0,1,2],[0,1,2,3]],\"opens_y\":[[],[1],[0,1],[0,1,2],[0,1,2,3]]}"},"fields":{"title":"Open Images versus Open Preimages","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]]. The complete total map has images=[1, 2, 0, 3] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).","answer":{"kind":"expression","value":"([1, 2, 0, 3], No, No, [[[], [], Yes], [[1], [0], No], [[0, 1], [0, 2], No], [[0, 1, 2], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[1], [2], No], [[1, 2], [0, 2], No], [[0, 1, 2], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations)","explanation":"Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 0, 3], No, No, [[[], [], Yes], [[1], [0], No], [[0, 1], [0, 2], No], [[0, 1, 2], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[1], [2], No], [[1, 2], [0, 2], No], [[0, 1, 2], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations)."},"skills":["Open Images versus Open Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0082","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97a9154ff3492ee3291eeb38dbba93c465184eaebf44ae785aeacf895b08b477","family_id":"qf-ext19-continuity-open-map-contrast","seed":202610193082,"semantic_template":"qf-ext19-continuity-open-map-contrast/{\"images\":[2,1,0,3],\"m\":4,\"n\":4,\"opens_x\":[[],[2],[1,2],[0,1,2],[0,1,2,3]],\"opens_y\":[[],[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]]}"},"fields":{"title":"Open Images versus Open Preimages","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [1, 2], [0, 1, 2], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[2, 1, 0, 3] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).","answer":{"kind":"expression","value":"([2, 1, 0, 3], No, Yes, [[[], [], Yes], [[0], [2], Yes], [[1], [1], No], [[0, 1], [1, 2], Yes], [[2], [0], No], [[0, 2], [0, 2], No], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1, 2], Yes], [[3], [3], No], [[0, 3], [2, 3], No], [[1, 3], [1, 3], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0, 3], No], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [0, 1, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[2], [0], Yes], [[1, 2], [0, 1], Yes], [[0, 1, 2], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ)","explanation":"Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 0, 3], No, Yes, [[[], [], Yes], [[0], [2], Yes], [[1], [1], No], [[0, 1], [1, 2], Yes], [[2], [0], No], [[0, 2], [0, 2], No], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1, 2], Yes], [[3], [3], No], [[0, 3], [2, 3], No], [[1, 3], [1, 3], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0, 3], No], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [0, 1, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[2], [0], Yes], [[1, 2], [0, 1], Yes], [[0, 1, 2], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ)."},"skills":["Open Images versus Open Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0083","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0499b93bb855418f8b4c587b46cae34958b48374690754ba4c8691ccd750b86","family_id":"qf-ext19-continuity-open-map-contrast","seed":202610193083,"semantic_template":"qf-ext19-continuity-open-map-contrast/{\"images\":[3,1,0,2],\"m\":4,\"n\":4,\"opens_x\":[[],[3],[1,3],[0,1,3],[0,1,2,3]],\"opens_y\":[[],[3],[2,3],[0,2,3],[0,1,2,3]]}"},"fields":{"title":"Open Images versus Open Preimages","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 1, 0, 2] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).","answer":{"kind":"expression","value":"([3, 1, 0, 2], No, No, [[[], [], Yes], [[3], [0], No], [[2, 3], [0, 3], No], [[0, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[3], [2], No], [[1, 3], [1, 2], No], [[0, 1, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations)","explanation":"Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 1, 0, 2], No, No, [[[], [], Yes], [[3], [0], No], [[2, 3], [0, 3], No], [[0, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[3], [2], No], [[1, 3], [1, 2], No], [[0, 1, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations)."},"skills":["Open Images versus Open Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0084","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61db3bd433be020c6c0e801a7acc39c522b7f229e388f0d5593d5004243f935e","family_id":"qf-ext19-continuity-open-map-contrast","seed":202610193084,"semantic_template":"qf-ext19-continuity-open-map-contrast/{\"images\":[0,3,1,2],\"m\":4,\"n\":4,\"opens_x\":[[],[0],[0,3],[0,1,3],[0,1,2,3]],\"opens_y\":[[],[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]]}"},"fields":{"title":"Open Images versus Open Preimages","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[0, 3, 1, 2] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).","answer":{"kind":"expression","value":"([0, 3, 1, 2], No, Yes, [[[], [], Yes], [[0], [0], Yes], [[1], [2], No], [[0, 1], [0, 2], No], [[2], [3], No], [[0, 2], [0, 3], Yes], [[1, 2], [2, 3], No], [[0, 1, 2], [0, 2, 3], No], [[3], [1], No], [[0, 3], [0, 1], No], [[1, 3], [1, 2], No], [[0, 1, 3], [0, 1, 2], No], [[2, 3], [1, 3], No], [[0, 2, 3], [0, 1, 3], Yes], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[0], [0], Yes], [[0, 3], [0, 2], Yes], [[0, 1, 3], [0, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ)","explanation":"Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 3, 1, 2], No, Yes, [[[], [], Yes], [[0], [0], Yes], [[1], [2], No], [[0, 1], [0, 2], No], [[2], [3], No], [[0, 2], [0, 3], Yes], [[1, 2], [2, 3], No], [[0, 1, 2], [0, 2, 3], No], [[3], [1], No], [[0, 3], [0, 1], No], [[1, 3], [1, 2], No], [[0, 1, 3], [0, 1, 2], No], [[2, 3], [1, 3], No], [[0, 2, 3], [0, 1, 3], Yes], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[0], [0], Yes], [[0, 3], [0, 2], Yes], [[0, 1, 3], [0, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ)."},"skills":["Open Images versus Open Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0085","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df8091764042502c7d5db94a76f9195c5abc08593c4b0742e1321885d97f3dc7","family_id":"qf-ext19-continuity-open-map-contrast","seed":202610193085,"semantic_template":"qf-ext19-continuity-open-map-contrast/{\"images\":[1,3,0,2],\"m\":4,\"n\":4,\"opens_x\":[[],[1],[1,3],[0,1,3],[0,1,2,3]],\"opens_y\":[[],[1],[1,3],[1,2,3],[0,1,2,3]]}"},"fields":{"title":"Open Images versus Open Preimages","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[1, 3, 0, 2] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).","answer":{"kind":"expression","value":"([1, 3, 0, 2], No, No, [[[], [], Yes], [[1], [0], No], [[1, 3], [0, 1], No], [[1, 2, 3], [0, 1, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[1], [3], No], [[1, 3], [2, 3], No], [[0, 1, 3], [1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations)","explanation":"Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 3, 0, 2], No, No, [[[], [], Yes], [[1], [0], No], [[1, 3], [0, 1], No], [[1, 2, 3], [0, 1, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[1], [3], No], [[1, 3], [2, 3], No], [[0, 1, 3], [1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations)."},"skills":["Open Images versus Open Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0086","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5bff1f26e9092a0700e0081f4513584c4c0b9eb6da2a8eaa790828db0e2e7c49","family_id":"qf-ext19-continuity-open-map-contrast","seed":202610193086,"semantic_template":"qf-ext19-continuity-open-map-contrast/{\"images\":[2,3,0,1],\"m\":4,\"n\":4,\"opens_x\":[[],[2],[2,3],[0,2,3],[0,1,2,3]],\"opens_y\":[[],[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]]}"},"fields":{"title":"Open Images versus Open Preimages","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[2, 3, 0, 1] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).","answer":{"kind":"expression","value":"([2, 3, 0, 1], No, Yes, [[[], [], Yes], [[0], [2], Yes], [[1], [3], No], [[0, 1], [2, 3], Yes], [[2], [0], No], [[0, 2], [0, 2], No], [[1, 2], [0, 3], No], [[0, 1, 2], [0, 2, 3], Yes], [[3], [1], No], [[0, 3], [1, 2], No], [[1, 3], [1, 3], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0, 1], No], [[0, 2, 3], [0, 1, 2], No], [[1, 2, 3], [0, 1, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[2], [0], Yes], [[2, 3], [0, 1], Yes], [[0, 2, 3], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ)","explanation":"Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 3, 0, 1], No, Yes, [[[], [], Yes], [[0], [2], Yes], [[1], [3], No], [[0, 1], [2, 3], Yes], [[2], [0], No], [[0, 2], [0, 2], No], [[1, 2], [0, 3], No], [[0, 1, 2], [0, 2, 3], Yes], [[3], [1], No], [[0, 3], [1, 2], No], [[1, 3], [1, 3], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0, 1], No], [[0, 2, 3], [0, 1, 2], No], [[1, 2, 3], [0, 1, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[2], [0], Yes], [[2, 3], [0, 1], Yes], [[0, 2, 3], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ)."},"skills":["Open Images versus Open Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0087","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78514c565f8765c274f18425242203cf5aaae0b607d4f7019f55ebcfa1dcb62c","family_id":"qf-ext19-continuity-open-map-contrast","seed":202610193087,"semantic_template":"qf-ext19-continuity-open-map-contrast/{\"images\":[3,2,0,1],\"m\":4,\"n\":4,\"opens_x\":[[],[3],[2,3],[0,2,3],[0,1,2,3]],\"opens_y\":[[],[3],[1,3],[0,1,3],[0,1,2,3]]}"},"fields":{"title":"Open Images versus Open Preimages","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 2, 0, 1] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).","answer":{"kind":"expression","value":"([3, 2, 0, 1], No, No, [[[], [], Yes], [[3], [0], No], [[1, 3], [0, 3], No], [[0, 1, 3], [0, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[3], [1], No], [[2, 3], [0, 1], No], [[0, 2, 3], [0, 1, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations)","explanation":"Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 2, 0, 1], No, No, [[[], [], Yes], [[3], [0], No], [[1, 3], [0, 3], No], [[0, 1, 3], [0, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[3], [1], No], [[2, 3], [0, 1], No], [[0, 2, 3], [0, 1, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations)."},"skills":["Open Images versus Open Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0088","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35ab5b57dec9fc1da04550c78c30a8c38db9fc2f5225a1a63f53398916aab827","family_id":"qf-ext19-continuity-open-map-contrast","seed":202610193088,"semantic_template":"qf-ext19-continuity-open-map-contrast/{\"images\":[0,1,3,2],\"m\":4,\"n\":4,\"opens_x\":[[],[0],[0,1],[0,1,3],[0,1,2,3]],\"opens_y\":[[],[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]]}"},"fields":{"title":"Open Images versus Open Preimages","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[0, 1, 3, 2] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).","answer":{"kind":"expression","value":"([0, 1, 3, 2], No, Yes, [[[], [], Yes], [[0], [0], Yes], [[1], [1], No], [[0, 1], [0, 1], Yes], [[2], [3], No], [[0, 2], [0, 3], No], [[1, 2], [1, 3], No], [[0, 1, 2], [0, 1, 3], Yes], [[3], [2], No], [[0, 3], [0, 2], No], [[1, 3], [1, 2], No], [[0, 1, 3], [0, 1, 2], No], [[2, 3], [2, 3], No], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[0], [0], Yes], [[0, 1], [0, 1], Yes], [[0, 1, 3], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ)","explanation":"Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 3, 2], No, Yes, [[[], [], Yes], [[0], [0], Yes], [[1], [1], No], [[0, 1], [0, 1], Yes], [[2], [3], No], [[0, 2], [0, 3], No], [[1, 2], [1, 3], No], [[0, 1, 2], [0, 1, 3], Yes], [[3], [2], No], [[0, 3], [0, 2], No], [[1, 3], [1, 2], No], [[0, 1, 3], [0, 1, 2], No], [[2, 3], [2, 3], No], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[0], [0], Yes], [[0, 1], [0, 1], Yes], [[0, 1, 3], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ)."},"skills":["Open Images versus Open Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0089","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"244b64dfdcfa56e658a2a16c0c6e34e8bafa5d21c908bff684c34e0de4ec0e97","family_id":"qf-ext19-continuity-open-map-contrast","seed":202610193089,"semantic_template":"qf-ext19-continuity-open-map-contrast/{\"images\":[1,0,3,2],\"m\":4,\"n\":4,\"opens_x\":[[],[1],[0,1],[0,1,3],[0,1,2,3]],\"opens_y\":[[],[1],[1,2],[1,2,3],[0,1,2,3]]}"},"fields":{"title":"Open Images versus Open Preimages","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[1, 0, 3, 2] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).","answer":{"kind":"expression","value":"([1, 0, 3, 2], No, No, [[[], [], Yes], [[1], [0], No], [[1, 2], [0, 3], No], [[1, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[1], [0], No], [[0, 1], [0, 1], No], [[0, 1, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations)","explanation":"Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 3, 2], No, No, [[[], [], Yes], [[1], [0], No], [[1, 2], [0, 3], No], [[1, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[1], [0], No], [[0, 1], [0, 1], No], [[0, 1, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations)."},"skills":["Open Images versus Open Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0090","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"404d1ceaef32eeb99985c5160c6b09649291463c6b927664d19e75439f5f3985","family_id":"qf-ext19-continuity-open-map-contrast","seed":202610193090,"semantic_template":"qf-ext19-continuity-open-map-contrast/{\"images\":[2,0,3,1],\"m\":4,\"n\":4,\"opens_x\":[[],[2],[0,2],[0,2,3],[0,1,2,3]],\"opens_y\":[[],[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]]}"},"fields":{"title":"Open Images versus Open Preimages","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[2, 0, 3, 1] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).","answer":{"kind":"expression","value":"([2, 0, 3, 1], No, Yes, [[[], [], Yes], [[0], [1], No], [[1], [3], No], [[0, 1], [1, 3], No], [[2], [0], No], [[0, 2], [0, 1], No], [[1, 2], [0, 3], No], [[0, 1, 2], [0, 1, 3], No], [[3], [2], Yes], [[0, 3], [1, 2], No], [[1, 3], [2, 3], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0, 2], Yes], [[0, 2, 3], [0, 1, 2], No], [[1, 2, 3], [0, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[2], [3], Yes], [[0, 2], [2, 3], Yes], [[0, 2, 3], [1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ)","explanation":"Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 3, 1], No, Yes, [[[], [], Yes], [[0], [1], No], [[1], [3], No], [[0, 1], [1, 3], No], [[2], [0], No], [[0, 2], [0, 1], No], [[1, 2], [0, 3], No], [[0, 1, 2], [0, 1, 3], No], [[3], [2], Yes], [[0, 3], [1, 2], No], [[1, 3], [2, 3], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0, 2], Yes], [[0, 2, 3], [0, 1, 2], No], [[1, 2, 3], [0, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[2], [3], Yes], [[0, 2], [2, 3], Yes], [[0, 2, 3], [1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ)."},"skills":["Open Images versus Open Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0091","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"249835f265f38e6b52298309594162c0f5081e3c7eb15ee2c8e211f9d290e021","family_id":"qf-ext19-continuity-open-map-contrast","seed":202610193091,"semantic_template":"qf-ext19-continuity-open-map-contrast/{\"images\":[3,0,2,1],\"m\":4,\"n\":4,\"opens_x\":[[],[3],[0,3],[0,2,3],[0,1,2,3]],\"opens_y\":[[],[3],[0,3],[0,1,3],[0,1,2,3]]}"},"fields":{"title":"Open Images versus Open Preimages","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 0, 2, 1] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).","answer":{"kind":"expression","value":"([3, 0, 2, 1], No, No, [[[], [], Yes], [[3], [0], No], [[0, 3], [0, 1], No], [[0, 1, 3], [0, 1, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[3], [1], No], [[0, 3], [1, 3], No], [[0, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations)","explanation":"Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 0, 2, 1], No, No, [[[], [], Yes], [[3], [0], No], [[0, 3], [0, 1], No], [[0, 1, 3], [0, 1, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[3], [1], No], [[0, 3], [1, 3], No], [[0, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations)."},"skills":["Open Images versus Open Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0092","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9acb10e9bfee7ac45b10a06a239af00c8a765403fb9f8bf95e3d3b2540dc66a9","family_id":"qf-ext19-continuity-open-map-contrast","seed":202610193092,"semantic_template":"qf-ext19-continuity-open-map-contrast/{\"images\":[0,2,3,1],\"m\":4,\"n\":4,\"opens_x\":[[],[0],[0,2],[0,2,3],[0,1,2,3]],\"opens_y\":[[],[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]]}"},"fields":{"title":"Open Images versus Open Preimages","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[0, 2, 3, 1] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).","answer":{"kind":"expression","value":"([0, 2, 3, 1], No, Yes, [[[], [], Yes], [[0], [0], Yes], [[1], [3], No], [[0, 1], [0, 3], No], [[2], [1], No], [[0, 2], [0, 1], No], [[1, 2], [1, 3], No], [[0, 1, 2], [0, 1, 3], No], [[3], [2], No], [[0, 3], [0, 2], Yes], [[1, 3], [2, 3], No], [[0, 1, 3], [0, 2, 3], Yes], [[2, 3], [1, 2], No], [[0, 2, 3], [0, 1, 2], No], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[0], [0], Yes], [[0, 2], [0, 3], Yes], [[0, 2, 3], [0, 1, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ)","explanation":"Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 3, 1], No, Yes, [[[], [], Yes], [[0], [0], Yes], [[1], [3], No], [[0, 1], [0, 3], No], [[2], [1], No], [[0, 2], [0, 1], No], [[1, 2], [1, 3], No], [[0, 1, 2], [0, 1, 3], No], [[3], [2], No], [[0, 3], [0, 2], Yes], [[1, 3], [2, 3], No], [[0, 1, 3], [0, 2, 3], Yes], [[2, 3], [1, 2], No], [[0, 2, 3], [0, 1, 2], No], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[0], [0], Yes], [[0, 2], [0, 3], Yes], [[0, 2, 3], [0, 1, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ)."},"skills":["Open Images versus Open Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0093","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99ef1508e91e751103504285be718dddb1ccc03634a8f75dfb11a20c51b06138","family_id":"qf-ext19-continuity-open-map-contrast","seed":202610193093,"semantic_template":"qf-ext19-continuity-open-map-contrast/{\"images\":[1,2,3,0],\"m\":4,\"n\":4,\"opens_x\":[[],[1],[1,2],[1,2,3],[0,1,2,3]],\"opens_y\":[[],[1],[0,1],[0,1,3],[0,1,2,3]]}"},"fields":{"title":"Open Images versus Open Preimages","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]]. The complete total map has images=[1, 2, 3, 0] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).","answer":{"kind":"expression","value":"([1, 2, 3, 0], No, No, [[[], [], Yes], [[1], [0], No], [[0, 1], [0, 3], No], [[0, 1, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[1], [2], No], [[1, 2], [2, 3], No], [[1, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations)","explanation":"Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 3, 0], No, No, [[[], [], Yes], [[1], [0], No], [[0, 1], [0, 3], No], [[0, 1, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[1], [2], No], [[1, 2], [2, 3], No], [[1, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations)."},"skills":["Open Images versus Open Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0094","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d786d44ac958ebbb21a1330b5a35be1545d34b2ad6987af5cf8a78aef06ab595","family_id":"qf-ext19-continuity-open-map-contrast","seed":202610193094,"semantic_template":"qf-ext19-continuity-open-map-contrast/{\"images\":[2,1,3,0],\"m\":4,\"n\":4,\"opens_x\":[[],[2],[1,2],[1,2,3],[0,1,2,3]],\"opens_y\":[[],[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]]}"},"fields":{"title":"Open Images versus Open Preimages","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[2, 1, 3, 0] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).","answer":{"kind":"expression","value":"([2, 1, 3, 0], No, Yes, [[[], [], Yes], [[0], [3], No], [[1], [1], No], [[0, 1], [1, 3], No], [[2], [0], No], [[0, 2], [0, 3], No], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1, 3], No], [[3], [2], Yes], [[0, 3], [2, 3], No], [[1, 3], [1, 2], Yes], [[0, 1, 3], [1, 2, 3], Yes], [[2, 3], [0, 2], No], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[2], [3], Yes], [[1, 2], [1, 3], Yes], [[1, 2, 3], [0, 1, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ)","explanation":"Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 3, 0], No, Yes, [[[], [], Yes], [[0], [3], No], [[1], [1], No], [[0, 1], [1, 3], No], [[2], [0], No], [[0, 2], [0, 3], No], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1, 3], No], [[3], [2], Yes], [[0, 3], [2, 3], No], [[1, 3], [1, 2], Yes], [[0, 1, 3], [1, 2, 3], Yes], [[2, 3], [0, 2], No], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[2], [3], Yes], [[1, 2], [1, 3], Yes], [[1, 2, 3], [0, 1, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ)."},"skills":["Open Images versus Open Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0095","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8fc0397c533deb0eb0ea9288a3a2591f66f584aa2765c80e6690a87fc10d4290","family_id":"qf-ext19-continuity-open-map-contrast","seed":202610193095,"semantic_template":"qf-ext19-continuity-open-map-contrast/{\"images\":[3,1,2,0],\"m\":4,\"n\":4,\"opens_x\":[[],[3],[1,3],[1,2,3],[0,1,2,3]],\"opens_y\":[[],[3],[2,3],[1,2,3],[0,1,2,3]]}"},"fields":{"title":"Open Images versus Open Preimages","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 1, 2, 0] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).","answer":{"kind":"expression","value":"([3, 1, 2, 0], No, No, [[[], [], Yes], [[3], [0], No], [[2, 3], [0, 2], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[3], [0], No], [[1, 3], [0, 1], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations)","explanation":"Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 1, 2, 0], No, No, [[[], [], Yes], [[3], [0], No], [[2, 3], [0, 2], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[3], [0], No], [[1, 3], [0, 1], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations)."},"skills":["Open Images versus Open Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0096","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"daab33e43e17c20a61276cd8e1aed3d40707f9dd3efbd643416e113eb2c2ee67","family_id":"qf-ext19-continuity-open-map-contrast","seed":202610193096,"semantic_template":"qf-ext19-continuity-open-map-contrast/{\"images\":[0,3,2,1],\"m\":4,\"n\":4,\"opens_x\":[[],[0],[0,3],[0,2,3],[0,1,2,3]],\"opens_y\":[[],[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]]}"},"fields":{"title":"Open Images versus Open Preimages","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[0, 3, 2, 1] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).","answer":{"kind":"expression","value":"([0, 3, 2, 1], No, Yes, [[[], [], Yes], [[0], [0], Yes], [[1], [3], No], [[0, 1], [0, 3], Yes], [[2], [2], No], [[0, 2], [0, 2], No], [[1, 2], [2, 3], No], [[0, 1, 2], [0, 2, 3], Yes], [[3], [1], No], [[0, 3], [0, 1], No], [[1, 3], [1, 3], No], [[0, 1, 3], [0, 1, 3], No], [[2, 3], [1, 2], No], [[0, 2, 3], [0, 1, 2], No], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[0], [0], Yes], [[0, 3], [0, 1], Yes], [[0, 2, 3], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ)","explanation":"Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 3, 2, 1], No, Yes, [[[], [], Yes], [[0], [0], Yes], [[1], [3], No], [[0, 1], [0, 3], Yes], [[2], [2], No], [[0, 2], [0, 2], No], [[1, 2], [2, 3], No], [[0, 1, 2], [0, 2, 3], Yes], [[3], [1], No], [[0, 3], [0, 1], No], [[1, 3], [1, 3], No], [[0, 1, 3], [0, 1, 3], No], [[2, 3], [1, 2], No], [[0, 2, 3], [0, 1, 2], No], [[1, 2, 3], [1, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[0], [0], Yes], [[0, 3], [0, 1], Yes], [[0, 2, 3], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ)."},"skills":["Open Images versus Open Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0097","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3bbabb0d58a2a6e548c48030982f3c5cd90f63acae565f64689500858def4358","family_id":"qf-ext19-continuity-open-map-contrast","seed":202610193097,"semantic_template":"qf-ext19-continuity-open-map-contrast/{\"images\":[1,3,2,0],\"m\":4,\"n\":4,\"opens_x\":[[],[1],[1,3],[1,2,3],[0,1,2,3]],\"opens_y\":[[],[1],[1,3],[0,1,3],[0,1,2,3]]}"},"fields":{"title":"Open Images versus Open Preimages","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]]. The complete total map has images=[1, 3, 2, 0] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).","answer":{"kind":"expression","value":"([1, 3, 2, 0], No, No, [[[], [], Yes], [[1], [0], No], [[1, 3], [0, 1], No], [[0, 1, 3], [0, 1, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[1], [3], No], [[1, 3], [0, 3], No], [[1, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations)","explanation":"Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 3, 2, 0], No, No, [[[], [], Yes], [[1], [0], No], [[1, 3], [0, 1], No], [[0, 1, 3], [0, 1, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[1], [3], No], [[1, 3], [0, 3], No], [[1, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations)."},"skills":["Open Images versus Open Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0098","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85142e0dc4650244f7b17c3a66d7ae192a3152451160c6a334a8a3a2a9093890","family_id":"qf-ext19-continuity-open-map-contrast","seed":202610193098,"semantic_template":"qf-ext19-continuity-open-map-contrast/{\"images\":[2,3,1,0],\"m\":4,\"n\":4,\"opens_x\":[[],[2],[2,3],[1,2,3],[0,1,2,3]],\"opens_y\":[[],[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]]}"},"fields":{"title":"Open Images versus Open Preimages","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [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]]. The complete total map has images=[2, 3, 1, 0] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).","answer":{"kind":"expression","value":"([2, 3, 1, 0], No, Yes, [[[], [], Yes], [[0], [3], No], [[1], [2], Yes], [[0, 1], [2, 3], Yes], [[2], [0], No], [[0, 2], [0, 3], No], [[1, 2], [0, 2], No], [[0, 1, 2], [0, 2, 3], No], [[3], [1], No], [[0, 3], [1, 3], No], [[1, 3], [1, 2], No], [[0, 1, 3], [1, 2, 3], Yes], [[2, 3], [0, 1], No], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[2], [1], Yes], [[2, 3], [0, 1], Yes], [[1, 2, 3], [0, 1, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ)","explanation":"Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 3, 1, 0], No, Yes, [[[], [], Yes], [[0], [3], No], [[1], [2], Yes], [[0, 1], [2, 3], Yes], [[2], [0], No], [[0, 2], [0, 3], No], [[1, 2], [0, 2], No], [[0, 1, 2], [0, 2, 3], No], [[3], [1], No], [[0, 3], [1, 3], No], [[1, 3], [1, 2], No], [[0, 1, 3], [1, 2, 3], Yes], [[2, 3], [0, 1], No], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[2], [1], Yes], [[2, 3], [0, 1], Yes], [[1, 2, 3], [0, 1, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], properties-differ)."},"skills":["Open Images versus Open Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0099","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5b188c96c1305f28781beeff9d9c4a733f41deaaa9ab745101bcdeb87a9f826","family_id":"qf-ext19-continuity-open-map-contrast","seed":202610193099,"semantic_template":"qf-ext19-continuity-open-map-contrast/{\"images\":[3,2,1,0],\"m\":4,\"n\":4,\"opens_x\":[[],[3],[2,3],[1,2,3],[0,1,2,3]],\"opens_y\":[[],[3],[1,3],[1,2,3],[0,1,2,3]]}"},"fields":{"title":"Open Images versus Open Preimages","stem":"Use complete finite topologies X=Fin(4)={0,…,3}, opens_X=[[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]] and Y=Fin(4)={0,…,3}, opens_Y=[[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. The complete total map has images=[3, 2, 1, 0] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).","answer":{"kind":"expression","value":"([3, 2, 1, 0], No, No, [[[], [], Yes], [[3], [0], No], [[1, 3], [0, 2], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[3], [0], No], [[2, 3], [0, 1], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations)","explanation":"Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 2, 1, 0], No, No, [[[], [], Yes], [[3], [0], No], [[1, 3], [0, 2], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], [[[], [], Yes], [[3], [0], No], [[2, 3], [0, 1], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], same-truth-value-but-different-obligations)."},"skills":["Open Images versus Open Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r03-topological-continuity-0100","batch_id":"batch-20261013-advance-20261019-r03-topological-continuity-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51f4e479a6027de1ec91d44f46056e7d50ecce2f654044712c1d3021a8c1fa87","family_id":"qf-ext19-continuity-open-map-contrast","seed":202610193100,"semantic_template":"qf-ext19-continuity-open-map-contrast/{\"images\":[0,1,2],\"m\":3,\"n\":3,\"opens_x\":[[],[0],[0,1],[0,1,2]],\"opens_y\":[[],[0],[1],[0,1],[2],[0,2],[1,2],[0,1,2]]}"},"fields":{"title":"Open Images versus Open Preimages","stem":"Use complete finite topologies X=Fin(3)={0,…,2}, opens_X=[[], [0], [0, 1], [0, 1, 2]] and Y=Fin(3)={0,…,2}, opens_Y=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2]]. The complete total map has images=[0, 1, 2] in source order. All entries must type-check; set preimages are not inverse functions. Give (images, continuous?, open map?, all target-open preimage rows, all source-open image rows, same-truth-value-but-different-obligations/properties-differ).","answer":{"kind":"expression","value":"([0, 1, 2], No, Yes, [[[], [], Yes], [[0], [0], Yes], [[1], [1], No], [[0, 1], [0, 1], Yes], [[2], [2], No], [[0, 2], [0, 2], No], [[1, 2], [1, 2], No], [[0, 1, 2], [0, 1, 2], Yes]], [[[], [], Yes], [[0], [0], Yes], [[0, 1], [0, 1], Yes], [[0, 1, 2], [0, 1, 2], Yes]], properties-differ)","explanation":"Audit all target opens by preimage and all source opens by image separately; equal truth values do not identify the two properties. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2], No, Yes, [[[], [], Yes], [[0], [0], Yes], [[1], [1], No], [[0, 1], [0, 1], Yes], [[2], [2], No], [[0, 2], [0, 2], No], [[1, 2], [1, 2], No], [[0, 1, 2], [0, 1, 2], Yes]], [[[], [], Yes], [[0], [0], Yes], [[0, 1], [0, 1], Yes], [[0, 1, 2], [0, 1, 2], Yes]], properties-differ)."},"skills":["Open Images versus Open Preimages"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}