{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0001","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ea12e84f0a35824ea699d3de6a7e5520c79d15b2c7cfbe3438b56377f47e634","family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055001,"semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,1,2,3,4],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]]}"},"fields":{"title":"A Homeomorphism Has Both Forward and Inverse Continuity","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]] and total candidate bijection map=[0, 1, 2, 3, 4], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)","explanation":"Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)."},"skills":["A Homeomorphism Has Both Forward and Inverse Continuity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0002","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"593efeaea7802b91483ebefe4b0631633fbe5dff3e4ed31802389b26157c3600","family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055002,"semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,1,2,4,3],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]]}"},"fields":{"title":"A Homeomorphism Has Both Forward and Inverse Continuity","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]] and total candidate bijection map=[0, 1, 2, 4, 3], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)","explanation":"Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)."},"skills":["A Homeomorphism Has Both Forward and Inverse Continuity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0003","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc8fa24f24065a85913ce52c728f452daafec5d027741ab75e076ac93f4f1845","family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055003,"semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,1,3,2,4],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,3],[0,2,4],[0],[1,2,3,4],[1,3],[2,4]]}"},"fields":{"title":"A Homeomorphism Has Both Forward and Inverse Continuity","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3], [0, 2, 4], [1, 2, 3, 4], [1, 3], [2, 4]] and total candidate bijection map=[0, 1, 3, 2, 4], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)","explanation":"Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)."},"skills":["A Homeomorphism Has Both Forward and Inverse Continuity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0004","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89d748eef8c44c931a46213d05610ec87138fa9ee7291d88d6708e652dd80dba","family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055004,"semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,1,3,4,2],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,3],[0,2,4],[0],[1,2,3,4],[1,3],[2,4]]}"},"fields":{"title":"A Homeomorphism Has Both Forward and Inverse Continuity","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3], [0, 2, 4], [1, 2, 3, 4], [1, 3], [2, 4]] and total candidate bijection map=[0, 1, 3, 4, 2], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)","explanation":"Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)."},"skills":["A Homeomorphism Has Both Forward and Inverse Continuity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0005","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29d6ae4161f7bedc8a458b2b5ac79202b89454ec514a4d60745ff1c14ceaa998","family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055005,"semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,1,4,2,3],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,4],[0,2,3],[0],[1,2,3,4],[1,4],[2,3]]}"},"fields":{"title":"A Homeomorphism Has Both Forward and Inverse Continuity","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 4], [0, 2, 3], [1, 2, 3, 4], [1, 4], [2, 3]] and total candidate bijection map=[0, 1, 4, 2, 3], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)","explanation":"Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)."},"skills":["A Homeomorphism Has Both Forward and Inverse Continuity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0006","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5ca987a28b5f6d3782833f8bffe49f1672e5457d0f5cfcf945f72936ec50cc2","family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055006,"semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,1,4,3,2],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,4],[0,2,3],[0],[1,2,3,4],[1,4],[2,3]]}"},"fields":{"title":"A Homeomorphism Has Both Forward and Inverse Continuity","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 4], [0, 2, 3], [1, 2, 3, 4], [1, 4], [2, 3]] and total candidate bijection map=[0, 1, 4, 3, 2], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)","explanation":"Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)."},"skills":["A Homeomorphism Has Both Forward and Inverse Continuity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0007","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05f51d64843c05a6453085b1d7b242ebc886a919e1d00ce986e6e4e97b1e7579","family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055007,"semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,2,1,3,4],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]]}"},"fields":{"title":"A Homeomorphism Has Both Forward and Inverse Continuity","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]] and total candidate bijection map=[0, 2, 1, 3, 4], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)","explanation":"Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)."},"skills":["A Homeomorphism Has Both Forward and Inverse Continuity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0008","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47b761b52e4f467b2e3fd4e2a8a909ff3fd15093ebf344349794d86590c02db4","family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055008,"semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,2,1,4,3],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]]}"},"fields":{"title":"A Homeomorphism Has Both Forward and Inverse Continuity","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]] and total candidate bijection map=[0, 2, 1, 4, 3], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)","explanation":"Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)."},"skills":["A Homeomorphism Has Both Forward and Inverse Continuity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0009","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"718b3609b60b46153a41ab8deb8792545efd599653ea3c86a589718d33d2dfff","family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055009,"semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,2,3,1,4],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,4],[0,2,3],[0],[1,2,3,4],[1,4],[2,3]]}"},"fields":{"title":"A Homeomorphism Has Both Forward and Inverse Continuity","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 4], [0, 2, 3], [1, 2, 3, 4], [1, 4], [2, 3]] and total candidate bijection map=[0, 2, 3, 1, 4], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)","explanation":"Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)."},"skills":["A Homeomorphism Has Both Forward and Inverse Continuity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0010","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14828a5e3d221a8a901dff0c1c01ceecfc25dcd5832a218a2a37e1da9086583f","family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055010,"semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,2,3,4,1],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,4],[0,2,3],[0],[1,2,3,4],[1,4],[2,3]]}"},"fields":{"title":"A Homeomorphism Has Both Forward and Inverse Continuity","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 4], [0, 2, 3], [1, 2, 3, 4], [1, 4], [2, 3]] and total candidate bijection map=[0, 2, 3, 4, 1], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)","explanation":"Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)."},"skills":["A Homeomorphism Has Both Forward and Inverse Continuity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0011","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f00111d87cb9629d6d6ce28c68df07dd54500b55939a3dacdae19c9a5b4ed527","family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055011,"semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,2,4,1,3],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,3],[0,2,4],[0],[1,2,3,4],[1,3],[2,4]]}"},"fields":{"title":"A Homeomorphism Has Both Forward and Inverse Continuity","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3], [0, 2, 4], [1, 2, 3, 4], [1, 3], [2, 4]] and total candidate bijection map=[0, 2, 4, 1, 3], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)","explanation":"Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)."},"skills":["A Homeomorphism Has Both Forward and Inverse Continuity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0012","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d95d3e24e38444515277cf3d94bcb031887f36b405beb2d791da4266e8dcee7","family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055012,"semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,2,4,3,1],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,3],[0,2,4],[0],[1,2,3,4],[1,3],[2,4]]}"},"fields":{"title":"A Homeomorphism Has Both Forward and Inverse Continuity","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3], [0, 2, 4], [1, 2, 3, 4], [1, 3], [2, 4]] and total candidate bijection map=[0, 2, 4, 3, 1], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)","explanation":"Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)."},"skills":["A Homeomorphism Has Both Forward and Inverse Continuity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0013","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1954f1e98fd849f9f8d55c49be3b83b4b3e7e382fc672a1a8351b3ec5ccd63f3","family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055013,"semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,3,1,2,4],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,3],[0,2,4],[0],[1,2,3,4],[1,3],[2,4]]}"},"fields":{"title":"A Homeomorphism Has Both Forward and Inverse Continuity","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3], [0, 2, 4], [1, 2, 3, 4], [1, 3], [2, 4]] and total candidate bijection map=[0, 3, 1, 2, 4], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)","explanation":"Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)."},"skills":["A Homeomorphism Has Both Forward and Inverse Continuity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0014","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8bc24367fb23b69105fb263585b97907ebef605cde5549a0f2768e77f2ae91b","family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055014,"semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,3,1,4,2],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,3],[0,2,4],[0],[1,2,3,4],[1,3],[2,4]]}"},"fields":{"title":"A Homeomorphism Has Both Forward and Inverse Continuity","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3], [0, 2, 4], [1, 2, 3, 4], [1, 3], [2, 4]] and total candidate bijection map=[0, 3, 1, 4, 2], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)","explanation":"Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)."},"skills":["A Homeomorphism Has Both Forward and Inverse Continuity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0015","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"763c45a87da5f06dc6b9005610a1055b2445f63b4740ae536414c6c1c4512267","family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055015,"semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,3,2,1,4],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,4],[0,2,3],[0],[1,2,3,4],[1,4],[2,3]]}"},"fields":{"title":"A Homeomorphism Has Both Forward and Inverse Continuity","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 4], [0, 2, 3], [1, 2, 3, 4], [1, 4], [2, 3]] and total candidate bijection map=[0, 3, 2, 1, 4], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)","explanation":"Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)."},"skills":["A Homeomorphism Has Both Forward and Inverse Continuity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0016","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76d8549cd4c50a614229b498d65033fbca566b55b6ac339bc32acf3b32dc9be4","family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055016,"semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,3,2,4,1],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,4],[0,2,3],[0],[1,2,3,4],[1,4],[2,3]]}"},"fields":{"title":"A Homeomorphism Has Both Forward and Inverse Continuity","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 4], [0, 2, 3], [1, 2, 3, 4], [1, 4], [2, 3]] and total candidate bijection map=[0, 3, 2, 4, 1], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)","explanation":"Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)."},"skills":["A Homeomorphism Has Both Forward and Inverse Continuity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0017","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd26a5fa816e5ae25f2fcedc59f56fc229a8f3381c8e88b233a5b462ec85da95","family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055017,"semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,3,4,1,2],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]]}"},"fields":{"title":"A Homeomorphism Has Both Forward and Inverse Continuity","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]] and total candidate bijection map=[0, 3, 4, 1, 2], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)","explanation":"Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)."},"skills":["A Homeomorphism Has Both Forward and Inverse Continuity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0018","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47a7ec2e493bfbf0d50de81697ca9c45c03ef88040402d0d736182b5cbde1559","family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055018,"semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,3,4,2,1],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]]}"},"fields":{"title":"A Homeomorphism Has Both Forward and Inverse Continuity","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]] and total candidate bijection map=[0, 3, 4, 2, 1], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)","explanation":"Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)."},"skills":["A Homeomorphism Has Both Forward and Inverse Continuity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0019","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a88dce185c0041ec4e0cdd7a0c3bf2cbaf22957794b423bba2b72da9de5f3f73","family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055019,"semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,4,1,2,3],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,4],[0,2,3],[0],[1,2,3,4],[1,4],[2,3]]}"},"fields":{"title":"A Homeomorphism Has Both Forward and Inverse Continuity","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 4], [0, 2, 3], [1, 2, 3, 4], [1, 4], [2, 3]] and total candidate bijection map=[0, 4, 1, 2, 3], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)","explanation":"Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)."},"skills":["A Homeomorphism Has Both Forward and Inverse Continuity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0020","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5934ce22a49ba19aca9694e434d8a4358566ed9e2585650dd7541cb10d8a8beb","family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055020,"semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,4,1,3,2],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,4],[0,2,3],[0],[1,2,3,4],[1,4],[2,3]]}"},"fields":{"title":"A Homeomorphism Has Both Forward and Inverse Continuity","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 4], [0, 2, 3], [1, 2, 3, 4], [1, 4], [2, 3]] and total candidate bijection map=[0, 4, 1, 3, 2], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)","explanation":"Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)."},"skills":["A Homeomorphism Has Both Forward and Inverse Continuity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0021","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"406dbd8711f8a4e21a7b92044119f5adf1ba89e2ae2d1d48d9c584f80c4e03dd","family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055021,"semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,4,2,1,3],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,3],[0,2,4],[0],[1,2,3,4],[1,3],[2,4]]}"},"fields":{"title":"A Homeomorphism Has Both Forward and Inverse Continuity","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3], [0, 2, 4], [1, 2, 3, 4], [1, 3], [2, 4]] and total candidate bijection map=[0, 4, 2, 1, 3], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)","explanation":"Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)."},"skills":["A Homeomorphism Has Both Forward and Inverse Continuity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0022","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"277964fcfe8c382cf08da9ce38de0a59a4fef60ef9832c7c49e24be5a6208a67","family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055022,"semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,4,2,3,1],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,3],[0,2,4],[0],[1,2,3,4],[1,3],[2,4]]}"},"fields":{"title":"A Homeomorphism Has Both Forward and Inverse Continuity","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3], [0, 2, 4], [1, 2, 3, 4], [1, 3], [2, 4]] and total candidate bijection map=[0, 4, 2, 3, 1], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)","explanation":"Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)."},"skills":["A Homeomorphism Has Both Forward and Inverse Continuity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0023","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d88ff66f554e15fcc948666cd95403d97ceb97aea43791abcc3798b0b1968c55","family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055023,"semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,4,3,1,2],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]]}"},"fields":{"title":"A Homeomorphism Has Both Forward and Inverse Continuity","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]] and total candidate bijection map=[0, 4, 3, 1, 2], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)","explanation":"Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)."},"skills":["A Homeomorphism Has Both Forward and Inverse Continuity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0024","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e4766f823e0d2ba7536626b49bad46b4c8ffe3ea65a695928eaa9af3503b552","family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055024,"semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[0,4,3,2,1],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]]}"},"fields":{"title":"A Homeomorphism Has Both Forward and Inverse Continuity","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]] and total candidate bijection map=[0, 4, 3, 2, 1], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)","explanation":"Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)."},"skills":["A Homeomorphism Has Both Forward and Inverse Continuity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0025","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4a31fd6f1905df53cb6a4afa39e3de51fc96b4cb1d17c77f4c4070f30e54631","family_id":"qf-mod36-homeo-finite-topology-audit","seed":202611055025,"semantic_template":"qf-mod36-homeo-finite-topology-audit/{\"map\":[1,0,2,3,4],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2],[0,3,4],[0],[1,2,3,4],[1,2],[3,4]],\"target_topology\":[[],[0,1,2,3,4],[0,1,2],[0,2,3,4],[0,2],[1,3,4],[1],[3,4]]}"},"fields":{"title":"A Homeomorphism Has Both Forward and Inverse Continuity","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 3, 4], [1, 2], [1, 2, 3, 4], [3, 4]], target_topology=[[], [0, 1, 2], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3, 4], [1], [1, 3, 4], [3, 4]] and total candidate bijection map=[1, 0, 2, 3, 4], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)","explanation":"Retain topology-validity and bijection flags, then test all open-set preimages in each direction. Only their conjunction supplies the homeomorphism flag for these complete finite spaces. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes, Yes, complete_open_set_preimage_audit)."},"skills":["A Homeomorphism Has Both Forward and Inverse Continuity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0026","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b730c1443228e0a841c3c6648b397bdab850bec3956a70f23aa1277fced9446","family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055026,"semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,1,2,3,4],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"},"fields":{"title":"A Bijection Must Still Pass Both Continuity Directions","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 1, 2, 3, 4], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)","explanation":"Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)."},"skills":["A Bijection Must Still Pass Both Continuity Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0027","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e11fe4de4c727d2fab6f7f74e906a5c652ed05ed5b1cfbcefe0692ef2118899d","family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055027,"semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,1,2,4,3],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"},"fields":{"title":"A Bijection Must Still Pass Both Continuity Directions","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 1, 2, 4, 3], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)","explanation":"Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)."},"skills":["A Bijection Must Still Pass Both Continuity Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0028","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae1b755f4ea8b0fc882ecf544a126f01a270043b34a98afead9c4915dbe5f96b","family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055028,"semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,1,3,2,4],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"},"fields":{"title":"A Bijection Must Still Pass Both Continuity Directions","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 1, 3, 2, 4], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)","explanation":"Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)."},"skills":["A Bijection Must Still Pass Both Continuity Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0029","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2edbbdbb7af03f434a31aee9e129f018c50acfaf03c48b26ea23cc7a41c4dc12","family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055029,"semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,1,3,4,2],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"},"fields":{"title":"A Bijection Must Still Pass Both Continuity Directions","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 1, 3, 4, 2], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)","explanation":"Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)."},"skills":["A Bijection Must Still Pass Both Continuity Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0030","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6bd54892d85a70bcbce1cc1222ba3a1fcbd61da0ecd83718c23430650e8b6422","family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055030,"semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,1,4,2,3],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"},"fields":{"title":"A Bijection Must Still Pass Both Continuity Directions","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 1, 4, 2, 3], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)","explanation":"Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)."},"skills":["A Bijection Must Still Pass Both Continuity Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0031","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f808129ab3f96f56495622e5ab9c41c74bab433198480ec4912c21d47fc2e819","family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055031,"semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,1,4,3,2],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"},"fields":{"title":"A Bijection Must Still Pass Both Continuity Directions","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 1, 4, 3, 2], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)","explanation":"Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)."},"skills":["A Bijection Must Still Pass Both Continuity Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0032","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71185e404727cdf20553739c49eabd0e4f58468b3a011124ffe9b9cef520c7ae","family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055032,"semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,2,1,3,4],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"},"fields":{"title":"A Bijection Must Still Pass Both Continuity Directions","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 2, 1, 3, 4], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)","explanation":"Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)."},"skills":["A Bijection Must Still Pass Both Continuity Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0033","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04e1689bb82458b4177071833bb0da0bf28ab0a053032d4703419c8889410818","family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055033,"semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,2,1,4,3],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"},"fields":{"title":"A Bijection Must Still Pass Both Continuity Directions","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 2, 1, 4, 3], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)","explanation":"Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)."},"skills":["A Bijection Must Still Pass Both Continuity Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0034","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba084a3b929da6bf759ceb968e08d3065d44439086cb0927f0f21547e5151af3","family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055034,"semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,2,3,1,4],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"},"fields":{"title":"A Bijection Must Still Pass Both Continuity Directions","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 2, 3, 1, 4], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)","explanation":"Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)."},"skills":["A Bijection Must Still Pass Both Continuity Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0035","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4714744811dea1adee132217c952ee3288f9a4f523531e019e486f124f51c88c","family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055035,"semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,2,3,4,1],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"},"fields":{"title":"A Bijection Must Still Pass Both Continuity Directions","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 2, 3, 4, 1], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)","explanation":"Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)."},"skills":["A Bijection Must Still Pass Both Continuity Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0036","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1702e7733c49c93960b90a796e685d71ecb5df765b1df83da4f9479a1e7639d3","family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055036,"semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,2,4,1,3],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"},"fields":{"title":"A Bijection Must Still Pass Both Continuity Directions","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 2, 4, 1, 3], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)","explanation":"Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)."},"skills":["A Bijection Must Still Pass Both Continuity Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0037","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4c6a574c78805da9865e47f93f333b016079d44ae28d295fe4dd6b06f5fad72","family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055037,"semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,2,4,3,1],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"},"fields":{"title":"A Bijection Must Still Pass Both Continuity Directions","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 2, 4, 3, 1], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)","explanation":"Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)."},"skills":["A Bijection Must Still Pass Both Continuity Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0038","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9369c11189321e296a264f63723e4929379a63c8c5cf09e1abf9d6615d7cf413","family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055038,"semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,3,1,2,4],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"},"fields":{"title":"A Bijection Must Still Pass Both Continuity Directions","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 3, 1, 2, 4], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)","explanation":"Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)."},"skills":["A Bijection Must Still Pass Both Continuity Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0039","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1886c180728a161d28e8c019462d175036b593244e18a8772b66ea660d5e69ab","family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055039,"semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,3,1,4,2],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"},"fields":{"title":"A Bijection Must Still Pass Both Continuity Directions","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 3, 1, 4, 2], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)","explanation":"Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)."},"skills":["A Bijection Must Still Pass Both Continuity Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0040","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7908e7c25f0710b3694d91e2d80b5346846a2e1a55f7a4fb951598351073762e","family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055040,"semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,3,2,1,4],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"},"fields":{"title":"A Bijection Must Still Pass Both Continuity Directions","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 3, 2, 1, 4], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)","explanation":"Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)."},"skills":["A Bijection Must Still Pass Both Continuity Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0041","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"356ee223536fbe28996b01e02d1c9705a9a8dceb9d0f43b5eccf0d227c978a2f","family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055041,"semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,3,2,4,1],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"},"fields":{"title":"A Bijection Must Still Pass Both Continuity Directions","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 3, 2, 4, 1], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)","explanation":"Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)."},"skills":["A Bijection Must Still Pass Both Continuity Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0042","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53ccd8ec993b16a006ef843a092a8e8b804989e1fdd3248a9a4a4a0aa397e2e6","family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055042,"semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,3,4,1,2],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"},"fields":{"title":"A Bijection Must Still Pass Both Continuity Directions","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 3, 4, 1, 2], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)","explanation":"Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)."},"skills":["A Bijection Must Still Pass Both Continuity Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0043","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e520bafb61606b2f60d8747b7889f844a46c90b52cea6e1fad49ce396a9ab3ca","family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055043,"semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,3,4,2,1],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"},"fields":{"title":"A Bijection Must Still Pass Both Continuity Directions","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 3, 4, 2, 1], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)","explanation":"Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)."},"skills":["A Bijection Must Still Pass Both Continuity Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0044","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5ac7d4334b8e4987e651c12138cd144219433997bec9059553d068255a72290","family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055044,"semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,4,1,2,3],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"},"fields":{"title":"A Bijection Must Still Pass Both Continuity Directions","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 4, 1, 2, 3], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)","explanation":"Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)."},"skills":["A Bijection Must Still Pass Both Continuity Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0045","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ccbf52167050b1e1f5740824c81b998b6f55a7eff97fdb3ef6f99f30e41bddf7","family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055045,"semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,4,1,3,2],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"},"fields":{"title":"A Bijection Must Still Pass Both Continuity Directions","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 4, 1, 3, 2], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)","explanation":"Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)."},"skills":["A Bijection Must Still Pass Both Continuity Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0046","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1def3399db21b529fd531d4db8e067d1830bc681ce0fa9f7efe9dbe40619fcbc","family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055046,"semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,4,2,1,3],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"},"fields":{"title":"A Bijection Must Still Pass Both Continuity Directions","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 4, 2, 1, 3], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)","explanation":"Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)."},"skills":["A Bijection Must Still Pass Both Continuity Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0047","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df7be481a86a1aa9bd24cb853928793a8b4340d2d424841ba2ef11c5881882c5","family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055047,"semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,4,2,3,1],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"},"fields":{"title":"A Bijection Must Still Pass Both Continuity Directions","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 4, 2, 3, 1], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)","explanation":"Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)."},"skills":["A Bijection Must Still Pass Both Continuity Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0048","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31161f6ee630c6c156b316a1ad8aad60c2e8172157243143ed6e9baaef27a3f2","family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055048,"semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,4,3,1,2],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"},"fields":{"title":"A Bijection Must Still Pass Both Continuity Directions","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 4, 3, 1, 2], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)","explanation":"Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)."},"skills":["A Bijection Must Still Pass Both Continuity Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0049","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c7e589dbee42611ecf4e237de81f64ea4b5ad9ddbc48b7f0fbf103dc5801c5b","family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055049,"semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[0,4,3,2,1],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"},"fields":{"title":"A Bijection Must Still Pass Both Continuity Directions","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[0, 4, 3, 2, 1], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)","explanation":"Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)."},"skills":["A Bijection Must Still Pass Both Continuity Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0050","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"510472d6a65bcc615cfb1c877f2bcc6d93ffd4065e516db87ebe09a015c9eb7f","family_id":"qf-mod36-homeo-continuous-bijection-boundary","seed":202611055050,"semantic_template":"qf-mod36-homeo-continuous-bijection-boundary/{\"map\":[1,0,2,3,4],\"n\":5,\"source_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]],\"target_topology\":[[],[0,1,2,3,4]]}"},"fields":{"title":"A Bijection Must Still Pass Both Continuity Directions","stem":"Use source and target on Fin(5)={0,…,4}, with complete source_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], target_topology=[[], [0, 1, 2, 3, 4]] and total candidate bijection map=[1, 0, 2, 3, 4], in source-index order. Check both complete topology axioms, bijection, every target-open-set preimage and every source-open-set inverse preimage separately. A continuous bijection alone need not have continuous inverse. Give (source_topology_valid, target_topology_valid, bijection, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)","explanation":"Compute both directions rather than assuming either one from the family name. A discrete-to-indiscrete continuous bijection can have a noncontinuous inverse; reversing its direction interchanges the two continuity obligations. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, No, No, continuous_bijection_inverse_continuity_required)."},"skills":["A Bijection Must Still Pass Both Continuity Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0051","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04f65fbbb780a05f039b90d06d8b9cb974ed9ccc85999f3c23b4e9763c521523","family_id":"qf-mod36-homeo-finite-discrete-bijection","seed":202611055051,"semantic_template":"qf-mod36-homeo-finite-discrete-bijection/{\"map\":[0,1,2,3,4],\"n\":5}"},"fields":{"title":"A Finite Discrete Bijection Is a Homeomorphism within That Scope","stem":"Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 1, 2, 3, 4]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(5, [0, 1, 2, 3, 4], Yes, Yes, Yes, finite_discrete_scope_only)","explanation":"Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 1, 2, 3, 4], Yes, Yes, Yes, finite_discrete_scope_only)."},"skills":["A Finite Discrete Bijection Is a Homeomorphism within That Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0052","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14456eea831f8a67b38475b2093f80ba69a06aa9ef0fe7f56ad004b722821052","family_id":"qf-mod36-homeo-finite-discrete-bijection","seed":202611055052,"semantic_template":"qf-mod36-homeo-finite-discrete-bijection/{\"map\":[0,1,2,4,3],\"n\":5}"},"fields":{"title":"A Finite Discrete Bijection Is a Homeomorphism within That Scope","stem":"Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 1, 2, 4, 3]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(5, [0, 1, 2, 4, 3], Yes, Yes, Yes, finite_discrete_scope_only)","explanation":"Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 1, 2, 4, 3], Yes, Yes, Yes, finite_discrete_scope_only)."},"skills":["A Finite Discrete Bijection Is a Homeomorphism within That Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0053","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7347f83704b9537856a8c45de82fd02d2107266cd670b340b107664502688c79","family_id":"qf-mod36-homeo-finite-discrete-bijection","seed":202611055053,"semantic_template":"qf-mod36-homeo-finite-discrete-bijection/{\"map\":[0,1,3,2,4],\"n\":5}"},"fields":{"title":"A Finite Discrete Bijection Is a Homeomorphism within That Scope","stem":"Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 1, 3, 2, 4]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(5, [0, 1, 3, 2, 4], Yes, Yes, Yes, finite_discrete_scope_only)","explanation":"Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 1, 3, 2, 4], Yes, Yes, Yes, finite_discrete_scope_only)."},"skills":["A Finite Discrete Bijection Is a Homeomorphism within That Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0054","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79100dc36d7eee42c62b6744fe17e763d35860aed6331295d307b0a80636608d","family_id":"qf-mod36-homeo-finite-discrete-bijection","seed":202611055054,"semantic_template":"qf-mod36-homeo-finite-discrete-bijection/{\"map\":[0,1,3,4,2],\"n\":5}"},"fields":{"title":"A Finite Discrete Bijection Is a Homeomorphism within That Scope","stem":"Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 1, 3, 4, 2]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(5, [0, 1, 3, 4, 2], Yes, Yes, Yes, finite_discrete_scope_only)","explanation":"Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 1, 3, 4, 2], Yes, Yes, Yes, finite_discrete_scope_only)."},"skills":["A Finite Discrete Bijection Is a Homeomorphism within That Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0055","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f19c5132e717e61b13ed93fc10baf913653b483e9235f7fe29d8700774a6ecb1","family_id":"qf-mod36-homeo-finite-discrete-bijection","seed":202611055055,"semantic_template":"qf-mod36-homeo-finite-discrete-bijection/{\"map\":[0,1,4,2,3],\"n\":5}"},"fields":{"title":"A Finite Discrete Bijection Is a Homeomorphism within That Scope","stem":"Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 1, 4, 2, 3]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(5, [0, 1, 4, 2, 3], Yes, Yes, Yes, finite_discrete_scope_only)","explanation":"Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 1, 4, 2, 3], Yes, Yes, Yes, finite_discrete_scope_only)."},"skills":["A Finite Discrete Bijection Is a Homeomorphism within That Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0056","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5b4a01a718b72dddafe7212fc3aa3e00efc518030b462fc9b09c30fa281e365","family_id":"qf-mod36-homeo-finite-discrete-bijection","seed":202611055056,"semantic_template":"qf-mod36-homeo-finite-discrete-bijection/{\"map\":[0,1,4,3,2],\"n\":5}"},"fields":{"title":"A Finite Discrete Bijection Is a Homeomorphism within That Scope","stem":"Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 1, 4, 3, 2]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(5, [0, 1, 4, 3, 2], Yes, Yes, Yes, finite_discrete_scope_only)","explanation":"Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 1, 4, 3, 2], Yes, Yes, Yes, finite_discrete_scope_only)."},"skills":["A Finite Discrete Bijection Is a Homeomorphism within That Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0057","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"186aac284e64a6f18dc82836551331fa40d9103925294d7d81c0c9992c855295","family_id":"qf-mod36-homeo-finite-discrete-bijection","seed":202611055057,"semantic_template":"qf-mod36-homeo-finite-discrete-bijection/{\"map\":[0,2,1,3,4],\"n\":5}"},"fields":{"title":"A Finite Discrete Bijection Is a Homeomorphism within That Scope","stem":"Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 2, 1, 3, 4]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(5, [0, 2, 1, 3, 4], Yes, Yes, Yes, finite_discrete_scope_only)","explanation":"Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 2, 1, 3, 4], Yes, Yes, Yes, finite_discrete_scope_only)."},"skills":["A Finite Discrete Bijection Is a Homeomorphism within That Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0058","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"237cd3e2865f8877333a19f3e0798294c41f930cf5b29140458eb67024ea3029","family_id":"qf-mod36-homeo-finite-discrete-bijection","seed":202611055058,"semantic_template":"qf-mod36-homeo-finite-discrete-bijection/{\"map\":[0,2,1,4,3],\"n\":5}"},"fields":{"title":"A Finite Discrete Bijection Is a Homeomorphism within That Scope","stem":"Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 2, 1, 4, 3]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(5, [0, 2, 1, 4, 3], Yes, Yes, Yes, finite_discrete_scope_only)","explanation":"Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 2, 1, 4, 3], Yes, Yes, Yes, finite_discrete_scope_only)."},"skills":["A Finite Discrete Bijection Is a Homeomorphism within That Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0059","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee82ddc421e7564c30d2e5a10b8f962724ebc286992913d4db42443cfc4c50c0","family_id":"qf-mod36-homeo-finite-discrete-bijection","seed":202611055059,"semantic_template":"qf-mod36-homeo-finite-discrete-bijection/{\"map\":[0,2,3,1,4],\"n\":5}"},"fields":{"title":"A Finite Discrete Bijection Is a Homeomorphism within That Scope","stem":"Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 2, 3, 1, 4]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(5, [0, 2, 3, 1, 4], Yes, Yes, Yes, finite_discrete_scope_only)","explanation":"Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 2, 3, 1, 4], Yes, Yes, Yes, finite_discrete_scope_only)."},"skills":["A Finite Discrete Bijection Is a Homeomorphism within That Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0060","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b517db27ecfbd060168d7d1600068b83c91bc66d19a38b4db1b26d62f04c3ace","family_id":"qf-mod36-homeo-finite-discrete-bijection","seed":202611055060,"semantic_template":"qf-mod36-homeo-finite-discrete-bijection/{\"map\":[0,2,3,4,1],\"n\":5}"},"fields":{"title":"A Finite Discrete Bijection Is a Homeomorphism within That Scope","stem":"Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 2, 3, 4, 1]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(5, [0, 2, 3, 4, 1], Yes, Yes, Yes, finite_discrete_scope_only)","explanation":"Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 2, 3, 4, 1], Yes, Yes, Yes, finite_discrete_scope_only)."},"skills":["A Finite Discrete Bijection Is a Homeomorphism within That Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0061","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c897da52033935268cd75bc3429ad1fd54451176d732996eeb698bdb965e9dde","family_id":"qf-mod36-homeo-finite-discrete-bijection","seed":202611055061,"semantic_template":"qf-mod36-homeo-finite-discrete-bijection/{\"map\":[0,2,4,1,3],\"n\":5}"},"fields":{"title":"A Finite Discrete Bijection Is a Homeomorphism within That Scope","stem":"Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 2, 4, 1, 3]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(5, [0, 2, 4, 1, 3], Yes, Yes, Yes, finite_discrete_scope_only)","explanation":"Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 2, 4, 1, 3], Yes, Yes, Yes, finite_discrete_scope_only)."},"skills":["A Finite Discrete Bijection Is a Homeomorphism within That Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0062","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7c65d47e4ee1e6aa38ac36c9984b29187068417997085c9a33cbac5d0c96f577","family_id":"qf-mod36-homeo-finite-discrete-bijection","seed":202611055062,"semantic_template":"qf-mod36-homeo-finite-discrete-bijection/{\"map\":[0,2,4,3,1],\"n\":5}"},"fields":{"title":"A Finite Discrete Bijection Is a Homeomorphism within That Scope","stem":"Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 2, 4, 3, 1]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(5, [0, 2, 4, 3, 1], Yes, Yes, Yes, finite_discrete_scope_only)","explanation":"Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 2, 4, 3, 1], Yes, Yes, Yes, finite_discrete_scope_only)."},"skills":["A Finite Discrete Bijection Is a Homeomorphism within That Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0063","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2f3ab4631dc35715efbde8869cfb6438417bcf7aeaa701c3f9c0d32cd54d207","family_id":"qf-mod36-homeo-finite-discrete-bijection","seed":202611055063,"semantic_template":"qf-mod36-homeo-finite-discrete-bijection/{\"map\":[0,3,1,2,4],\"n\":5}"},"fields":{"title":"A Finite Discrete Bijection Is a Homeomorphism within That Scope","stem":"Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 3, 1, 2, 4]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(5, [0, 3, 1, 2, 4], Yes, Yes, Yes, finite_discrete_scope_only)","explanation":"Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 3, 1, 2, 4], Yes, Yes, Yes, finite_discrete_scope_only)."},"skills":["A Finite Discrete Bijection Is a Homeomorphism within That Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0064","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d19ca5bc2a6c51080fd47fad67bd1f9a24aa9f28a25b35c13502fe4e235d7123","family_id":"qf-mod36-homeo-finite-discrete-bijection","seed":202611055064,"semantic_template":"qf-mod36-homeo-finite-discrete-bijection/{\"map\":[0,3,1,4,2],\"n\":5}"},"fields":{"title":"A Finite Discrete Bijection Is a Homeomorphism within That Scope","stem":"Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 3, 1, 4, 2]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(5, [0, 3, 1, 4, 2], Yes, Yes, Yes, finite_discrete_scope_only)","explanation":"Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 3, 1, 4, 2], Yes, Yes, Yes, finite_discrete_scope_only)."},"skills":["A Finite Discrete Bijection Is a Homeomorphism within That Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0065","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d288a2dae9d47ee2bb87ec397a6d926cf4fea17dc89e288eb6f599bb2fba18cd","family_id":"qf-mod36-homeo-finite-discrete-bijection","seed":202611055065,"semantic_template":"qf-mod36-homeo-finite-discrete-bijection/{\"map\":[0,3,2,1,4],\"n\":5}"},"fields":{"title":"A Finite Discrete Bijection Is a Homeomorphism within That Scope","stem":"Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 3, 2, 1, 4]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(5, [0, 3, 2, 1, 4], Yes, Yes, Yes, finite_discrete_scope_only)","explanation":"Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 3, 2, 1, 4], Yes, Yes, Yes, finite_discrete_scope_only)."},"skills":["A Finite Discrete Bijection Is a Homeomorphism within That Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0066","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65bdab4d975b4d812e7adab079111e581681663c355ee37c5f58753d9ce06194","family_id":"qf-mod36-homeo-finite-discrete-bijection","seed":202611055066,"semantic_template":"qf-mod36-homeo-finite-discrete-bijection/{\"map\":[0,3,2,4,1],\"n\":5}"},"fields":{"title":"A Finite Discrete Bijection Is a Homeomorphism within That Scope","stem":"Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 3, 2, 4, 1]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(5, [0, 3, 2, 4, 1], Yes, Yes, Yes, finite_discrete_scope_only)","explanation":"Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 3, 2, 4, 1], Yes, Yes, Yes, finite_discrete_scope_only)."},"skills":["A Finite Discrete Bijection Is a Homeomorphism within That Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0067","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ee1447b99c7e003e7ca36518e8facabd53c65ccecb7dccd2d388d6c927462c3","family_id":"qf-mod36-homeo-finite-discrete-bijection","seed":202611055067,"semantic_template":"qf-mod36-homeo-finite-discrete-bijection/{\"map\":[0,3,4,1,2],\"n\":5}"},"fields":{"title":"A Finite Discrete Bijection Is a Homeomorphism within That Scope","stem":"Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 3, 4, 1, 2]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(5, [0, 3, 4, 1, 2], Yes, Yes, Yes, finite_discrete_scope_only)","explanation":"Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 3, 4, 1, 2], Yes, Yes, Yes, finite_discrete_scope_only)."},"skills":["A Finite Discrete Bijection Is a Homeomorphism within That Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0068","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b69051325c186d7b925f389ef2d6a49e24deb537e3165d177852beac27bfe6b","family_id":"qf-mod36-homeo-finite-discrete-bijection","seed":202611055068,"semantic_template":"qf-mod36-homeo-finite-discrete-bijection/{\"map\":[0,3,4,2,1],\"n\":5}"},"fields":{"title":"A Finite Discrete Bijection Is a Homeomorphism within That Scope","stem":"Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 3, 4, 2, 1]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(5, [0, 3, 4, 2, 1], Yes, Yes, Yes, finite_discrete_scope_only)","explanation":"Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 3, 4, 2, 1], Yes, Yes, Yes, finite_discrete_scope_only)."},"skills":["A Finite Discrete Bijection Is a Homeomorphism within That Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0069","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"914a42911738ca1611abb526ac2965df1b85f2d5dc980710fb52c6aff965928f","family_id":"qf-mod36-homeo-finite-discrete-bijection","seed":202611055069,"semantic_template":"qf-mod36-homeo-finite-discrete-bijection/{\"map\":[0,4,1,2,3],\"n\":5}"},"fields":{"title":"A Finite Discrete Bijection Is a Homeomorphism within That Scope","stem":"Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 4, 1, 2, 3]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(5, [0, 4, 1, 2, 3], Yes, Yes, Yes, finite_discrete_scope_only)","explanation":"Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 4, 1, 2, 3], Yes, Yes, Yes, finite_discrete_scope_only)."},"skills":["A Finite Discrete Bijection Is a Homeomorphism within That Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0070","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35ef173d53b1f308761f7956880e800500c533f3d99c81d92e789f82150bae4b","family_id":"qf-mod36-homeo-finite-discrete-bijection","seed":202611055070,"semantic_template":"qf-mod36-homeo-finite-discrete-bijection/{\"map\":[0,4,1,3,2],\"n\":5}"},"fields":{"title":"A Finite Discrete Bijection Is a Homeomorphism within That Scope","stem":"Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 4, 1, 3, 2]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(5, [0, 4, 1, 3, 2], Yes, Yes, Yes, finite_discrete_scope_only)","explanation":"Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 4, 1, 3, 2], Yes, Yes, Yes, finite_discrete_scope_only)."},"skills":["A Finite Discrete Bijection Is a Homeomorphism within That Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0071","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2d2ee3fdcf7b157aeeb55630b6f04313f1591309bc3337311eb387c361819c2","family_id":"qf-mod36-homeo-finite-discrete-bijection","seed":202611055071,"semantic_template":"qf-mod36-homeo-finite-discrete-bijection/{\"map\":[0,4,2,1,3],\"n\":5}"},"fields":{"title":"A Finite Discrete Bijection Is a Homeomorphism within That Scope","stem":"Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 4, 2, 1, 3]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(5, [0, 4, 2, 1, 3], Yes, Yes, Yes, finite_discrete_scope_only)","explanation":"Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 4, 2, 1, 3], Yes, Yes, Yes, finite_discrete_scope_only)."},"skills":["A Finite Discrete Bijection Is a Homeomorphism within That Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0072","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53f15a9cc4fddcdd419f256520913b5a1b9d006847c3b1fd66e9dfabdadb4e80","family_id":"qf-mod36-homeo-finite-discrete-bijection","seed":202611055072,"semantic_template":"qf-mod36-homeo-finite-discrete-bijection/{\"map\":[0,4,2,3,1],\"n\":5}"},"fields":{"title":"A Finite Discrete Bijection Is a Homeomorphism within That Scope","stem":"Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 4, 2, 3, 1]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(5, [0, 4, 2, 3, 1], Yes, Yes, Yes, finite_discrete_scope_only)","explanation":"Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 4, 2, 3, 1], Yes, Yes, Yes, finite_discrete_scope_only)."},"skills":["A Finite Discrete Bijection Is a Homeomorphism within That Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0073","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9ee0890be97db2c244e68aaa4d4a2c11cb37c75f05e6748a1e46a2c1acf0e1c","family_id":"qf-mod36-homeo-finite-discrete-bijection","seed":202611055073,"semantic_template":"qf-mod36-homeo-finite-discrete-bijection/{\"map\":[0,4,3,1,2],\"n\":5}"},"fields":{"title":"A Finite Discrete Bijection Is a Homeomorphism within That Scope","stem":"Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 4, 3, 1, 2]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(5, [0, 4, 3, 1, 2], Yes, Yes, Yes, finite_discrete_scope_only)","explanation":"Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 4, 3, 1, 2], Yes, Yes, Yes, finite_discrete_scope_only)."},"skills":["A Finite Discrete Bijection Is a Homeomorphism within That Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0074","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b05d722b088c132b5b1825542abe55d0f9e818ef59f46ab4506df59c165d0fc1","family_id":"qf-mod36-homeo-finite-discrete-bijection","seed":202611055074,"semantic_template":"qf-mod36-homeo-finite-discrete-bijection/{\"map\":[0,4,3,2,1],\"n\":5}"},"fields":{"title":"A Finite Discrete Bijection Is a Homeomorphism within That Scope","stem":"Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[0, 4, 3, 2, 1]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(5, [0, 4, 3, 2, 1], Yes, Yes, Yes, finite_discrete_scope_only)","explanation":"Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [0, 4, 3, 2, 1], Yes, Yes, Yes, finite_discrete_scope_only)."},"skills":["A Finite Discrete Bijection Is a Homeomorphism within That Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0075","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fabbf64cf9a301d1413ca94e083e13030dc4d5b084f950239ea5822cc5fc6fac","family_id":"qf-mod36-homeo-finite-discrete-bijection","seed":202611055075,"semantic_template":"qf-mod36-homeo-finite-discrete-bijection/{\"map\":[1,0,2,3,4],\"n\":5}"},"fields":{"title":"A Finite Discrete Bijection Is a Homeomorphism within That Scope","stem":"Use Fin(5)={0,…,4} with discrete topology on both sides and complete map=[1, 0, 2, 3, 4]. Give (vertex_count, map, forward_continuous, inverse_continuous, homeomorphism, boundary).","answer":{"kind":"expression","value":"(5, [1, 0, 2, 3, 4], Yes, Yes, Yes, finite_discrete_scope_only)","explanation":"Validate the total bijection. Every subset is open in each discrete space, so both directions are continuous. Do not extrapolate to arbitrary topologies or infer metric or differentiable structure. The exact result, retaining the requested order, is (5, [1, 0, 2, 3, 4], Yes, Yes, Yes, finite_discrete_scope_only)."},"skills":["A Finite Discrete Bijection Is a Homeomorphism within That Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0076","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b1cd10c93df38c235662fc12a1e0c00eb1f384ca7a4467868f0bca610578ec1","family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055076,"semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,1,2,3,4],\"n\":5,\"source_scale\":1,\"target_scale\":2}"},"fields":{"title":"Equal Scale Parameters Are Not an Unqualified Isometry Test","stem":"Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 1, 2, 3, 4], positive source_scale=1 and target_scale=2. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).","answer":{"kind":"expression","value":"(5, Yes, No, 1, 2, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation":"The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 1, 2, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)."},"skills":["Equal Scale Parameters Are Not an Unqualified Isometry Test"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0077","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"132f5ee325a23f88639618f37021469516115f5634ebcf66d5a8b8c343e70a27","family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055077,"semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,1,2,4,3],\"n\":5,\"source_scale\":2,\"target_scale\":3}"},"fields":{"title":"Equal Scale Parameters Are Not an Unqualified Isometry Test","stem":"Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 1, 2, 4, 3], positive source_scale=2 and target_scale=3. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).","answer":{"kind":"expression","value":"(5, Yes, No, 2, 3, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation":"The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 2, 3, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)."},"skills":["Equal Scale Parameters Are Not an Unqualified Isometry Test"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0078","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6544df993c2effd748e88343ff273474492de4452690240c7a4968cb987e7c27","family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055078,"semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,1,3,2,4],\"n\":5,\"source_scale\":3,\"target_scale\":4}"},"fields":{"title":"Equal Scale Parameters Are Not an Unqualified Isometry Test","stem":"Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 1, 3, 2, 4], positive source_scale=3 and target_scale=4. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).","answer":{"kind":"expression","value":"(5, Yes, No, 3, 4, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation":"The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 3, 4, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)."},"skills":["Equal Scale Parameters Are Not an Unqualified Isometry Test"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0079","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60c6c39fabc8f0a032ebc664453297bbe14651c10a8d60bdad3400dd887a6a70","family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055079,"semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,1,3,4,2],\"n\":5,\"source_scale\":4,\"target_scale\":5}"},"fields":{"title":"Equal Scale Parameters Are Not an Unqualified Isometry Test","stem":"Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 1, 3, 4, 2], positive source_scale=4 and target_scale=5. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).","answer":{"kind":"expression","value":"(5, Yes, No, 4, 5, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation":"The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 4, 5, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)."},"skills":["Equal Scale Parameters Are Not an Unqualified Isometry Test"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0080","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"140cf395310de8052fbaed29ff8e321fb7612a2ff8e3ff0031261daddc84ad01","family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055080,"semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,1,4,2,3],\"n\":5,\"source_scale\":5,\"target_scale\":6}"},"fields":{"title":"Equal Scale Parameters Are Not an Unqualified Isometry Test","stem":"Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 1, 4, 2, 3], positive source_scale=5 and target_scale=6. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).","answer":{"kind":"expression","value":"(5, Yes, No, 5, 6, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation":"The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 5, 6, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)."},"skills":["Equal Scale Parameters Are Not an Unqualified Isometry Test"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0081","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"183fafcb9f06d7770207535c54728f41a9c494c50674ba8f4487b1656229db68","family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055081,"semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,1,4,3,2],\"n\":5,\"source_scale\":6,\"target_scale\":7}"},"fields":{"title":"Equal Scale Parameters Are Not an Unqualified Isometry Test","stem":"Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 1, 4, 3, 2], positive source_scale=6 and target_scale=7. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).","answer":{"kind":"expression","value":"(5, Yes, No, 6, 7, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation":"The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 6, 7, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)."},"skills":["Equal Scale Parameters Are Not an Unqualified Isometry Test"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0082","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a57b52c08f213f16cac0cea1460af0700349b334f524aafef873300520a45f93","family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055082,"semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,2,1,3,4],\"n\":5,\"source_scale\":7,\"target_scale\":8}"},"fields":{"title":"Equal Scale Parameters Are Not an Unqualified Isometry Test","stem":"Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 2, 1, 3, 4], positive source_scale=7 and target_scale=8. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).","answer":{"kind":"expression","value":"(5, Yes, No, 7, 8, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation":"The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 7, 8, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)."},"skills":["Equal Scale Parameters Are Not an Unqualified Isometry Test"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0083","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef644968dc9d44c7e0c622aaa5fe449cc94870a68560804ddeca3079238e3e4c","family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055083,"semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,2,1,4,3],\"n\":5,\"source_scale\":8,\"target_scale\":9}"},"fields":{"title":"Equal Scale Parameters Are Not an Unqualified Isometry Test","stem":"Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 2, 1, 4, 3], positive source_scale=8 and target_scale=9. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).","answer":{"kind":"expression","value":"(5, Yes, No, 8, 9, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation":"The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 8, 9, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)."},"skills":["Equal Scale Parameters Are Not an Unqualified Isometry Test"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0084","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c8573c70b13559486593e9b43d0fa64e94e6c3cf4452c7d6e4b181c73527442","family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055084,"semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,2,3,1,4],\"n\":5,\"source_scale\":9,\"target_scale\":10}"},"fields":{"title":"Equal Scale Parameters Are Not an Unqualified Isometry Test","stem":"Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 2, 3, 1, 4], positive source_scale=9 and target_scale=10. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).","answer":{"kind":"expression","value":"(5, Yes, No, 9, 10, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation":"The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 9, 10, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)."},"skills":["Equal Scale Parameters Are Not an Unqualified Isometry Test"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0085","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4a76dcd48e2658a0844139b3a31eeb293848cc34139cbdd371f22e1b2591322","family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055085,"semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,2,3,4,1],\"n\":5,\"source_scale\":10,\"target_scale\":11}"},"fields":{"title":"Equal Scale Parameters Are Not an Unqualified Isometry Test","stem":"Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 2, 3, 4, 1], positive source_scale=10 and target_scale=11. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).","answer":{"kind":"expression","value":"(5, Yes, No, 10, 11, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation":"The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 10, 11, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)."},"skills":["Equal Scale Parameters Are Not an Unqualified Isometry Test"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0086","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8683f3bac0282d9f2611ed0736f52f55290c1b165de95a9e0204f5a4f467368d","family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055086,"semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,2,4,1,3],\"n\":5,\"source_scale\":11,\"target_scale\":12}"},"fields":{"title":"Equal Scale Parameters Are Not an Unqualified Isometry Test","stem":"Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 2, 4, 1, 3], positive source_scale=11 and target_scale=12. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).","answer":{"kind":"expression","value":"(5, Yes, No, 11, 12, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation":"The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 11, 12, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)."},"skills":["Equal Scale Parameters Are Not an Unqualified Isometry Test"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0087","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c16376bdc59c7666dc62946674f9d3c59a2663add5f16b2ec801a74db6ab3bf","family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055087,"semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,2,4,3,1],\"n\":5,\"source_scale\":12,\"target_scale\":13}"},"fields":{"title":"Equal Scale Parameters Are Not an Unqualified Isometry Test","stem":"Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 2, 4, 3, 1], positive source_scale=12 and target_scale=13. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).","answer":{"kind":"expression","value":"(5, Yes, No, 12, 13, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation":"The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 12, 13, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)."},"skills":["Equal Scale Parameters Are Not an Unqualified Isometry Test"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0088","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c79e1b5f9475ae7fc3f6973254144ae694b23ff81e939979cfb0e171d87774fa","family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055088,"semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,3,1,2,4],\"n\":5,\"source_scale\":13,\"target_scale\":14}"},"fields":{"title":"Equal Scale Parameters Are Not an Unqualified Isometry Test","stem":"Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 3, 1, 2, 4], positive source_scale=13 and target_scale=14. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).","answer":{"kind":"expression","value":"(5, Yes, No, 13, 14, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation":"The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 13, 14, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)."},"skills":["Equal Scale Parameters Are Not an Unqualified Isometry Test"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0089","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16756cd52ac27cef6c90285493a2af9fa633e95d389f61a4452860b09758fd01","family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055089,"semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,3,1,4,2],\"n\":5,\"source_scale\":14,\"target_scale\":15}"},"fields":{"title":"Equal Scale Parameters Are Not an Unqualified Isometry Test","stem":"Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 3, 1, 4, 2], positive source_scale=14 and target_scale=15. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).","answer":{"kind":"expression","value":"(5, Yes, No, 14, 15, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation":"The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 14, 15, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)."},"skills":["Equal Scale Parameters Are Not an Unqualified Isometry Test"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0090","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c91cfae89b7aa865e980cddf6a93e90273bfcf32301763892290618759f1696","family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055090,"semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,3,2,1,4],\"n\":5,\"source_scale\":15,\"target_scale\":16}"},"fields":{"title":"Equal Scale Parameters Are Not an Unqualified Isometry Test","stem":"Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 3, 2, 1, 4], positive source_scale=15 and target_scale=16. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).","answer":{"kind":"expression","value":"(5, Yes, No, 15, 16, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation":"The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 15, 16, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)."},"skills":["Equal Scale Parameters Are Not an Unqualified Isometry Test"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0091","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1aab5f904bea0a1ad9a817bccc0d5f5b398fe8d0cb647fe2322fb38d5edcad0c","family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055091,"semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,3,2,4,1],\"n\":5,\"source_scale\":16,\"target_scale\":17}"},"fields":{"title":"Equal Scale Parameters Are Not an Unqualified Isometry Test","stem":"Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 3, 2, 4, 1], positive source_scale=16 and target_scale=17. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).","answer":{"kind":"expression","value":"(5, Yes, No, 16, 17, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation":"The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 16, 17, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)."},"skills":["Equal Scale Parameters Are Not an Unqualified Isometry Test"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0092","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"322f36ffe9d83d366f38fc5ddc8f14d3b8b52553426c3ff3f18ba437e55683b8","family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055092,"semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,3,4,1,2],\"n\":5,\"source_scale\":17,\"target_scale\":18}"},"fields":{"title":"Equal Scale Parameters Are Not an Unqualified Isometry Test","stem":"Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 3, 4, 1, 2], positive source_scale=17 and target_scale=18. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).","answer":{"kind":"expression","value":"(5, Yes, No, 17, 18, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation":"The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 17, 18, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)."},"skills":["Equal Scale Parameters Are Not an Unqualified Isometry Test"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0093","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37114d8326d169ffd40c693031cdd68bad6f5f6f4cf6188b3b071a41280abb2d","family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055093,"semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,3,4,2,1],\"n\":5,\"source_scale\":18,\"target_scale\":19}"},"fields":{"title":"Equal Scale Parameters Are Not an Unqualified Isometry Test","stem":"Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 3, 4, 2, 1], positive source_scale=18 and target_scale=19. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).","answer":{"kind":"expression","value":"(5, Yes, No, 18, 19, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation":"The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 18, 19, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)."},"skills":["Equal Scale Parameters Are Not an Unqualified Isometry Test"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0094","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f9bf5a904fa2ef9f644003f6776e87dd6e9568f483b07837d6952c5ca02caf2","family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055094,"semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,4,1,2,3],\"n\":5,\"source_scale\":19,\"target_scale\":20}"},"fields":{"title":"Equal Scale Parameters Are Not an Unqualified Isometry Test","stem":"Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 4, 1, 2, 3], positive source_scale=19 and target_scale=20. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).","answer":{"kind":"expression","value":"(5, Yes, No, 19, 20, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation":"The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 19, 20, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)."},"skills":["Equal Scale Parameters Are Not an Unqualified Isometry Test"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0095","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ee022d3caa1174760de157f86306b1ec89dcd982593115154a5e94233749c18","family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055095,"semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,4,1,3,2],\"n\":5,\"source_scale\":20,\"target_scale\":21}"},"fields":{"title":"Equal Scale Parameters Are Not an Unqualified Isometry Test","stem":"Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 4, 1, 3, 2], positive source_scale=20 and target_scale=21. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).","answer":{"kind":"expression","value":"(5, Yes, No, 20, 21, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation":"The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 20, 21, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)."},"skills":["Equal Scale Parameters Are Not an Unqualified Isometry Test"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0096","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6bff3e8524e009eeae21b2479bf2605786d05f2f615a74389b74e3c3af355c99","family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055096,"semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,4,2,1,3],\"n\":5,\"source_scale\":21,\"target_scale\":22}"},"fields":{"title":"Equal Scale Parameters Are Not an Unqualified Isometry Test","stem":"Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 4, 2, 1, 3], positive source_scale=21 and target_scale=22. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).","answer":{"kind":"expression","value":"(5, Yes, No, 21, 22, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation":"The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 21, 22, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)."},"skills":["Equal Scale Parameters Are Not an Unqualified Isometry Test"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0097","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ec670de4895b39337597b4de83c79c073914cce485e0cb1cb20b51d7c12e01c","family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055097,"semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,4,2,3,1],\"n\":5,\"source_scale\":22,\"target_scale\":23}"},"fields":{"title":"Equal Scale Parameters Are Not an Unqualified Isometry Test","stem":"Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 4, 2, 3, 1], positive source_scale=22 and target_scale=23. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).","answer":{"kind":"expression","value":"(5, Yes, No, 22, 23, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation":"The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 22, 23, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)."},"skills":["Equal Scale Parameters Are Not an Unqualified Isometry Test"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0098","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"33ea10936be237386eb8bc9da843ab8dc1fadc0319523db467778964b550ee81","family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055098,"semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,4,3,1,2],\"n\":5,\"source_scale\":23,\"target_scale\":24}"},"fields":{"title":"Equal Scale Parameters Are Not an Unqualified Isometry Test","stem":"Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 4, 3, 1, 2], positive source_scale=23 and target_scale=24. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).","answer":{"kind":"expression","value":"(5, Yes, No, 23, 24, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation":"The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 23, 24, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)."},"skills":["Equal Scale Parameters Are Not an Unqualified Isometry Test"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0099","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24158d054aa1f4ee42bac0a3e8b99e7a49273c68b6eb4e86d9cc5ec6b17e060b","family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055099,"semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[0,4,3,2,1],\"n\":5,\"source_scale\":24,\"target_scale\":25}"},"fields":{"title":"Equal Scale Parameters Are Not an Unqualified Isometry Test","stem":"Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 4, 3, 2, 1], positive source_scale=24 and target_scale=25. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).","answer":{"kind":"expression","value":"(5, Yes, No, 24, 25, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation":"The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 24, 25, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)."},"skills":["Equal Scale Parameters Are Not an Unqualified Isometry Test"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261105-r05-homeomorphism-0100","batch_id":"batch-20261014-advance-20261105-r05-homeomorphism-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1164a0a23dda79f4f275e5c195e13c8808d962e587cf8c439f25d440eded92e3","family_id":"qf-mod36-homeo-metric-nonimplication","seed":202611055100,"semantic_template":"qf-mod36-homeo-metric-nonimplication/{\"map\":[1,0,2,3,4],\"n\":5,\"source_scale\":25,\"target_scale\":26}"},"fields":{"title":"Equal Scale Parameters Are Not an Unqualified Isometry Test","stem":"Use discrete spaces on Fin(5)={0,…,4}, bijection map=[1, 0, 2, 3, 4], positive source_scale=25 and target_scale=26. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).","answer":{"kind":"expression","value":"(5, Yes, No, 25, 26, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)","explanation":"The map is a homeomorphism. The next flag compares the two scale parameters exactly. For at least two points unequal scales witness non-isometry in this model; for a singleton no nonzero distance is tested. Do not turn the scale-comparison flag into a general isometry, diffeomorphism or congruence certificate. The exact result, retaining the requested order, is (5, Yes, No, 25, 26, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)."},"skills":["Equal Scale Parameters Are Not an Unqualified Isometry Test"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}