{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r03-open-cover-and-subcover-0001","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a5e8dad773f5433e2248562aba3f3db87ca9479b1c5d46863e7cbd2f234a537","family_id":"qf-mod32-cover-audit","seed":202611013001,"semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[0]]}"},"fields":{"title":"Coverage and Openness Are Independent Original-Family Obligations","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], Yes, Yes)","explanation":"List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes)."},"skills":["Coverage and Openness Are Independent Original-Family Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0002","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a3db960b89201df481d70c664bd3487b94eb23b1807f7e3a647711faa95173d","family_id":"qf-mod32-cover-audit","seed":202611013002,"semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,1],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,1],[0]]}"},"fields":{"title":"Coverage and Openness Are Independent Original-Family Obligations","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 1], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], Yes, Yes)","explanation":"List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes)."},"skills":["Coverage and Openness Are Independent Original-Family Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0003","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2f973154c0ae18ea8614adfacfd1e9e4c87a0d4f74ed1ae771c85248d068340","family_id":"qf-mod32-cover-audit","seed":202611013003,"semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,1],[0]]}"},"fields":{"title":"Coverage and Openness Are Independent Original-Family Obligations","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], Yes, Yes)","explanation":"List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes)."},"skills":["Coverage and Openness Are Independent Original-Family Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0004","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"acc47c23d6957cf560d7753425297fa8fd247cd784f891e82481a4c421a323be","family_id":"qf-mod32-cover-audit","seed":202611013004,"semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,1],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,1],[0]]}"},"fields":{"title":"Coverage and Openness Are Independent Original-Family Obligations","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 1], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], Yes, Yes)","explanation":"List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes)."},"skills":["Coverage and Openness Are Independent Original-Family Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0005","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a2e981a10c39b251dbc7bca61164067d7a86359a46141d6b1794fc97296469f","family_id":"qf-mod32-cover-audit","seed":202611013005,"semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,1],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,1],[0]]}"},"fields":{"title":"Coverage and Openness Are Independent Original-Family Obligations","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 1], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], Yes, Yes)","explanation":"List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes)."},"skills":["Coverage and Openness Are Independent Original-Family Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0006","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dca8b3a8a5c65fafdb7d9d214acdec0ee9077ef31cb080d67cb3df65dc420bb8","family_id":"qf-mod32-cover-audit","seed":202611013006,"semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,1],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,1],[0]]}"},"fields":{"title":"Coverage and Openness Are Independent Original-Family Obligations","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 1], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], Yes, Yes)","explanation":"List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes)."},"skills":["Coverage and Openness Are Independent Original-Family Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0007","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b505e5c095101c78b29b035847bcde018ba98e47e86f12f9024b6a97f4ec050f","family_id":"qf-mod32-cover-audit","seed":202611013007,"semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,2],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,2],[0]]}"},"fields":{"title":"Coverage and Openness Are Independent Original-Family Obligations","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], Yes, Yes)","explanation":"List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes)."},"skills":["Coverage and Openness Are Independent Original-Family Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0008","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d30de3df72164d1fb031f3418b878567021f363aad8367583dfaaa6ecfd6bb42","family_id":"qf-mod32-cover-audit","seed":202611013008,"semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,2],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,2],[0]]}"},"fields":{"title":"Coverage and Openness Are Independent Original-Family Obligations","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], Yes, Yes)","explanation":"List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes)."},"skills":["Coverage and Openness Are Independent Original-Family Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0009","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"875d7bad9f4f680b8585aaafb34372358bf7ee742ced33bd39b80eb80e8373e1","family_id":"qf-mod32-cover-audit","seed":202611013009,"semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,2],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,2],[0]]}"},"fields":{"title":"Coverage and Openness Are Independent Original-Family Obligations","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], Yes, Yes)","explanation":"List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes)."},"skills":["Coverage and Openness Are Independent Original-Family Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0010","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9a333d46ca29c86fb9c97abd9e76a575124c135cc2a3250cba02c8857a3c34d","family_id":"qf-mod32-cover-audit","seed":202611013010,"semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,2],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,2],[0]]}"},"fields":{"title":"Coverage and Openness Are Independent Original-Family Obligations","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 2], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], Yes, Yes)","explanation":"List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes)."},"skills":["Coverage and Openness Are Independent Original-Family Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0011","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6d6a72196f81a3bdb8aa9db0949b2c4059e4ed7098b77eb31101c3433ff850c","family_id":"qf-mod32-cover-audit","seed":202611013011,"semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,2],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,2],[0]]}"},"fields":{"title":"Coverage and Openness Are Independent Original-Family Obligations","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], Yes, Yes)","explanation":"List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes)."},"skills":["Coverage and Openness Are Independent Original-Family Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0012","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8977184bd0cea21a598a61ea0fde35177fa7898a69a0509b1a36cc4a2a68a4f8","family_id":"qf-mod32-cover-audit","seed":202611013012,"semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,2],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,2],[0]]}"},"fields":{"title":"Coverage and Openness Are Independent Original-Family Obligations","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 2], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], Yes, Yes)","explanation":"List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes)."},"skills":["Coverage and Openness Are Independent Original-Family Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0013","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e615e1e3c4d07105ab37326ee0da0343ebe1f10d3d120efa6c7d486fed02294","family_id":"qf-mod32-cover-audit","seed":202611013013,"semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,3],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,3],[0]]}"},"fields":{"title":"Coverage and Openness Are Independent Original-Family Obligations","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], Yes, Yes)","explanation":"List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes)."},"skills":["Coverage and Openness Are Independent Original-Family Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0014","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"904dba35c5c154bf630ac5d8454083b870456c8ad5dae7bc7ce17623d9534edb","family_id":"qf-mod32-cover-audit","seed":202611013014,"semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,3],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,3],[0]]}"},"fields":{"title":"Coverage and Openness Are Independent Original-Family Obligations","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], Yes, Yes)","explanation":"List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes)."},"skills":["Coverage and Openness Are Independent Original-Family Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0015","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49861ad0244a05cc8a1e37385601e85584f5c96d397347bd173c1333b8169a0a","family_id":"qf-mod32-cover-audit","seed":202611013015,"semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,3],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,3],[0]]}"},"fields":{"title":"Coverage and Openness Are Independent Original-Family Obligations","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], Yes, Yes)","explanation":"List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes)."},"skills":["Coverage and Openness Are Independent Original-Family Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0016","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fda02801dfa82532d16e083fb4b88eee65a7ca4db712f879388355c5829a2dce","family_id":"qf-mod32-cover-audit","seed":202611013016,"semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,3],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,3],[0]]}"},"fields":{"title":"Coverage and Openness Are Independent Original-Family Obligations","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], Yes, Yes)","explanation":"List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes)."},"skills":["Coverage and Openness Are Independent Original-Family Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0017","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e027b53a64adc5d7030fb89808b1ccf52007947e6c299fd3506f1a6ccdbe374e","family_id":"qf-mod32-cover-audit","seed":202611013017,"semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,3],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,3],[0]]}"},"fields":{"title":"Coverage and Openness Are Independent Original-Family Obligations","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], Yes, Yes)","explanation":"List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes)."},"skills":["Coverage and Openness Are Independent Original-Family Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0018","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48d4f85c349864628aaeff7d9b047282c4692aa25dc523e7663c2989ec6c92ac","family_id":"qf-mod32-cover-audit","seed":202611013018,"semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,3],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,3],[0]]}"},"fields":{"title":"Coverage and Openness Are Independent Original-Family Obligations","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], Yes, Yes)","explanation":"List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes)."},"skills":["Coverage and Openness Are Independent Original-Family Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0019","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec6c8eabd11fc5cd364bb15ff63af6fbc19e9bc8823e8f5accbede69a28b82c8","family_id":"qf-mod32-cover-audit","seed":202611013019,"semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,4],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,4],[0]]}"},"fields":{"title":"Coverage and Openness Are Independent Original-Family Obligations","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], Yes, Yes)","explanation":"List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes)."},"skills":["Coverage and Openness Are Independent Original-Family Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0020","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf92bcd1b10f90423bed8ba38785580b9bcedf2fad9331908829a76a11d02331","family_id":"qf-mod32-cover-audit","seed":202611013020,"semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,4],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,4],[0]]}"},"fields":{"title":"Coverage and Openness Are Independent Original-Family Obligations","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], Yes, Yes)","explanation":"List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes)."},"skills":["Coverage and Openness Are Independent Original-Family Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0021","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c0cf10fca4c3ac4a38288670fbd65c0e376a7b29f9126ece92feeb6c60403a3","family_id":"qf-mod32-cover-audit","seed":202611013021,"semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,4],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,4],[0]]}"},"fields":{"title":"Coverage and Openness Are Independent Original-Family Obligations","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], Yes, Yes)","explanation":"List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes)."},"skills":["Coverage and Openness Are Independent Original-Family Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0022","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00b600d0a6959ba9e3a98b65021b583e90eee618af295dce8690a66cd2157a0a","family_id":"qf-mod32-cover-audit","seed":202611013022,"semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,4],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,4],[0]]}"},"fields":{"title":"Coverage and Openness Are Independent Original-Family Obligations","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], Yes, Yes)","explanation":"List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes)."},"skills":["Coverage and Openness Are Independent Original-Family Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0023","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5dd1cc81ec86c0e96242f97bbc8c5e63a35e33ffdb90d71305d033e93c34cdc8","family_id":"qf-mod32-cover-audit","seed":202611013023,"semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,4],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,4],[0]]}"},"fields":{"title":"Coverage and Openness Are Independent Original-Family Obligations","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], Yes, Yes)","explanation":"List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes)."},"skills":["Coverage and Openness Are Independent Original-Family Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0024","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"688dfa7760649314bc4d53edded9e2f70c3213bbfdbd6ce43b596750d6581c84","family_id":"qf-mod32-cover-audit","seed":202611013024,"semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,4],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,4],[0]]}"},"fields":{"title":"Coverage and Openness Are Independent Original-Family Obligations","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], Yes, Yes)","explanation":"List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes)."},"skills":["Coverage and Openness Are Independent Original-Family Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0025","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d869e3a30b4e9e95f39c7d5dd076fa70afbce019b0c46065fa945b740d69f2f6","family_id":"qf-mod32-cover-audit","seed":202611013025,"semantic_template":"qf-mod32-cover-audit/{\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[1]]}"},"fields":{"title":"Coverage and Openness Are Independent Original-Family Obligations","stem":"Use the complete finite topology=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. Give (invalid_open_indexes,covered_union,covers_target?,valid_open_cover?), with invalid indices increasing and the union in carrier order.","answer":{"kind":"expression","value":"([], [0, 1, 2, 3, 4], Yes, Yes)","explanation":"List every supplied nonopen member even if the family covers the target. Conversely, all members being open does not establish coverage. Only their conjunction qualifies this original family. The exact result, retaining the requested order, is ([], [0, 1, 2, 3, 4], Yes, Yes)."},"skills":["Coverage and Openness Are Independent Original-Family Obligations"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0026","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91015420a5401ee4858969c90a84a9576537e58db93882d5be5b3ae5cc9a96d5","family_id":"qf-mod32-cover-subcover-audit","seed":202611013026,"semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[0]]}"},"fields":{"title":"A Certified Subcover Selects Only from the Qualified Original Cover","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).","answer":{"kind":"expression","value":"([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes)","explanation":"Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes)."},"skills":["A Certified Subcover Selects Only from the Qualified Original Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0027","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bac6016b35c12726f0dfc3b47eff14fc16df3922ca23fb52faa41a4988c607e3","family_id":"qf-mod32-cover-subcover-audit","seed":202611013027,"semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,1],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,1],[0]]}"},"fields":{"title":"A Certified Subcover Selects Only from the Qualified Original Cover","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 1], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).","answer":{"kind":"expression","value":"([1], [[0, 1, 2, 4]], [0, 1, 2, 4], Yes)","explanation":"Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 2, 4]], [0, 1, 2, 4], Yes)."},"skills":["A Certified Subcover Selects Only from the Qualified Original Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0028","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62932c672fe9ea0cdd684f359d02e6317110f814bf2f1669ac1f8298c1de3148","family_id":"qf-mod32-cover-subcover-audit","seed":202611013028,"semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,1],[0]]}"},"fields":{"title":"A Certified Subcover Selects Only from the Qualified Original Cover","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).","answer":{"kind":"expression","value":"([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes)","explanation":"Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes)."},"skills":["A Certified Subcover Selects Only from the Qualified Original Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0029","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c9cbbd0bd5b6fb93de89a5577387c46c9f6f7ec6e58914931e6e873204e4cb2","family_id":"qf-mod32-cover-subcover-audit","seed":202611013029,"semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,1],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,1],[0]]}"},"fields":{"title":"A Certified Subcover Selects Only from the Qualified Original Cover","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 1], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).","answer":{"kind":"expression","value":"([1], [[0, 1, 3, 4]], [0, 1, 3, 4], Yes)","explanation":"Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 3, 4]], [0, 1, 3, 4], Yes)."},"skills":["A Certified Subcover Selects Only from the Qualified Original Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0030","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8b0f9be394dd7f15a02c6cddbc8132bbd876557c1abd1b1ce21e743e273ef8a","family_id":"qf-mod32-cover-subcover-audit","seed":202611013030,"semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,1],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,1],[0]]}"},"fields":{"title":"A Certified Subcover Selects Only from the Qualified Original Cover","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 1], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).","answer":{"kind":"expression","value":"([1], [[0, 1, 2, 4]], [0, 1, 2, 4], Yes)","explanation":"Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 2, 4]], [0, 1, 2, 4], Yes)."},"skills":["A Certified Subcover Selects Only from the Qualified Original Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0031","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0e4f58c46dc33eef8cc3369097c5d25773a4ac1f13d36bcd1f72b0a1ad38f92","family_id":"qf-mod32-cover-subcover-audit","seed":202611013031,"semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,1],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,1],[0]]}"},"fields":{"title":"A Certified Subcover Selects Only from the Qualified Original Cover","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 1], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).","answer":{"kind":"expression","value":"([1], [[0, 1, 3, 4]], [0, 1, 3, 4], Yes)","explanation":"Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 3, 4]], [0, 1, 3, 4], Yes)."},"skills":["A Certified Subcover Selects Only from the Qualified Original Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0032","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63ccb783a9e8046d19e66b65461ba218d0d9ed52cbf64a04711178820e835e86","family_id":"qf-mod32-cover-subcover-audit","seed":202611013032,"semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,2],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,2],[0]]}"},"fields":{"title":"A Certified Subcover Selects Only from the Qualified Original Cover","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).","answer":{"kind":"expression","value":"([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes)","explanation":"Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes)."},"skills":["A Certified Subcover Selects Only from the Qualified Original Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0033","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"447ddbdc095e196c91ce3ac46914c05cbe5d4d332fab5d0cbc2033911dc0d75f","family_id":"qf-mod32-cover-subcover-audit","seed":202611013033,"semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,2],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,2],[0]]}"},"fields":{"title":"A Certified Subcover Selects Only from the Qualified Original Cover","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).","answer":{"kind":"expression","value":"([1], [[0, 1, 2, 4]], [0, 1, 2, 4], Yes)","explanation":"Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 2, 4]], [0, 1, 2, 4], Yes)."},"skills":["A Certified Subcover Selects Only from the Qualified Original Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0034","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90d70025140bede2c510f007b0a3aea52891df70fe53fe5e58dde67c024b7eb0","family_id":"qf-mod32-cover-subcover-audit","seed":202611013034,"semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,2],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,2],[0]]}"},"fields":{"title":"A Certified Subcover Selects Only from the Qualified Original Cover","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).","answer":{"kind":"expression","value":"([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes)","explanation":"Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes)."},"skills":["A Certified Subcover Selects Only from the Qualified Original Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0035","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"33cea1a467938c02880a8673db21adbf4db216973cb460542e34445ebce6e764","family_id":"qf-mod32-cover-subcover-audit","seed":202611013035,"semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,2],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,2],[0]]}"},"fields":{"title":"A Certified Subcover Selects Only from the Qualified Original Cover","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 2], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).","answer":{"kind":"expression","value":"([1], [[0, 2, 3, 4]], [0, 2, 3, 4], Yes)","explanation":"Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 2, 3, 4]], [0, 2, 3, 4], Yes)."},"skills":["A Certified Subcover Selects Only from the Qualified Original Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0036","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c81964b5ebb36434b2d5cc7f108e6d7bb2c8a70c8f318226a02af0c68315428","family_id":"qf-mod32-cover-subcover-audit","seed":202611013036,"semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,2],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,2],[0]]}"},"fields":{"title":"A Certified Subcover Selects Only from the Qualified Original Cover","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).","answer":{"kind":"expression","value":"([1], [[0, 1, 2, 4]], [0, 1, 2, 4], Yes)","explanation":"Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 2, 4]], [0, 1, 2, 4], Yes)."},"skills":["A Certified Subcover Selects Only from the Qualified Original Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0037","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"decd14da84345e446e762141624efd9599bbee330b0dd039bfd3d88e4a9abfd4","family_id":"qf-mod32-cover-subcover-audit","seed":202611013037,"semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,2],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,2],[0]]}"},"fields":{"title":"A Certified Subcover Selects Only from the Qualified Original Cover","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 2], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).","answer":{"kind":"expression","value":"([1], [[0, 2, 3, 4]], [0, 2, 3, 4], Yes)","explanation":"Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 2, 3, 4]], [0, 2, 3, 4], Yes)."},"skills":["A Certified Subcover Selects Only from the Qualified Original Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0038","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0d6e6755d536a77a33f43d6c6e56977c7241aa038a356d8f1a3e535082d9752","family_id":"qf-mod32-cover-subcover-audit","seed":202611013038,"semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,3],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,3],[0]]}"},"fields":{"title":"A Certified Subcover Selects Only from the Qualified Original Cover","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).","answer":{"kind":"expression","value":"([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes)","explanation":"Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes)."},"skills":["A Certified Subcover Selects Only from the Qualified Original Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0039","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0de5eaa5780802b8ad4c6cc202b0ce7713e670c7ffddf330b9dfc9982ae5e3b4","family_id":"qf-mod32-cover-subcover-audit","seed":202611013039,"semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,3],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,3],[0]]}"},"fields":{"title":"A Certified Subcover Selects Only from the Qualified Original Cover","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).","answer":{"kind":"expression","value":"([1], [[0, 1, 3, 4]], [0, 1, 3, 4], Yes)","explanation":"Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 3, 4]], [0, 1, 3, 4], Yes)."},"skills":["A Certified Subcover Selects Only from the Qualified Original Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0040","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d22a2e1eba71e7291fcf8da4cbb5be358c817dfbb05f3dce4a3d1bce870f1ac5","family_id":"qf-mod32-cover-subcover-audit","seed":202611013040,"semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,3],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,3],[0]]}"},"fields":{"title":"A Certified Subcover Selects Only from the Qualified Original Cover","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).","answer":{"kind":"expression","value":"([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes)","explanation":"Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes)."},"skills":["A Certified Subcover Selects Only from the Qualified Original Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0041","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b6524f25e8269acea2ea71e0b996977d2f3082061d19588a3e99b71ec719b3cc","family_id":"qf-mod32-cover-subcover-audit","seed":202611013041,"semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,3],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,3],[0]]}"},"fields":{"title":"A Certified Subcover Selects Only from the Qualified Original Cover","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).","answer":{"kind":"expression","value":"([1], [[0, 2, 3, 4]], [0, 2, 3, 4], Yes)","explanation":"Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 2, 3, 4]], [0, 2, 3, 4], Yes)."},"skills":["A Certified Subcover Selects Only from the Qualified Original Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0042","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"510fdd9f5fd007dbf609c5aef4b4320cb51aa444f1f6d0f7c0f23c178508dd43","family_id":"qf-mod32-cover-subcover-audit","seed":202611013042,"semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,3],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,3],[0]]}"},"fields":{"title":"A Certified Subcover Selects Only from the Qualified Original Cover","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).","answer":{"kind":"expression","value":"([1], [[0, 1, 3, 4]], [0, 1, 3, 4], Yes)","explanation":"Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 3, 4]], [0, 1, 3, 4], Yes)."},"skills":["A Certified Subcover Selects Only from the Qualified Original Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0043","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b6175288563c43a9209a2b26e1d975039c6aa07f8b4b3b4e2d73f7b93087d9d","family_id":"qf-mod32-cover-subcover-audit","seed":202611013043,"semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,3],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,3],[0]]}"},"fields":{"title":"A Certified Subcover Selects Only from the Qualified Original Cover","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).","answer":{"kind":"expression","value":"([1], [[0, 2, 3, 4]], [0, 2, 3, 4], Yes)","explanation":"Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 2, 3, 4]], [0, 2, 3, 4], Yes)."},"skills":["A Certified Subcover Selects Only from the Qualified Original Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0044","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"998b82b80a6f4c23611dca4f80234ff633d925f7d1177fc8c94df8d295b24517","family_id":"qf-mod32-cover-subcover-audit","seed":202611013044,"semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,4],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,4],[0]]}"},"fields":{"title":"A Certified Subcover Selects Only from the Qualified Original Cover","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).","answer":{"kind":"expression","value":"([1], [[0, 1, 2, 4]], [0, 1, 2, 4], Yes)","explanation":"Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 2, 4]], [0, 1, 2, 4], Yes)."},"skills":["A Certified Subcover Selects Only from the Qualified Original Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0045","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8c1e9fa21aad4d26433d3f2859b2542741dc9a58f02943131d1fc139f0ffbae","family_id":"qf-mod32-cover-subcover-audit","seed":202611013045,"semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,4],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,4],[0]]}"},"fields":{"title":"A Certified Subcover Selects Only from the Qualified Original Cover","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).","answer":{"kind":"expression","value":"([1], [[0, 1, 3, 4]], [0, 1, 3, 4], Yes)","explanation":"Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 3, 4]], [0, 1, 3, 4], Yes)."},"skills":["A Certified Subcover Selects Only from the Qualified Original Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0046","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04086775aee58fe7baff296de735bb26b429e7d363f1ae8ce709e685fc3da19b","family_id":"qf-mod32-cover-subcover-audit","seed":202611013046,"semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,4],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,4],[0]]}"},"fields":{"title":"A Certified Subcover Selects Only from the Qualified Original Cover","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).","answer":{"kind":"expression","value":"([1], [[0, 1, 2, 4]], [0, 1, 2, 4], Yes)","explanation":"Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 2, 4]], [0, 1, 2, 4], Yes)."},"skills":["A Certified Subcover Selects Only from the Qualified Original Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0047","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86a9c33beb2579ff7a0717d1fa46571b1fc37340d88d696bd75d2ebb535a985d","family_id":"qf-mod32-cover-subcover-audit","seed":202611013047,"semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,4],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,4],[0]]}"},"fields":{"title":"A Certified Subcover Selects Only from the Qualified Original Cover","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).","answer":{"kind":"expression","value":"([1], [[0, 2, 3, 4]], [0, 2, 3, 4], Yes)","explanation":"Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 2, 3, 4]], [0, 2, 3, 4], Yes)."},"skills":["A Certified Subcover Selects Only from the Qualified Original Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0048","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"648fd942e8026053b6c1b0bbb04691bc0c3e6d4269b099ff422fa4f01ea8adbf","family_id":"qf-mod32-cover-subcover-audit","seed":202611013048,"semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,4],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,4],[0]]}"},"fields":{"title":"A Certified Subcover Selects Only from the Qualified Original Cover","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).","answer":{"kind":"expression","value":"([1], [[0, 1, 3, 4]], [0, 1, 3, 4], Yes)","explanation":"Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 3, 4]], [0, 1, 3, 4], Yes)."},"skills":["A Certified Subcover Selects Only from the Qualified Original Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0049","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c8f2b177e0093b4cac874cd3a14365c0886d01b5c3cc31e819510fe45d9bc7e","family_id":"qf-mod32-cover-subcover-audit","seed":202611013049,"semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,4],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,4],[0]]}"},"fields":{"title":"A Certified Subcover Selects Only from the Qualified Original Cover","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).","answer":{"kind":"expression","value":"([1], [[0, 2, 3, 4]], [0, 2, 3, 4], Yes)","explanation":"Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 2, 3, 4]], [0, 2, 3, 4], Yes)."},"skills":["A Certified Subcover Selects Only from the Qualified Original Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0050","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b753a5dc6641b52636ec2c3a52ca4df25d2646abc1b5e6aa78309fb7797536c6","family_id":"qf-mod32-cover-subcover-audit","seed":202611013050,"semantic_template":"qf-mod32-cover-subcover-audit/{\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[1]]}"},"fields":{"title":"A Certified Subcover Selects Only from the Qualified Original Cover","stem":"Use the complete finite topology=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (sorted_selected_indexes,selected_sets,covered_union,valid_subcover?).","answer":{"kind":"expression","value":"([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes)","explanation":"Select the original indexed sets and test the same target. This protocol additionally requires the whole original family to be a valid open cover, so a good selected subfamily does not erase an unselected invalid member of that original input. The exact result, retaining the requested order, is ([1], [[0, 1, 2, 3]], [0, 1, 2, 3], Yes)."},"skills":["A Certified Subcover Selects Only from the Qualified Original Cover"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0051","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a5e707ee5c359dfb9e908e2f31f14f7b1eeb5f20aea23e507b1198c97719948","family_id":"qf-mod32-cover-minimal-subcover","seed":202611013051,"semantic_template":"qf-mod32-cover-minimal-subcover/{\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[0]]}"},"fields":{"title":"Inclusion-Minimal Does Not Mean Minimum Cardinality","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.","answer":{"kind":"expression","value":"([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)","explanation":"Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)."},"skills":["Inclusion-Minimal Does Not Mean Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0052","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fafd702ec3c3a20eeb4e11ffcab6931bf410ac89a8fabb66f5af9195a018c89e","family_id":"qf-mod32-cover-minimal-subcover","seed":202611013052,"semantic_template":"qf-mod32-cover-minimal-subcover/{\"cover\":[[0,1],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,1],[0]]}"},"fields":{"title":"Inclusion-Minimal Does Not Mean Minimum Cardinality","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 1], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.","answer":{"kind":"expression","value":"([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)","explanation":"Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)."},"skills":["Inclusion-Minimal Does Not Mean Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0053","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1840935cbab7e03bba272b72409926975166f7b4df5f6575f66e86e297a252e3","family_id":"qf-mod32-cover-minimal-subcover","seed":202611013053,"semantic_template":"qf-mod32-cover-minimal-subcover/{\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,1],[0]]}"},"fields":{"title":"Inclusion-Minimal Does Not Mean Minimum Cardinality","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.","answer":{"kind":"expression","value":"([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)","explanation":"Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)."},"skills":["Inclusion-Minimal Does Not Mean Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0054","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b810509db78de1a36faac749b2a59e19d6ce1355d9091daf90bea59a610d30f","family_id":"qf-mod32-cover-minimal-subcover","seed":202611013054,"semantic_template":"qf-mod32-cover-minimal-subcover/{\"cover\":[[0,1],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,1],[0]]}"},"fields":{"title":"Inclusion-Minimal Does Not Mean Minimum Cardinality","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 1], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.","answer":{"kind":"expression","value":"([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)","explanation":"Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)."},"skills":["Inclusion-Minimal Does Not Mean Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0055","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb884213258c0d70230909d550ae5970092489cdc152d516088cb2f5a3c418ac","family_id":"qf-mod32-cover-minimal-subcover","seed":202611013055,"semantic_template":"qf-mod32-cover-minimal-subcover/{\"cover\":[[0,1],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,1],[0]]}"},"fields":{"title":"Inclusion-Minimal Does Not Mean Minimum Cardinality","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 1], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.","answer":{"kind":"expression","value":"([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)","explanation":"Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)."},"skills":["Inclusion-Minimal Does Not Mean Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0056","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"973b063487aa7a2f0e0d27a6d90c652d55c6dde6e8accd04e143f30b68e44054","family_id":"qf-mod32-cover-minimal-subcover","seed":202611013056,"semantic_template":"qf-mod32-cover-minimal-subcover/{\"cover\":[[0,1],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,1],[0]]}"},"fields":{"title":"Inclusion-Minimal Does Not Mean Minimum Cardinality","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 1], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.","answer":{"kind":"expression","value":"([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)","explanation":"Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)."},"skills":["Inclusion-Minimal Does Not Mean Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0057","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1db08be4d812b070bbb8d6df5fece2deb0c27841d9ed9e95098ddff6e5c0bd3a","family_id":"qf-mod32-cover-minimal-subcover","seed":202611013057,"semantic_template":"qf-mod32-cover-minimal-subcover/{\"cover\":[[0,2],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,2],[0]]}"},"fields":{"title":"Inclusion-Minimal Does Not Mean Minimum Cardinality","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.","answer":{"kind":"expression","value":"([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)","explanation":"Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)."},"skills":["Inclusion-Minimal Does Not Mean Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0058","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30021d430806a22f1ef4a517c842bf628d8b0e5580d54fc76e8e95a25ec12cf1","family_id":"qf-mod32-cover-minimal-subcover","seed":202611013058,"semantic_template":"qf-mod32-cover-minimal-subcover/{\"cover\":[[0,2],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,2],[0]]}"},"fields":{"title":"Inclusion-Minimal Does Not Mean Minimum Cardinality","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.","answer":{"kind":"expression","value":"([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)","explanation":"Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)."},"skills":["Inclusion-Minimal Does Not Mean Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0059","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67894e7146c680f3eccbd14fbbb2cf31e3892fa7a6fd9481a9658a8926bedf7d","family_id":"qf-mod32-cover-minimal-subcover","seed":202611013059,"semantic_template":"qf-mod32-cover-minimal-subcover/{\"cover\":[[0,2],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,2],[0]]}"},"fields":{"title":"Inclusion-Minimal Does Not Mean Minimum Cardinality","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.","answer":{"kind":"expression","value":"([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)","explanation":"Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)."},"skills":["Inclusion-Minimal Does Not Mean Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0060","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"230fe9fb408e2b4833cc5178782f34f955471d397b11c5cd86834a84678dcf30","family_id":"qf-mod32-cover-minimal-subcover","seed":202611013060,"semantic_template":"qf-mod32-cover-minimal-subcover/{\"cover\":[[0,2],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,2],[0]]}"},"fields":{"title":"Inclusion-Minimal Does Not Mean Minimum Cardinality","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 2], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.","answer":{"kind":"expression","value":"([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)","explanation":"Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)."},"skills":["Inclusion-Minimal Does Not Mean Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0061","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"328cba70fd76cef74e188a10b522d1f0d3aaf9f9b4fce19d5af3c4d6541b0aa8","family_id":"qf-mod32-cover-minimal-subcover","seed":202611013061,"semantic_template":"qf-mod32-cover-minimal-subcover/{\"cover\":[[0,2],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,2],[0]]}"},"fields":{"title":"Inclusion-Minimal Does Not Mean Minimum Cardinality","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.","answer":{"kind":"expression","value":"([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)","explanation":"Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)."},"skills":["Inclusion-Minimal Does Not Mean Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0062","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f0399122b271868c1bd211f5fc4519d32d9c9d7c564449b6c1f7b8fe9cb0281","family_id":"qf-mod32-cover-minimal-subcover","seed":202611013062,"semantic_template":"qf-mod32-cover-minimal-subcover/{\"cover\":[[0,2],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,2],[0]]}"},"fields":{"title":"Inclusion-Minimal Does Not Mean Minimum Cardinality","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 2], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.","answer":{"kind":"expression","value":"([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)","explanation":"Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)."},"skills":["Inclusion-Minimal Does Not Mean Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0063","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0eb76c9bcac500c9a81779c3a96e43f440654ca6b8482facaf1de04327fbfd2a","family_id":"qf-mod32-cover-minimal-subcover","seed":202611013063,"semantic_template":"qf-mod32-cover-minimal-subcover/{\"cover\":[[0,3],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,3],[0]]}"},"fields":{"title":"Inclusion-Minimal Does Not Mean Minimum Cardinality","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.","answer":{"kind":"expression","value":"([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)","explanation":"Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)."},"skills":["Inclusion-Minimal Does Not Mean Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0064","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c3031b1faaef8e4f39026989e1c85dcefccc21c68a7e3c82107be5f2ee8cd367","family_id":"qf-mod32-cover-minimal-subcover","seed":202611013064,"semantic_template":"qf-mod32-cover-minimal-subcover/{\"cover\":[[0,3],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,3],[0]]}"},"fields":{"title":"Inclusion-Minimal Does Not Mean Minimum Cardinality","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.","answer":{"kind":"expression","value":"([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)","explanation":"Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)."},"skills":["Inclusion-Minimal Does Not Mean Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0065","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a8c6d8b3639aea7f7a54ae59410ddc79c961471fda8ac965ab8f6a28b77e7be","family_id":"qf-mod32-cover-minimal-subcover","seed":202611013065,"semantic_template":"qf-mod32-cover-minimal-subcover/{\"cover\":[[0,3],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,3],[0]]}"},"fields":{"title":"Inclusion-Minimal Does Not Mean Minimum Cardinality","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.","answer":{"kind":"expression","value":"([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)","explanation":"Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)."},"skills":["Inclusion-Minimal Does Not Mean Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0066","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6767b64c9cc51e86d8490a4ec007005645dc657b3730817ee96b7d09a21162a","family_id":"qf-mod32-cover-minimal-subcover","seed":202611013066,"semantic_template":"qf-mod32-cover-minimal-subcover/{\"cover\":[[0,3],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,3],[0]]}"},"fields":{"title":"Inclusion-Minimal Does Not Mean Minimum Cardinality","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.","answer":{"kind":"expression","value":"([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)","explanation":"Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)."},"skills":["Inclusion-Minimal Does Not Mean Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0067","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7869b374f0ff46fcbd60a25399863b7726739001abbb2074ec039b914a205c8","family_id":"qf-mod32-cover-minimal-subcover","seed":202611013067,"semantic_template":"qf-mod32-cover-minimal-subcover/{\"cover\":[[0,3],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,3],[0]]}"},"fields":{"title":"Inclusion-Minimal Does Not Mean Minimum Cardinality","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.","answer":{"kind":"expression","value":"([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)","explanation":"Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)."},"skills":["Inclusion-Minimal Does Not Mean Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0068","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c07ae4e81dc5c904f12f9391ff864b2890f41020aefa53c6b653d0d1ef4079d2","family_id":"qf-mod32-cover-minimal-subcover","seed":202611013068,"semantic_template":"qf-mod32-cover-minimal-subcover/{\"cover\":[[0,3],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,3],[0]]}"},"fields":{"title":"Inclusion-Minimal Does Not Mean Minimum Cardinality","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.","answer":{"kind":"expression","value":"([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)","explanation":"Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)."},"skills":["Inclusion-Minimal Does Not Mean Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0069","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1943bc6d619cf3a9cb933de064691be08eb561b1014366e4247aabd255fc27ab","family_id":"qf-mod32-cover-minimal-subcover","seed":202611013069,"semantic_template":"qf-mod32-cover-minimal-subcover/{\"cover\":[[0,4],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,4],[0]]}"},"fields":{"title":"Inclusion-Minimal Does Not Mean Minimum Cardinality","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.","answer":{"kind":"expression","value":"([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)","explanation":"Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)."},"skills":["Inclusion-Minimal Does Not Mean Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0070","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"211b82cf87a07979a278d68ddea3e64c23040e0bffa16769552616ccaf19534b","family_id":"qf-mod32-cover-minimal-subcover","seed":202611013070,"semantic_template":"qf-mod32-cover-minimal-subcover/{\"cover\":[[0,4],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,4],[0]]}"},"fields":{"title":"Inclusion-Minimal Does Not Mean Minimum Cardinality","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.","answer":{"kind":"expression","value":"([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)","explanation":"Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)."},"skills":["Inclusion-Minimal Does Not Mean Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0071","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb347435223746e14da655af9287e9baa9684e69f6860862b92e9a118217271e","family_id":"qf-mod32-cover-minimal-subcover","seed":202611013071,"semantic_template":"qf-mod32-cover-minimal-subcover/{\"cover\":[[0,4],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,4],[0]]}"},"fields":{"title":"Inclusion-Minimal Does Not Mean Minimum Cardinality","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.","answer":{"kind":"expression","value":"([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)","explanation":"Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)."},"skills":["Inclusion-Minimal Does Not Mean Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0072","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee32d79d5689a22335e575c8ec978eb879901928c710925eebc7de48205cb13e","family_id":"qf-mod32-cover-minimal-subcover","seed":202611013072,"semantic_template":"qf-mod32-cover-minimal-subcover/{\"cover\":[[0,4],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,4],[0]]}"},"fields":{"title":"Inclusion-Minimal Does Not Mean Minimum Cardinality","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.","answer":{"kind":"expression","value":"([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)","explanation":"Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)."},"skills":["Inclusion-Minimal Does Not Mean Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0073","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5d3dd2bce8cf6e5133f05da3ff118e7473ee9ba74ac0d3846fd31cadeec8eea","family_id":"qf-mod32-cover-minimal-subcover","seed":202611013073,"semantic_template":"qf-mod32-cover-minimal-subcover/{\"cover\":[[0,4],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,4],[0]]}"},"fields":{"title":"Inclusion-Minimal Does Not Mean Minimum Cardinality","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.","answer":{"kind":"expression","value":"([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)","explanation":"Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)."},"skills":["Inclusion-Minimal Does Not Mean Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0074","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0efa8f56db721fe21f7e84314c45781954ca1b0bbeb639c907b081ceb57f9c98","family_id":"qf-mod32-cover-minimal-subcover","seed":202611013074,"semantic_template":"qf-mod32-cover-minimal-subcover/{\"cover\":[[0,4],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,4],[0]]}"},"fields":{"title":"Inclusion-Minimal Does Not Mean Minimum Cardinality","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.","answer":{"kind":"expression","value":"([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)","explanation":"Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)."},"skills":["Inclusion-Minimal Does Not Mean Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0075","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c2ea31a15e071d0b07605eb10ef7dab990946f03b6157199abec867ecc46b9c","family_id":"qf-mod32-cover-minimal-subcover","seed":202611013075,"semantic_template":"qf-mod32-cover-minimal-subcover/{\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[1]]}"},"fields":{"title":"Inclusion-Minimal Does Not Mean Minimum Cardinality","stem":"Use the complete finite topology=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,removal_rows,inclusion_minimal?,inclusion_minimal_not_minimum_cardinality). Each removal row is [removed_original_index,remaining_selected_indexes,still_covers_target?] in sorted selected-index order.","answer":{"kind":"expression","value":"([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)","explanation":"Starting from a valid selected subcover, remove each chosen member separately and recompute target coverage. It is inclusion-minimal when every such removal fails. Another subcover of the original family may still use fewer sets. The exact result, retaining the requested order, is ([1], Yes, [[1, [], No]], Yes, inclusion_minimal_not_minimum_cardinality)."},"skills":["Inclusion-Minimal Does Not Mean Minimum Cardinality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0076","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b09ef36140087e362973ebd1fd98d3e8ce31dfd49f6404468702a146b4271f9","family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013076,"semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[0]]}"},"fields":{"title":"One Supplied Finite Subcover Is Not the Universal Compactness Definition","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).","answer":{"kind":"expression","value":"([1], Yes, No, one_finite_subcover_does_not_prove_compactness)","explanation":"The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness)."},"skills":["One Supplied Finite Subcover Is Not the Universal Compactness Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0077","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e0e3288e67da2c1d59d397acf6847633f7b82065fb37135570f71bab1a4113b","family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013077,"semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,1],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,1],[0]]}"},"fields":{"title":"One Supplied Finite Subcover Is Not the Universal Compactness Definition","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 1], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).","answer":{"kind":"expression","value":"([1], Yes, No, one_finite_subcover_does_not_prove_compactness)","explanation":"The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness)."},"skills":["One Supplied Finite Subcover Is Not the Universal Compactness Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0078","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54f7413433eaa9e6ce994a939df550193e2d82e740a364709b5ccc7b275ee321","family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013078,"semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,1],[0]]}"},"fields":{"title":"One Supplied Finite Subcover Is Not the Universal Compactness Definition","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).","answer":{"kind":"expression","value":"([1], Yes, No, one_finite_subcover_does_not_prove_compactness)","explanation":"The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness)."},"skills":["One Supplied Finite Subcover Is Not the Universal Compactness Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0079","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"502cae4b660febb78210b1efb300d0655a5c77c6c77c57c1447213644a45223a","family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013079,"semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,1],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,1],[0]]}"},"fields":{"title":"One Supplied Finite Subcover Is Not the Universal Compactness Definition","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 1], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).","answer":{"kind":"expression","value":"([1], Yes, No, one_finite_subcover_does_not_prove_compactness)","explanation":"The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness)."},"skills":["One Supplied Finite Subcover Is Not the Universal Compactness Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0080","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a88468614055887ef2d2c1ced1fc54393ff20d94e9eacf5fc27e60df1181f140","family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013080,"semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,1],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,1],[0]]}"},"fields":{"title":"One Supplied Finite Subcover Is Not the Universal Compactness Definition","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 1], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).","answer":{"kind":"expression","value":"([1], Yes, No, one_finite_subcover_does_not_prove_compactness)","explanation":"The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness)."},"skills":["One Supplied Finite Subcover Is Not the Universal Compactness Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0081","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8d6c2a89d60ac654552d8c62096e03a49aa4e75e3d76d3a6dd60d809c64ffcf","family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013081,"semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,1],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,1],[0]]}"},"fields":{"title":"One Supplied Finite Subcover Is Not the Universal Compactness Definition","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 1], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).","answer":{"kind":"expression","value":"([1], Yes, No, one_finite_subcover_does_not_prove_compactness)","explanation":"The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness)."},"skills":["One Supplied Finite Subcover Is Not the Universal Compactness Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0082","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f24a62cfde4b653b137844f937614fc49ad31eb4b4409178034d549a9e4472a7","family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013082,"semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,2],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,2],[0]]}"},"fields":{"title":"One Supplied Finite Subcover Is Not the Universal Compactness Definition","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).","answer":{"kind":"expression","value":"([1], Yes, No, one_finite_subcover_does_not_prove_compactness)","explanation":"The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness)."},"skills":["One Supplied Finite Subcover Is Not the Universal Compactness Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0083","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4caecd31a72eaed3a2ca5c182b871deafddf8e18d9d47df172687173dad3177a","family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013083,"semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,2],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,2],[0]]}"},"fields":{"title":"One Supplied Finite Subcover Is Not the Universal Compactness Definition","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).","answer":{"kind":"expression","value":"([1], Yes, No, one_finite_subcover_does_not_prove_compactness)","explanation":"The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness)."},"skills":["One Supplied Finite Subcover Is Not the Universal Compactness Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0084","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f101bfcc1525dca73aa1064007b644575cccd7709a6ee30938f0c0fe98520d6a","family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013084,"semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,2],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,2],[0]]}"},"fields":{"title":"One Supplied Finite Subcover Is Not the Universal Compactness Definition","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).","answer":{"kind":"expression","value":"([1], Yes, No, one_finite_subcover_does_not_prove_compactness)","explanation":"The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness)."},"skills":["One Supplied Finite Subcover Is Not the Universal Compactness Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0085","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ac8deb3dd26b4db0138cc129919dac298d643f21623b8c22f7f1c32f1810f44","family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013085,"semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,2],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,2],[0]]}"},"fields":{"title":"One Supplied Finite Subcover Is Not the Universal Compactness Definition","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 2], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).","answer":{"kind":"expression","value":"([1], Yes, No, one_finite_subcover_does_not_prove_compactness)","explanation":"The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness)."},"skills":["One Supplied Finite Subcover Is Not the Universal Compactness Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0086","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dedf23066303324546081dcfebb7e916cd2b1aabacd285661a5d164d10ed1227","family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013086,"semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,2],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,2],[0]]}"},"fields":{"title":"One Supplied Finite Subcover Is Not the Universal Compactness Definition","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).","answer":{"kind":"expression","value":"([1], Yes, No, one_finite_subcover_does_not_prove_compactness)","explanation":"The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness)."},"skills":["One Supplied Finite Subcover Is Not the Universal Compactness Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0087","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08d2ab44295d3210945cddaa403a58e31725f2708444822f899547b50157f3fe","family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013087,"semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,2],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,2],[0]]}"},"fields":{"title":"One Supplied Finite Subcover Is Not the Universal Compactness Definition","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 2], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).","answer":{"kind":"expression","value":"([1], Yes, No, one_finite_subcover_does_not_prove_compactness)","explanation":"The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness)."},"skills":["One Supplied Finite Subcover Is Not the Universal Compactness Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0088","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6846716f82957c326fe68dac77f799ea3ca4ec47f6fbd54e70b06d92a288abee","family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013088,"semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,3],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,3],[0]]}"},"fields":{"title":"One Supplied Finite Subcover Is Not the Universal Compactness Definition","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).","answer":{"kind":"expression","value":"([1], Yes, No, one_finite_subcover_does_not_prove_compactness)","explanation":"The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness)."},"skills":["One Supplied Finite Subcover Is Not the Universal Compactness Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0089","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a119bdf83e6d21d5c513392656cb3e3f321bdf93bbc481e491933c622ff9ef08","family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013089,"semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,3],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,3],[0]]}"},"fields":{"title":"One Supplied Finite Subcover Is Not the Universal Compactness Definition","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).","answer":{"kind":"expression","value":"([1], Yes, No, one_finite_subcover_does_not_prove_compactness)","explanation":"The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness)."},"skills":["One Supplied Finite Subcover Is Not the Universal Compactness Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0090","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2255209c65ae5b614cf1a1d7e9dec5491b68d6a008374920ae092a64bb8af8e3","family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013090,"semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,3],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,3],[0]]}"},"fields":{"title":"One Supplied Finite Subcover Is Not the Universal Compactness Definition","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).","answer":{"kind":"expression","value":"([1], Yes, No, one_finite_subcover_does_not_prove_compactness)","explanation":"The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness)."},"skills":["One Supplied Finite Subcover Is Not the Universal Compactness Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0091","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1d56e412bce413e2c1bb670ea04c92a89a0b6ebc5e630e2953b11671d323e8e","family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013091,"semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,3],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,3],[0]]}"},"fields":{"title":"One Supplied Finite Subcover Is Not the Universal Compactness Definition","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).","answer":{"kind":"expression","value":"([1], Yes, No, one_finite_subcover_does_not_prove_compactness)","explanation":"The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness)."},"skills":["One Supplied Finite Subcover Is Not the Universal Compactness Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0092","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6021bc978526ca23fc93e097cb9ff554b091160ed9845ce1caeb55edfd85df18","family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013092,"semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,3],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,3],[0]]}"},"fields":{"title":"One Supplied Finite Subcover Is Not the Universal Compactness Definition","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).","answer":{"kind":"expression","value":"([1], Yes, No, one_finite_subcover_does_not_prove_compactness)","explanation":"The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness)."},"skills":["One Supplied Finite Subcover Is Not the Universal Compactness Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0093","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3610cb2dbdd7fcdc6a750b77ee1cd56b92f6a93d233c293af664234e524cbef6","family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013093,"semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,3],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,3],[0]]}"},"fields":{"title":"One Supplied Finite Subcover Is Not the Universal Compactness Definition","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).","answer":{"kind":"expression","value":"([1], Yes, No, one_finite_subcover_does_not_prove_compactness)","explanation":"The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness)."},"skills":["One Supplied Finite Subcover Is Not the Universal Compactness Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0094","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24017bdd386d42f310f76996a51f0ca640686a07b72e563fa2f3cde376fac7ec","family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013094,"semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,4],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,4],[0]]}"},"fields":{"title":"One Supplied Finite Subcover Is Not the Universal Compactness Definition","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).","answer":{"kind":"expression","value":"([1], Yes, No, one_finite_subcover_does_not_prove_compactness)","explanation":"The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness)."},"skills":["One Supplied Finite Subcover Is Not the Universal Compactness Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0095","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"219f3b6522c3d7819dbdeaa4498eeb69f736912621de623e04c77cfa83eacb40","family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013095,"semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,4],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,4],[0]]}"},"fields":{"title":"One Supplied Finite Subcover Is Not the Universal Compactness Definition","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).","answer":{"kind":"expression","value":"([1], Yes, No, one_finite_subcover_does_not_prove_compactness)","explanation":"The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness)."},"skills":["One Supplied Finite Subcover Is Not the Universal Compactness Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0096","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6ea55c728ea71b20771314c8b338c7fa3a98f480fd6f7a6fb9e99113ecd4f27","family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013096,"semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,4],[0,1,2,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,4],[0]]}"},"fields":{"title":"One Supplied Finite Subcover Is Not the Universal Compactness Definition","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 4], ordered original cover family=[[0, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).","answer":{"kind":"expression","value":"([1], Yes, No, one_finite_subcover_does_not_prove_compactness)","explanation":"The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness)."},"skills":["One Supplied Finite Subcover Is Not the Universal Compactness Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0097","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1850e64cbe27f006f2ea8be302a9804c6fe19eca197516fa72bf65086119a194","family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013097,"semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,4],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,4],[0]]}"},"fields":{"title":"One Supplied Finite Subcover Is Not the Universal Compactness Definition","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).","answer":{"kind":"expression","value":"([1], Yes, No, one_finite_subcover_does_not_prove_compactness)","explanation":"The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness)."},"skills":["One Supplied Finite Subcover Is Not the Universal Compactness Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0098","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc658c8fce53a24b2b87a97c9d38ca8ce31dbe5c38f80d4840be508cf5e1f8b4","family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013098,"semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,4],[0,1,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,4],[0]]}"},"fields":{"title":"One Supplied Finite Subcover Is Not the Universal Compactness Definition","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 3, 4], ordered original cover family=[[0, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).","answer":{"kind":"expression","value":"([1], Yes, No, one_finite_subcover_does_not_prove_compactness)","explanation":"The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness)."},"skills":["One Supplied Finite Subcover Is Not the Universal Compactness Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0099","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2f1bb1f6d6eebdb5cf8ccd65721dee3ee99a7e23ac598cfc1958a16c73d4a23","family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013099,"semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,4],[0,2,3,4],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,2,3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,4],[0]]}"},"fields":{"title":"One Supplied Finite Subcover Is Not the Universal Compactness Definition","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 2, 3, 4], ordered original cover family=[[0, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).","answer":{"kind":"expression","value":"([1], Yes, No, one_finite_subcover_does_not_prove_compactness)","explanation":"The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness)."},"skills":["One Supplied Finite Subcover Is Not the Universal Compactness Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261101-r03-open-cover-and-subcover-0100","batch_id":"batch-20261014-advance-20261101-r03-open-cover-and-subcover-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9edaf4df82ba7540069dffc900e0deb4672a9e22fe4ee85778dbe5ef5b0d7c0","family_id":"qf-mod32-cover-single-cover-boundary","seed":202611013100,"semantic_template":"qf-mod32-cover-single-cover-boundary/{\"cover\":[[0,1],[0,1,2,3],[0,1,2,3,4]],\"n\":5,\"selected\":[1],\"target\":[0,1,2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[1]]}"},"fields":{"title":"One Supplied Finite Subcover Is Not the Universal Compactness Definition","stem":"Use the complete finite topology=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Every subset is an increasing duplicate-free carrier list. First check the unique empty/whole opens and all union/intersection closure obligations; finite topology suffices for exhaustive closure checks. Complements refer to this carrier. No metric is silently supplied. Target=[0, 1, 2, 3], ordered original cover family=[[0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4]]. Every original family member must be open and their union must include the target before a selected subcover is certified. Cover indices begin at 0; selection only uses those original members, sorts indices and rejects duplicates. Keep original family order, including any equal sets at different indices. selected=[1]. Give (selected_indexes,valid_subcover?,proves_compactness?,one_finite_subcover_does_not_prove_compactness).","answer":{"kind":"expression","value":"([1], Yes, No, one_finite_subcover_does_not_prove_compactness)","explanation":"The final No concerns what checking this single supplied cover establishes. It is not a claim that the specified finite space is noncompact: finiteness supports a separate compactness argument, while the definition quantifies over every open cover. The exact result, retaining the requested order, is ([1], Yes, No, one_finite_subcover_does_not_prove_compactness)."},"skills":["One Supplied Finite Subcover Is Not the Universal Compactness Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}