{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r03-topological-basis-0001","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"94fc3f0f194e6918086cbea7eebd0153b335cc0c8e1077abefd77ab8045220e9","family_id":"qf-mod38-topbasis-basis-qualification","seed":202611073001,"semantic_template":"qf-mod38-topbasis-basis-qualification/{\"basis\":[[0],[1],[2],[3],[4]],\"n\":5}"},"fields":{"title":"Coverage and Pointwise Intersection Refinement Are Separate Axioms","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).","answer":{"kind":"expression","value":"(Yes, Yes, null, Yes)","explanation":"Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes)."},"skills":["Coverage and Pointwise Intersection Refinement Are Separate Axioms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0002","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c33f48ce672d14ed254c3bb69e146abccad3cd22118e01f3f53129fd061a289","family_id":"qf-mod38-topbasis-basis-qualification","seed":202611073002,"semantic_template":"qf-mod38-topbasis-basis-qualification/{\"basis\":[[0,1],[0],[1],[2],[3],[4]],\"n\":5}"},"fields":{"title":"Coverage and Pointwise Intersection Refinement Are Separate Axioms","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [1], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).","answer":{"kind":"expression","value":"(Yes, Yes, null, Yes)","explanation":"Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes)."},"skills":["Coverage and Pointwise Intersection Refinement Are Separate Axioms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0003","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08b81333f42c2c0ed3c42b2a5f0b41b87a65f035d492d5f659746e7ed915605a","family_id":"qf-mod38-topbasis-basis-qualification","seed":202611073003,"semantic_template":"qf-mod38-topbasis-basis-qualification/{\"basis\":[[0,2],[0],[1],[2],[3],[4]],\"n\":5}"},"fields":{"title":"Coverage and Pointwise Intersection Refinement Are Separate Axioms","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 2], [1], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).","answer":{"kind":"expression","value":"(Yes, Yes, null, Yes)","explanation":"Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes)."},"skills":["Coverage and Pointwise Intersection Refinement Are Separate Axioms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0004","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99a229260377481c6599b03046c99f63e7220bbb96796b99fb2690d67f81a349","family_id":"qf-mod38-topbasis-basis-qualification","seed":202611073004,"semantic_template":"qf-mod38-topbasis-basis-qualification/{\"basis\":[[0,1],[0,2],[0],[1],[2],[3],[4]],\"n\":5}"},"fields":{"title":"Coverage and Pointwise Intersection Refinement Are Separate Axioms","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [0, 2], [1], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).","answer":{"kind":"expression","value":"(Yes, Yes, null, Yes)","explanation":"Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes)."},"skills":["Coverage and Pointwise Intersection Refinement Are Separate Axioms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0005","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ac0576671654dcc29104f97c1b3cca52775046b6653ed326d96e85604119169","family_id":"qf-mod38-topbasis-basis-qualification","seed":202611073005,"semantic_template":"qf-mod38-topbasis-basis-qualification/{\"basis\":[[0],[1,2],[1],[2],[3],[4]],\"n\":5}"},"fields":{"title":"Coverage and Pointwise Intersection Refinement Are Separate Axioms","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [1, 2], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).","answer":{"kind":"expression","value":"(Yes, Yes, null, Yes)","explanation":"Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes)."},"skills":["Coverage and Pointwise Intersection Refinement Are Separate Axioms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0006","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b4658519134b911c15eef20be7c83295641521e9466ed1d165b14781900dd39","family_id":"qf-mod38-topbasis-basis-qualification","seed":202611073006,"semantic_template":"qf-mod38-topbasis-basis-qualification/{\"basis\":[[0,1],[0],[1,2],[1],[2],[3],[4]],\"n\":5}"},"fields":{"title":"Coverage and Pointwise Intersection Refinement Are Separate Axioms","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [1], [1, 2], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).","answer":{"kind":"expression","value":"(Yes, Yes, null, Yes)","explanation":"Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes)."},"skills":["Coverage and Pointwise Intersection Refinement Are Separate Axioms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0007","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e6b147733229f561b31180ea3c0f34c157f206184a81a885e97fcef3b39b3e1","family_id":"qf-mod38-topbasis-basis-qualification","seed":202611073007,"semantic_template":"qf-mod38-topbasis-basis-qualification/{\"basis\":[[0,2],[0],[1,2],[1],[2],[3],[4]],\"n\":5}"},"fields":{"title":"Coverage and Pointwise Intersection Refinement Are Separate Axioms","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 2], [1], [1, 2], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).","answer":{"kind":"expression","value":"(Yes, Yes, null, Yes)","explanation":"Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes)."},"skills":["Coverage and Pointwise Intersection Refinement Are Separate Axioms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0008","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8f73844eb17b19e899e8eb5678e9cda153933fdb855bc0ec713be5b7d113ca3","family_id":"qf-mod38-topbasis-basis-qualification","seed":202611073008,"semantic_template":"qf-mod38-topbasis-basis-qualification/{\"basis\":[[0,1],[0,2],[0],[1,2],[1],[2],[3],[4]],\"n\":5}"},"fields":{"title":"Coverage and Pointwise Intersection Refinement Are Separate Axioms","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [0, 2], [1], [1, 2], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).","answer":{"kind":"expression","value":"(Yes, Yes, null, Yes)","explanation":"Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes)."},"skills":["Coverage and Pointwise Intersection Refinement Are Separate Axioms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0009","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71ce6b73808d12d23137767e9c1169e9c00279d6e98d9cb4fbc7850072dc0acd","family_id":"qf-mod38-topbasis-basis-qualification","seed":202611073009,"semantic_template":"qf-mod38-topbasis-basis-qualification/{\"basis\":[[0],[1],[2,3],[2],[3],[4]],\"n\":5}"},"fields":{"title":"Coverage and Pointwise Intersection Refinement Are Separate Axioms","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).","answer":{"kind":"expression","value":"(Yes, Yes, null, Yes)","explanation":"Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes)."},"skills":["Coverage and Pointwise Intersection Refinement Are Separate Axioms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0010","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b03c95f84784c0b9cd8f0a941f47202d9044c94455563ada821b40d851a35372","family_id":"qf-mod38-topbasis-basis-qualification","seed":202611073010,"semantic_template":"qf-mod38-topbasis-basis-qualification/{\"basis\":[[0,1],[0],[1],[2,3],[2],[3],[4]],\"n\":5}"},"fields":{"title":"Coverage and Pointwise Intersection Refinement Are Separate Axioms","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [1], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).","answer":{"kind":"expression","value":"(Yes, Yes, null, Yes)","explanation":"Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes)."},"skills":["Coverage and Pointwise Intersection Refinement Are Separate Axioms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0011","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"09886aba7906f9a985938861511b5332ad25f254ec88df389f1343530e89829d","family_id":"qf-mod38-topbasis-basis-qualification","seed":202611073011,"semantic_template":"qf-mod38-topbasis-basis-qualification/{\"basis\":[[0,2],[0],[1],[2,3],[2],[3],[4]],\"n\":5}"},"fields":{"title":"Coverage and Pointwise Intersection Refinement Are Separate Axioms","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 2], [1], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).","answer":{"kind":"expression","value":"(Yes, Yes, null, Yes)","explanation":"Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes)."},"skills":["Coverage and Pointwise Intersection Refinement Are Separate Axioms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0012","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da4954594586f3b0c28074893721976012981c3c0d47e1952582ec5696a3cee6","family_id":"qf-mod38-topbasis-basis-qualification","seed":202611073012,"semantic_template":"qf-mod38-topbasis-basis-qualification/{\"basis\":[[0,1],[0,2],[0],[1],[2,3],[2],[3],[4]],\"n\":5}"},"fields":{"title":"Coverage and Pointwise Intersection Refinement Are Separate Axioms","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [0, 2], [1], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).","answer":{"kind":"expression","value":"(Yes, Yes, null, Yes)","explanation":"Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes)."},"skills":["Coverage and Pointwise Intersection Refinement Are Separate Axioms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0013","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"904cb03c9c5d649cd3e5fc8fb135998665ea47da3c4abd7baa33b5756e32920d","family_id":"qf-mod38-topbasis-basis-qualification","seed":202611073013,"semantic_template":"qf-mod38-topbasis-basis-qualification/{\"basis\":[[0],[1,2],[1],[2,3],[2],[3],[4]],\"n\":5}"},"fields":{"title":"Coverage and Pointwise Intersection Refinement Are Separate Axioms","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [1, 2], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).","answer":{"kind":"expression","value":"(Yes, Yes, null, Yes)","explanation":"Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes)."},"skills":["Coverage and Pointwise Intersection Refinement Are Separate Axioms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0014","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8771680b39bbb6b344cfe11ef245a654ced8102eef8332af6c592503188de78","family_id":"qf-mod38-topbasis-basis-qualification","seed":202611073014,"semantic_template":"qf-mod38-topbasis-basis-qualification/{\"basis\":[[0,1],[0],[1,2],[1],[2,3],[2],[3],[4]],\"n\":5}"},"fields":{"title":"Coverage and Pointwise Intersection Refinement Are Separate Axioms","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [1], [1, 2], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).","answer":{"kind":"expression","value":"(Yes, Yes, null, Yes)","explanation":"Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes)."},"skills":["Coverage and Pointwise Intersection Refinement Are Separate Axioms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0015","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e37ac5cbf0ac7d64e2c1fa590507627e5ebf9608751b568a92b42a9717708160","family_id":"qf-mod38-topbasis-basis-qualification","seed":202611073015,"semantic_template":"qf-mod38-topbasis-basis-qualification/{\"basis\":[[0,2],[0],[1,2],[1],[2,3],[2],[3],[4]],\"n\":5}"},"fields":{"title":"Coverage and Pointwise Intersection Refinement Are Separate Axioms","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 2], [1], [1, 2], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).","answer":{"kind":"expression","value":"(Yes, Yes, null, Yes)","explanation":"Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes)."},"skills":["Coverage and Pointwise Intersection Refinement Are Separate Axioms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0016","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a2af05fcb0c321c8cbff6a7d701419a6ffbf6c3f2e3eb883a6964e1b4b73a9b","family_id":"qf-mod38-topbasis-basis-qualification","seed":202611073016,"semantic_template":"qf-mod38-topbasis-basis-qualification/{\"basis\":[[0,1],[0,2],[0],[1,2],[1],[2,3],[2],[3],[4]],\"n\":5}"},"fields":{"title":"Coverage and Pointwise Intersection Refinement Are Separate Axioms","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [0, 2], [1], [1, 2], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).","answer":{"kind":"expression","value":"(Yes, Yes, null, Yes)","explanation":"Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes)."},"skills":["Coverage and Pointwise Intersection Refinement Are Separate Axioms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0017","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4b056684e65f23d40c54ef43ca868b3c56ad0680c966842a1b98886f30d23a8","family_id":"qf-mod38-topbasis-basis-qualification","seed":202611073017,"semantic_template":"qf-mod38-topbasis-basis-qualification/{\"basis\":[[0],[1],[2],[3,4],[3],[4]],\"n\":5}"},"fields":{"title":"Coverage and Pointwise Intersection Refinement Are Separate Axioms","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).","answer":{"kind":"expression","value":"(Yes, Yes, null, Yes)","explanation":"Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes)."},"skills":["Coverage and Pointwise Intersection Refinement Are Separate Axioms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0018","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d6cbfc78eaca723f4441fc280ef7804fe630ba8bd51ae21a951b3322d84f500","family_id":"qf-mod38-topbasis-basis-qualification","seed":202611073018,"semantic_template":"qf-mod38-topbasis-basis-qualification/{\"basis\":[[0,1],[0],[1],[2],[3,4],[3],[4]],\"n\":5}"},"fields":{"title":"Coverage and Pointwise Intersection Refinement Are Separate Axioms","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [1], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).","answer":{"kind":"expression","value":"(Yes, Yes, null, Yes)","explanation":"Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes)."},"skills":["Coverage and Pointwise Intersection Refinement Are Separate Axioms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0019","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5334977214c37d2ec6c5a387dbec2010ee9b7e29d1f8cf254856f321650c5ed4","family_id":"qf-mod38-topbasis-basis-qualification","seed":202611073019,"semantic_template":"qf-mod38-topbasis-basis-qualification/{\"basis\":[[0,2],[0],[1],[2],[3,4],[3],[4]],\"n\":5}"},"fields":{"title":"Coverage and Pointwise Intersection Refinement Are Separate Axioms","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 2], [1], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).","answer":{"kind":"expression","value":"(Yes, Yes, null, Yes)","explanation":"Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes)."},"skills":["Coverage and Pointwise Intersection Refinement Are Separate Axioms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0020","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a764a2eeea99c60eb054a6968fa46c539d9b211505f0e4267793ae80035b75e","family_id":"qf-mod38-topbasis-basis-qualification","seed":202611073020,"semantic_template":"qf-mod38-topbasis-basis-qualification/{\"basis\":[[0,1],[0,2],[0],[1],[2],[3,4],[3],[4]],\"n\":5}"},"fields":{"title":"Coverage and Pointwise Intersection Refinement Are Separate Axioms","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [0, 2], [1], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).","answer":{"kind":"expression","value":"(Yes, Yes, null, Yes)","explanation":"Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes)."},"skills":["Coverage and Pointwise Intersection Refinement Are Separate Axioms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0021","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3948afef9e0db85aeb69a1cbd5038ee880177f0d9b38de7414b3e8faa45e9451","family_id":"qf-mod38-topbasis-basis-qualification","seed":202611073021,"semantic_template":"qf-mod38-topbasis-basis-qualification/{\"basis\":[[0],[1,2],[1],[2],[3,4],[3],[4]],\"n\":5}"},"fields":{"title":"Coverage and Pointwise Intersection Refinement Are Separate Axioms","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [1, 2], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).","answer":{"kind":"expression","value":"(Yes, Yes, null, Yes)","explanation":"Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes)."},"skills":["Coverage and Pointwise Intersection Refinement Are Separate Axioms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0022","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"610c73c9e745f0a5266cfe2ba47cd92da1627e58c9f412339d63ff27d6caa3ec","family_id":"qf-mod38-topbasis-basis-qualification","seed":202611073022,"semantic_template":"qf-mod38-topbasis-basis-qualification/{\"basis\":[[0,1],[0],[1,2],[1],[2],[3,4],[3],[4]],\"n\":5}"},"fields":{"title":"Coverage and Pointwise Intersection Refinement Are Separate Axioms","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [1], [1, 2], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).","answer":{"kind":"expression","value":"(Yes, Yes, null, Yes)","explanation":"Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes)."},"skills":["Coverage and Pointwise Intersection Refinement Are Separate Axioms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0023","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8cf17ed89fe25a758fb86edc6df0c7cc3f83816d81904c4e944fec2578d6ba8b","family_id":"qf-mod38-topbasis-basis-qualification","seed":202611073023,"semantic_template":"qf-mod38-topbasis-basis-qualification/{\"basis\":[[0,2],[0],[1,2],[1],[2],[3,4],[3],[4]],\"n\":5}"},"fields":{"title":"Coverage and Pointwise Intersection Refinement Are Separate Axioms","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 2], [1], [1, 2], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).","answer":{"kind":"expression","value":"(Yes, Yes, null, Yes)","explanation":"Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes)."},"skills":["Coverage and Pointwise Intersection Refinement Are Separate Axioms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0024","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b02b2d0944cf36ca461e80b83042baa2b17c27d57aa4277752020364041140d","family_id":"qf-mod38-topbasis-basis-qualification","seed":202611073024,"semantic_template":"qf-mod38-topbasis-basis-qualification/{\"basis\":[[0,1],[0,2],[0],[1,2],[1],[2],[3,4],[3],[4]],\"n\":5}"},"fields":{"title":"Coverage and Pointwise Intersection Refinement Are Separate Axioms","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [0, 2], [1], [1, 2], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).","answer":{"kind":"expression","value":"(Yes, Yes, null, Yes)","explanation":"Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes)."},"skills":["Coverage and Pointwise Intersection Refinement Are Separate Axioms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0025","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29f1c8b4b4ae1eac7e9ed8c60e9b069fb12addeddac7bc58cc0c2376b18d13be","family_id":"qf-mod38-topbasis-basis-qualification","seed":202611073025,"semantic_template":"qf-mod38-topbasis-basis-qualification/{\"basis\":[[0],[1],[2,3],[2],[3,4],[3],[4]],\"n\":5}"},"fields":{"title":"Coverage and Pointwise Intersection Refinement Are Separate Axioms","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [2], [2, 3], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, first_failure, is_basis).","answer":{"kind":"expression","value":"(Yes, Yes, null, Yes)","explanation":"Return coverage, full local refinement, the first failed [left_family_index,right_family_index,point,intersection] or null, then their joint basis-validity flag. A covering family can fail refinement. The exact result, retaining the requested order, is (Yes, Yes, null, Yes)."},"skills":["Coverage and Pointwise Intersection Refinement Are Separate Axioms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0026","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12feae5bd3f0cc636b18108a10959d492004a7f166829704d7b1e95eee5d644f","family_id":"qf-mod38-topbasis-generated-topology","seed":202611073026,"semantic_template":"qf-mod38-topbasis-generated-topology/{\"basis\":[[0],[1],[2],[3],[4]],\"n\":5}"},"fields":{"title":"Generate All Unions Only after the Basis Axioms Pass","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)","explanation":"Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)."},"skills":["Generate All Unions Only after the Basis Axioms Pass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0027","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22627e66608aa5fdd3f43c45d70f62caac43399afaa81c7c4f7e2b7f1a0c41ae","family_id":"qf-mod38-topbasis-generated-topology","seed":202611073027,"semantic_template":"qf-mod38-topbasis-generated-topology/{\"basis\":[[0,1],[0],[1],[2],[3],[4]],\"n\":5}"},"fields":{"title":"Generate All Unions Only after the Basis Axioms Pass","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [1], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)","explanation":"Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)."},"skills":["Generate All Unions Only after the Basis Axioms Pass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0028","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6ac65203fe1b0b4b130d715007cac324735391f7dc6353a7d90e91f1e5532d2","family_id":"qf-mod38-topbasis-generated-topology","seed":202611073028,"semantic_template":"qf-mod38-topbasis-generated-topology/{\"basis\":[[0,2],[0],[1],[2],[3],[4]],\"n\":5}"},"fields":{"title":"Generate All Unions Only after the Basis Axioms Pass","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 2], [1], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)","explanation":"Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)."},"skills":["Generate All Unions Only after the Basis Axioms Pass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0029","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75cdf5db18b5f0d4b22548c8cde87e241a619caad3d18f5f5a9eacf5644cd52a","family_id":"qf-mod38-topbasis-generated-topology","seed":202611073029,"semantic_template":"qf-mod38-topbasis-generated-topology/{\"basis\":[[0,1],[0,2],[0],[1],[2],[3],[4]],\"n\":5}"},"fields":{"title":"Generate All Unions Only after the Basis Axioms Pass","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [0, 2], [1], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)","explanation":"Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)."},"skills":["Generate All Unions Only after the Basis Axioms Pass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0030","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8bb3c2b938c8e4513ba499c1db2650dda7bb9301900b1dde7a11ec0b249ba8e0","family_id":"qf-mod38-topbasis-generated-topology","seed":202611073030,"semantic_template":"qf-mod38-topbasis-generated-topology/{\"basis\":[[0],[1,2],[1],[2],[3],[4]],\"n\":5}"},"fields":{"title":"Generate All Unions Only after the Basis Axioms Pass","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [1, 2], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)","explanation":"Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)."},"skills":["Generate All Unions Only after the Basis Axioms Pass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0031","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"728ab7e3a2306eb06ccd57ed21e8597ba61be722ea2055decca9ed8fe113d686","family_id":"qf-mod38-topbasis-generated-topology","seed":202611073031,"semantic_template":"qf-mod38-topbasis-generated-topology/{\"basis\":[[0,1],[0],[1,2],[1],[2],[3],[4]],\"n\":5}"},"fields":{"title":"Generate All Unions Only after the Basis Axioms Pass","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [1], [1, 2], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)","explanation":"Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)."},"skills":["Generate All Unions Only after the Basis Axioms Pass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0032","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"56c516b6cc4b913e8e00d84de2ad503a62fbd095e7eadeb0532d20ef5b58abdd","family_id":"qf-mod38-topbasis-generated-topology","seed":202611073032,"semantic_template":"qf-mod38-topbasis-generated-topology/{\"basis\":[[0,2],[0],[1,2],[1],[2],[3],[4]],\"n\":5}"},"fields":{"title":"Generate All Unions Only after the Basis Axioms Pass","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 2], [1], [1, 2], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)","explanation":"Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)."},"skills":["Generate All Unions Only after the Basis Axioms Pass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0033","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7578033d65fdc69b4bd38d02562d5698bce75a506af7cdbc73f993fb8c4786b5","family_id":"qf-mod38-topbasis-generated-topology","seed":202611073033,"semantic_template":"qf-mod38-topbasis-generated-topology/{\"basis\":[[0,1],[0,2],[0],[1,2],[1],[2],[3],[4]],\"n\":5}"},"fields":{"title":"Generate All Unions Only after the Basis Axioms Pass","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [0, 2], [1], [1, 2], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)","explanation":"Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)."},"skills":["Generate All Unions Only after the Basis Axioms Pass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0034","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19a9eae19a37bfc4dab6d02522d5c4108a5afd3e369a45ec2da7ada3838a11ff","family_id":"qf-mod38-topbasis-generated-topology","seed":202611073034,"semantic_template":"qf-mod38-topbasis-generated-topology/{\"basis\":[[0],[1],[2,3],[2],[3],[4]],\"n\":5}"},"fields":{"title":"Generate All Unions Only after the Basis Axioms Pass","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)","explanation":"Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)."},"skills":["Generate All Unions Only after the Basis Axioms Pass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0035","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8b8d47e9dcd27f57cc353ae6d33fe20d30ec23a024d5e30a4ac7c1a998d98d6","family_id":"qf-mod38-topbasis-generated-topology","seed":202611073035,"semantic_template":"qf-mod38-topbasis-generated-topology/{\"basis\":[[0,1],[0],[1],[2,3],[2],[3],[4]],\"n\":5}"},"fields":{"title":"Generate All Unions Only after the Basis Axioms Pass","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [1], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)","explanation":"Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)."},"skills":["Generate All Unions Only after the Basis Axioms Pass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0036","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e76bc239e9d8b251c09b5885f4130a8a045502f3305c8c7762beb75dbf64e97","family_id":"qf-mod38-topbasis-generated-topology","seed":202611073036,"semantic_template":"qf-mod38-topbasis-generated-topology/{\"basis\":[[0,2],[0],[1],[2,3],[2],[3],[4]],\"n\":5}"},"fields":{"title":"Generate All Unions Only after the Basis Axioms Pass","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 2], [1], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)","explanation":"Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)."},"skills":["Generate All Unions Only after the Basis Axioms Pass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0037","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41bdebf4ae18f7e13d660d0e9feedfb218dff65a6be1391b66866d9acfc2e2f9","family_id":"qf-mod38-topbasis-generated-topology","seed":202611073037,"semantic_template":"qf-mod38-topbasis-generated-topology/{\"basis\":[[0,1],[0,2],[0],[1],[2,3],[2],[3],[4]],\"n\":5}"},"fields":{"title":"Generate All Unions Only after the Basis Axioms Pass","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [0, 2], [1], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)","explanation":"Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)."},"skills":["Generate All Unions Only after the Basis Axioms Pass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0038","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd5fff09f9dd274fd28f49897489a54f43220c5fed5a87102d2b54c1af6dde61","family_id":"qf-mod38-topbasis-generated-topology","seed":202611073038,"semantic_template":"qf-mod38-topbasis-generated-topology/{\"basis\":[[0],[1,2],[1],[2,3],[2],[3],[4]],\"n\":5}"},"fields":{"title":"Generate All Unions Only after the Basis Axioms Pass","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [1, 2], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)","explanation":"Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)."},"skills":["Generate All Unions Only after the Basis Axioms Pass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0039","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d897fc3997b6305fb817a16e219f86546d92bcfbbd9c8a96a14e3d0b909c933","family_id":"qf-mod38-topbasis-generated-topology","seed":202611073039,"semantic_template":"qf-mod38-topbasis-generated-topology/{\"basis\":[[0,1],[0],[1,2],[1],[2,3],[2],[3],[4]],\"n\":5}"},"fields":{"title":"Generate All Unions Only after the Basis Axioms Pass","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [1], [1, 2], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)","explanation":"Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)."},"skills":["Generate All Unions Only after the Basis Axioms Pass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0040","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e728abc7f6f2309c325d71fbe258fc8f422ecb65e3c630713cf36b4b6fbdd01e","family_id":"qf-mod38-topbasis-generated-topology","seed":202611073040,"semantic_template":"qf-mod38-topbasis-generated-topology/{\"basis\":[[0,2],[0],[1,2],[1],[2,3],[2],[3],[4]],\"n\":5}"},"fields":{"title":"Generate All Unions Only after the Basis Axioms Pass","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 2], [1], [1, 2], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)","explanation":"Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)."},"skills":["Generate All Unions Only after the Basis Axioms Pass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0041","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7f2851141c207550d48bf990a7fb2fc2216bcbec8b4b94c161afb3674d42ed5","family_id":"qf-mod38-topbasis-generated-topology","seed":202611073041,"semantic_template":"qf-mod38-topbasis-generated-topology/{\"basis\":[[0,1],[0,2],[0],[1,2],[1],[2,3],[2],[3],[4]],\"n\":5}"},"fields":{"title":"Generate All Unions Only after the Basis Axioms Pass","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [0, 2], [1], [1, 2], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)","explanation":"Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)."},"skills":["Generate All Unions Only after the Basis Axioms Pass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0042","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aabb8399141ff3631a04eab37a5b5292feabf537fec6029e5d6562308c1688f5","family_id":"qf-mod38-topbasis-generated-topology","seed":202611073042,"semantic_template":"qf-mod38-topbasis-generated-topology/{\"basis\":[[0],[1],[2],[3,4],[3],[4]],\"n\":5}"},"fields":{"title":"Generate All Unions Only after the Basis Axioms Pass","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)","explanation":"Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)."},"skills":["Generate All Unions Only after the Basis Axioms Pass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0043","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"833a188e30dbfc2c7308a75a0149dd5ad7a756377618f2565d6355727cbcb8a6","family_id":"qf-mod38-topbasis-generated-topology","seed":202611073043,"semantic_template":"qf-mod38-topbasis-generated-topology/{\"basis\":[[0,1],[0],[1],[2],[3,4],[3],[4]],\"n\":5}"},"fields":{"title":"Generate All Unions Only after the Basis Axioms Pass","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [1], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)","explanation":"Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)."},"skills":["Generate All Unions Only after the Basis Axioms Pass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0044","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d425be7f713f11108ed49312c5feaa5752f7f6ef7146f3ca268e81794eba9ab","family_id":"qf-mod38-topbasis-generated-topology","seed":202611073044,"semantic_template":"qf-mod38-topbasis-generated-topology/{\"basis\":[[0,2],[0],[1],[2],[3,4],[3],[4]],\"n\":5}"},"fields":{"title":"Generate All Unions Only after the Basis Axioms Pass","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 2], [1], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)","explanation":"Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)."},"skills":["Generate All Unions Only after the Basis Axioms Pass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0045","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0581a8551c795fa52161328fb642f8ee84c6602a47943211a57a75442b10113","family_id":"qf-mod38-topbasis-generated-topology","seed":202611073045,"semantic_template":"qf-mod38-topbasis-generated-topology/{\"basis\":[[0,1],[0,2],[0],[1],[2],[3,4],[3],[4]],\"n\":5}"},"fields":{"title":"Generate All Unions Only after the Basis Axioms Pass","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [0, 2], [1], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)","explanation":"Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)."},"skills":["Generate All Unions Only after the Basis Axioms Pass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0046","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45b98ecd6e78ebbeeed198f82d0c6c371fe055019bb03708b2b0fdd17b3d0501","family_id":"qf-mod38-topbasis-generated-topology","seed":202611073046,"semantic_template":"qf-mod38-topbasis-generated-topology/{\"basis\":[[0],[1,2],[1],[2],[3,4],[3],[4]],\"n\":5}"},"fields":{"title":"Generate All Unions Only after the Basis Axioms Pass","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [1, 2], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)","explanation":"Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)."},"skills":["Generate All Unions Only after the Basis Axioms Pass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0047","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"538c459c383aea485a572051040c7e0c854134eb9b7a31b552e2f9e592dfe87a","family_id":"qf-mod38-topbasis-generated-topology","seed":202611073047,"semantic_template":"qf-mod38-topbasis-generated-topology/{\"basis\":[[0,1],[0],[1,2],[1],[2],[3,4],[3],[4]],\"n\":5}"},"fields":{"title":"Generate All Unions Only after the Basis Axioms Pass","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [1], [1, 2], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)","explanation":"Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)."},"skills":["Generate All Unions Only after the Basis Axioms Pass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0048","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ab5379e765c5204e1dddb863e012e02ce726374d6f3aa34e7af9afe687b5bf2","family_id":"qf-mod38-topbasis-generated-topology","seed":202611073048,"semantic_template":"qf-mod38-topbasis-generated-topology/{\"basis\":[[0,2],[0],[1,2],[1],[2],[3,4],[3],[4]],\"n\":5}"},"fields":{"title":"Generate All Unions Only after the Basis Axioms Pass","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 2], [1], [1, 2], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)","explanation":"Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)."},"skills":["Generate All Unions Only after the Basis Axioms Pass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0049","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e601a101e7b7dced6c4fc2faacd12087317e23d987aa16f713b75dc0db81e820","family_id":"qf-mod38-topbasis-generated-topology","seed":202611073049,"semantic_template":"qf-mod38-topbasis-generated-topology/{\"basis\":[[0,1],[0,2],[0],[1,2],[1],[2],[3,4],[3],[4]],\"n\":5}"},"fields":{"title":"Generate All Unions Only after the Basis Axioms Pass","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [0, 2], [1], [1, 2], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)","explanation":"Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)."},"skills":["Generate All Unions Only after the Basis Axioms Pass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0050","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d04c60fa8da78e1d091b4808c418e62915c21f1c3161564cf3c66ca6679af336","family_id":"qf-mod38-topbasis-generated-topology","seed":202611073050,"semantic_template":"qf-mod38-topbasis-generated-topology/{\"basis\":[[0],[1],[2,3],[2],[3,4],[3],[4]],\"n\":5}"},"fields":{"title":"Generate All Unions Only after the Basis Axioms Pass","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [2], [2, 3], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (topology, open_count, topology_certificate, certificate).","answer":{"kind":"expression","value":"([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)","explanation":"Require a valid basis and enumerate every distinct union, including the empty union. Retain the source canonical set-family order and exact open-set count. This is an exact certificate for the stated finite carrier, not an arbitrary-space sampling proof. The exact result, retaining the requested order, is ([[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]], 32, Yes, all_unions_of_basis_elements)."},"skills":["Generate All Unions Only after the Basis Axioms Pass"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0051","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2ebf724dae445d64b85251948e0ff86db3898c39d2fc8ee690d7abd64f177ad","family_id":"qf-mod38-topbasis-existing-topology-audit","seed":202611073051,"semantic_template":"qf-mod38-topbasis-existing-topology-audit/{\"basis\":[[0],[1],[2],[3],[4]],\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]]}"},"fields":{"title":"A Basis for Some Topology Need Not Generate This Given Topology","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes)","explanation":"Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes)."},"skills":["A Basis for Some Topology Need Not Generate This Given Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0052","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2da8440bb1759a02eb963c7aab3e480bc9a410cd28fe111e5bfb9f94ed834a6b","family_id":"qf-mod38-topbasis-existing-topology-audit","seed":202611073052,"semantic_template":"qf-mod38-topbasis-existing-topology-audit/{\"basis\":[[0,1],[0],[1],[2],[3],[4]],\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]]}"},"fields":{"title":"A Basis for Some Topology Need Not Generate This Given Topology","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [1], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes)","explanation":"Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes)."},"skills":["A Basis for Some Topology Need Not Generate This Given Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0053","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"846745d24f133b981cab2a935ddf84e5a2f8d56bff06607ff4c5ca62f6af0c61","family_id":"qf-mod38-topbasis-existing-topology-audit","seed":202611073053,"semantic_template":"qf-mod38-topbasis-existing-topology-audit/{\"basis\":[[0,2],[0],[1],[2],[3],[4]],\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]]}"},"fields":{"title":"A Basis for Some Topology Need Not Generate This Given Topology","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 2], [1], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes)","explanation":"Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes)."},"skills":["A Basis for Some Topology Need Not Generate This Given Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0054","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64579a61e284156e930f008492f2edd5a35762ff43e3f34345bf398c4fdd4e1f","family_id":"qf-mod38-topbasis-existing-topology-audit","seed":202611073054,"semantic_template":"qf-mod38-topbasis-existing-topology-audit/{\"basis\":[[0,1],[0,2],[0],[1],[2],[3],[4]],\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]]}"},"fields":{"title":"A Basis for Some Topology Need Not Generate This Given Topology","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [0, 2], [1], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes)","explanation":"Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes)."},"skills":["A Basis for Some Topology Need Not Generate This Given Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0055","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88f12010e97cb8beff91e3e687775cf4049a24e1243f927838796dcdbb7735e9","family_id":"qf-mod38-topbasis-existing-topology-audit","seed":202611073055,"semantic_template":"qf-mod38-topbasis-existing-topology-audit/{\"basis\":[[0],[1,2],[1],[2],[3],[4]],\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]]}"},"fields":{"title":"A Basis for Some Topology Need Not Generate This Given Topology","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [1, 2], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes)","explanation":"Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes)."},"skills":["A Basis for Some Topology Need Not Generate This Given Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0056","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1d35982916727d80a876d0ef3726d87ee9eb3c698bb385f7d4be007fb0df226","family_id":"qf-mod38-topbasis-existing-topology-audit","seed":202611073056,"semantic_template":"qf-mod38-topbasis-existing-topology-audit/{\"basis\":[[0,1],[0],[1,2],[1],[2],[3],[4]],\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]]}"},"fields":{"title":"A Basis for Some Topology Need Not Generate This Given Topology","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [1], [1, 2], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes)","explanation":"Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes)."},"skills":["A Basis for Some Topology Need Not Generate This Given Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0057","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78e6d16108670a32e14f0b97a0a5c2ed2afd5605da47743bce2b4ff6e42980bf","family_id":"qf-mod38-topbasis-existing-topology-audit","seed":202611073057,"semantic_template":"qf-mod38-topbasis-existing-topology-audit/{\"basis\":[[0,2],[0],[1,2],[1],[2],[3],[4]],\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]]}"},"fields":{"title":"A Basis for Some Topology Need Not Generate This Given Topology","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 2], [1], [1, 2], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes)","explanation":"Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes)."},"skills":["A Basis for Some Topology Need Not Generate This Given Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0058","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"407260cdc8952d29baebf6e06da76029ff548477cd6d99e5528562d98e97cdcc","family_id":"qf-mod38-topbasis-existing-topology-audit","seed":202611073058,"semantic_template":"qf-mod38-topbasis-existing-topology-audit/{\"basis\":[[0,1],[0,2],[0],[1,2],[1],[2],[3],[4]],\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]]}"},"fields":{"title":"A Basis for Some Topology Need Not Generate This Given Topology","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [0, 2], [1], [1, 2], [2], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes)","explanation":"Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes)."},"skills":["A Basis for Some Topology Need Not Generate This Given Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0059","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2729396024098f16092a21244c01aa7afaa5a54926b53ca4293fea9c901d766f","family_id":"qf-mod38-topbasis-existing-topology-audit","seed":202611073059,"semantic_template":"qf-mod38-topbasis-existing-topology-audit/{\"basis\":[[0],[1],[2,3],[2],[3],[4]],\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]]}"},"fields":{"title":"A Basis for Some Topology Need Not Generate This Given Topology","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes)","explanation":"Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes)."},"skills":["A Basis for Some Topology Need Not Generate This Given Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0060","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae3ee0e36b1e281b794f403786996e9803afc992d164412d4e0d1ba464018ffc","family_id":"qf-mod38-topbasis-existing-topology-audit","seed":202611073060,"semantic_template":"qf-mod38-topbasis-existing-topology-audit/{\"basis\":[[0,1],[0],[1],[2,3],[2],[3],[4]],\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]]}"},"fields":{"title":"A Basis for Some Topology Need Not Generate This Given Topology","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [1], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes)","explanation":"Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes)."},"skills":["A Basis for Some Topology Need Not Generate This Given Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0061","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e40ab97843b8e3ba0995d1309c588f8948b0bf821253ce4e988f5e1d5abba1f","family_id":"qf-mod38-topbasis-existing-topology-audit","seed":202611073061,"semantic_template":"qf-mod38-topbasis-existing-topology-audit/{\"basis\":[[0,2],[0],[1],[2,3],[2],[3],[4]],\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]]}"},"fields":{"title":"A Basis for Some Topology Need Not Generate This Given Topology","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 2], [1], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes)","explanation":"Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes)."},"skills":["A Basis for Some Topology Need Not Generate This Given Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0062","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb2a61ac50c7af90f1b4273c3b9192e7a0d88765f2c0f49312578b40425c3d6d","family_id":"qf-mod38-topbasis-existing-topology-audit","seed":202611073062,"semantic_template":"qf-mod38-topbasis-existing-topology-audit/{\"basis\":[[0,1],[0,2],[0],[1],[2,3],[2],[3],[4]],\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]]}"},"fields":{"title":"A Basis for Some Topology Need Not Generate This Given Topology","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [0, 2], [1], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes)","explanation":"Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes)."},"skills":["A Basis for Some Topology Need Not Generate This Given Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0063","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95b63b6cbeee59b47841dd2fe077a4d6db0b7844b1302681f8cc54b185376d05","family_id":"qf-mod38-topbasis-existing-topology-audit","seed":202611073063,"semantic_template":"qf-mod38-topbasis-existing-topology-audit/{\"basis\":[[0],[1,2],[1],[2,3],[2],[3],[4]],\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]]}"},"fields":{"title":"A Basis for Some Topology Need Not Generate This Given Topology","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [1, 2], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes)","explanation":"Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes)."},"skills":["A Basis for Some Topology Need Not Generate This Given Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0064","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2446ceddcddd91a868c460a6c083f7830bf1870a43d423ffc4fc83233c8fa0fa","family_id":"qf-mod38-topbasis-existing-topology-audit","seed":202611073064,"semantic_template":"qf-mod38-topbasis-existing-topology-audit/{\"basis\":[[0,1],[0],[1,2],[1],[2,3],[2],[3],[4]],\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]]}"},"fields":{"title":"A Basis for Some Topology Need Not Generate This Given Topology","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [1], [1, 2], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes)","explanation":"Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes)."},"skills":["A Basis for Some Topology Need Not Generate This Given Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0065","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd575bf0989edb69bcb935c7e047d506a246f862b7cd4a5bd5d0f12c886c742a","family_id":"qf-mod38-topbasis-existing-topology-audit","seed":202611073065,"semantic_template":"qf-mod38-topbasis-existing-topology-audit/{\"basis\":[[0,2],[0],[1,2],[1],[2,3],[2],[3],[4]],\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]]}"},"fields":{"title":"A Basis for Some Topology Need Not Generate This Given Topology","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 2], [1], [1, 2], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes)","explanation":"Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes)."},"skills":["A Basis for Some Topology Need Not Generate This Given Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0066","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e6445589b889366dea995a7a5f007df929e4b98a74cad071df0762a8aef10f8","family_id":"qf-mod38-topbasis-existing-topology-audit","seed":202611073066,"semantic_template":"qf-mod38-topbasis-existing-topology-audit/{\"basis\":[[0,1],[0,2],[0],[1,2],[1],[2,3],[2],[3],[4]],\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]]}"},"fields":{"title":"A Basis for Some Topology Need Not Generate This Given Topology","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [0, 2], [1], [1, 2], [2], [2, 3], [3], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes)","explanation":"Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes)."},"skills":["A Basis for Some Topology Need Not Generate This Given Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0067","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd8a4b8c89ab456b0d6fa78cf795d4568836fb7ba02daa13e006afa1cbe6e153","family_id":"qf-mod38-topbasis-existing-topology-audit","seed":202611073067,"semantic_template":"qf-mod38-topbasis-existing-topology-audit/{\"basis\":[[0],[1],[2],[3,4],[3],[4]],\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]]}"},"fields":{"title":"A Basis for Some Topology Need Not Generate This Given Topology","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes)","explanation":"Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes)."},"skills":["A Basis for Some Topology Need Not Generate This Given Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0068","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c0a47c7ff56b6d54db884a635ac4a16355551625e1304d3858d3789072ac828","family_id":"qf-mod38-topbasis-existing-topology-audit","seed":202611073068,"semantic_template":"qf-mod38-topbasis-existing-topology-audit/{\"basis\":[[0,1],[0],[1],[2],[3,4],[3],[4]],\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]]}"},"fields":{"title":"A Basis for Some Topology Need Not Generate This Given Topology","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [1], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes)","explanation":"Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes)."},"skills":["A Basis for Some Topology Need Not Generate This Given Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0069","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50d481ff0f86f274ff262c5fb564f92af922a6fa27ee54dacf79299f889563b6","family_id":"qf-mod38-topbasis-existing-topology-audit","seed":202611073069,"semantic_template":"qf-mod38-topbasis-existing-topology-audit/{\"basis\":[[0,2],[0],[1],[2],[3,4],[3],[4]],\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]]}"},"fields":{"title":"A Basis for Some Topology Need Not Generate This Given Topology","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 2], [1], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes)","explanation":"Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes)."},"skills":["A Basis for Some Topology Need Not Generate This Given Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0070","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0fdbf781dae94233f37d707dbf9243a86fe05b26c35b718ef9d7d5d6c812f560","family_id":"qf-mod38-topbasis-existing-topology-audit","seed":202611073070,"semantic_template":"qf-mod38-topbasis-existing-topology-audit/{\"basis\":[[0,1],[0,2],[0],[1],[2],[3,4],[3],[4]],\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]]}"},"fields":{"title":"A Basis for Some Topology Need Not Generate This Given Topology","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [0, 2], [1], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes)","explanation":"Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes)."},"skills":["A Basis for Some Topology Need Not Generate This Given Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0071","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05ec14679e25a3843267b11cc55ccf1ea90c129fbd8f4e7dd0562f7d4906a9ff","family_id":"qf-mod38-topbasis-existing-topology-audit","seed":202611073071,"semantic_template":"qf-mod38-topbasis-existing-topology-audit/{\"basis\":[[0],[1,2],[1],[2],[3,4],[3],[4]],\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]]}"},"fields":{"title":"A Basis for Some Topology Need Not Generate This Given Topology","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [1, 2], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes)","explanation":"Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes)."},"skills":["A Basis for Some Topology Need Not Generate This Given Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0072","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"921402531362790cf607c1b386799bca0293ec4a783f8ffef0dd1d2639d7378d","family_id":"qf-mod38-topbasis-existing-topology-audit","seed":202611073072,"semantic_template":"qf-mod38-topbasis-existing-topology-audit/{\"basis\":[[0,1],[0],[1,2],[1],[2],[3,4],[3],[4]],\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]]}"},"fields":{"title":"A Basis for Some Topology Need Not Generate This Given Topology","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [1], [1, 2], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes)","explanation":"Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes)."},"skills":["A Basis for Some Topology Need Not Generate This Given Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0073","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e122001de20367e3e3291fa008bddd9fd18a16360d9184c2aa01931e2571b05b","family_id":"qf-mod38-topbasis-existing-topology-audit","seed":202611073073,"semantic_template":"qf-mod38-topbasis-existing-topology-audit/{\"basis\":[[0,2],[0],[1,2],[1],[2],[3,4],[3],[4]],\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]]}"},"fields":{"title":"A Basis for Some Topology Need Not Generate This Given Topology","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 2], [1], [1, 2], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes)","explanation":"Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes)."},"skills":["A Basis for Some Topology Need Not Generate This Given Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0074","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2fafa5fc750880d03611fbbb9e080f3067c5058900d46a8128632d14c308ce0b","family_id":"qf-mod38-topbasis-existing-topology-audit","seed":202611073074,"semantic_template":"qf-mod38-topbasis-existing-topology-audit/{\"basis\":[[0,1],[0,2],[0],[1,2],[1],[2],[3,4],[3],[4]],\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]]}"},"fields":{"title":"A Basis for Some Topology Need Not Generate This Given Topology","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [0, 1], [0, 2], [1], [1, 2], [2], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes)","explanation":"Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes)."},"skills":["A Basis for Some Topology Need Not Generate This Given Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0075","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19f41a0bf7d8d4cfbe4813456cacd90a58143dab16ceafe4417b1f303166bba5","family_id":"qf-mod38-topbasis-existing-topology-audit","seed":202611073075,"semantic_template":"qf-mod38-topbasis-existing-topology-audit/{\"basis\":[[0],[1],[2,3],[2],[3,4],[3],[4]],\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2],[0,1,3,4],[0,1,3],[0,1,4],[0,1],[0,2,3,4],[0,2,3],[0,2,4],[0,2],[0,3,4],[0,3],[0,4],[0],[1,2,3,4],[1,2,3],[1,2,4],[1,2],[1,3,4],[1,3],[1,4],[1],[2,3,4],[2,3],[2,4],[2],[3,4],[3],[4]]}"},"fields":{"title":"A Basis for Some Topology Need Not Generate This Given Topology","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0], [1], [2], [2, 3], [3], [3, 4], [4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Supplied complete topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 3], [0, 1, 3, 4], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]. Give (topology_valid, abstract_basis_valid, basis_elements_open, every_open_is_basis_union, basis_for_given_topology).","answer":{"kind":"expression","value":"(Yes, Yes, Yes, Yes, Yes)","explanation":"Check topology axioms, abstract basis axioms, openness of every basis element, and that every supplied open set is a basis union. Their conjunction establishes a basis for this topology; the abstract basis flag alone does not. The exact result, retaining the requested order, is (Yes, Yes, Yes, Yes, Yes)."},"skills":["A Basis for Some Topology Need Not Generate This Given Topology"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0076","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00b86998f572167264deb31aa7101663a6040e11164f67fadebca878dce13829","family_id":"qf-mod38-topbasis-subbasis-boundary","seed":202611073076,"semantic_template":"qf-mod38-topbasis-subbasis-boundary/{\"basis\":[[0,1],[1,2],[2,3],[3,4]],\"n\":5}"},"fields":{"title":"A Cover Alone Does Not Supply the Local Basis Axiom","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [1, 2], [2, 3], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)","explanation":"Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)."},"skills":["A Cover Alone Does Not Supply the Local Basis Axiom"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0077","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e5738aeaa7d4c42153d7584f99754801d206996dd62266bb6a119c3a6c158b0","family_id":"qf-mod38-topbasis-subbasis-boundary","seed":202611073077,"semantic_template":"qf-mod38-topbasis-subbasis-boundary/{\"basis\":[[0,1],[0,2],[1,2],[2,3],[3,4]],\"n\":5}"},"fields":{"title":"A Cover Alone Does Not Supply the Local Basis Axiom","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [0, 2], [1, 2], [2, 3], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)","explanation":"Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)."},"skills":["A Cover Alone Does Not Supply the Local Basis Axiom"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0078","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"040d31ba063830d38dd8bb7e52a05b55b1de0df1466a9193de347e31470107c7","family_id":"qf-mod38-topbasis-subbasis-boundary","seed":202611073078,"semantic_template":"qf-mod38-topbasis-subbasis-boundary/{\"basis\":[[0,1],[1,2],[1,3],[2,3],[3,4]],\"n\":5}"},"fields":{"title":"A Cover Alone Does Not Supply the Local Basis Axiom","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [1, 2], [1, 3], [2, 3], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)","explanation":"Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)."},"skills":["A Cover Alone Does Not Supply the Local Basis Axiom"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0079","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3be29fcb9c34eeb03ff0aa463eab8da2390124f9a62b60edfeb4a3a1097abe87","family_id":"qf-mod38-topbasis-subbasis-boundary","seed":202611073079,"semantic_template":"qf-mod38-topbasis-subbasis-boundary/{\"basis\":[[0,1],[0,2],[1,2],[1,3],[2,3],[3,4]],\"n\":5}"},"fields":{"title":"A Cover Alone Does Not Supply the Local Basis Axiom","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)","explanation":"Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)."},"skills":["A Cover Alone Does Not Supply the Local Basis Axiom"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0080","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29bd2491b837dc2d7ec3305f39e82131b361fc7246785dacfecfb4ca7cdc069a","family_id":"qf-mod38-topbasis-subbasis-boundary","seed":202611073080,"semantic_template":"qf-mod38-topbasis-subbasis-boundary/{\"basis\":[[0,1],[1,2],[2,3],[2,4],[3,4]],\"n\":5}"},"fields":{"title":"A Cover Alone Does Not Supply the Local Basis Axiom","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [1, 2], [2, 3], [2, 4], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)","explanation":"Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)."},"skills":["A Cover Alone Does Not Supply the Local Basis Axiom"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0081","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"abf20fc32f6e11cd2875b1be15c3d9ed1333a4029672d43e33781f929f09fc22","family_id":"qf-mod38-topbasis-subbasis-boundary","seed":202611073081,"semantic_template":"qf-mod38-topbasis-subbasis-boundary/{\"basis\":[[0,1],[0,2],[1,2],[2,3],[2,4],[3,4]],\"n\":5}"},"fields":{"title":"A Cover Alone Does Not Supply the Local Basis Axiom","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [0, 2], [1, 2], [2, 3], [2, 4], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)","explanation":"Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)."},"skills":["A Cover Alone Does Not Supply the Local Basis Axiom"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0082","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2e240c2db691aefb465ac909fe110e725d146b83f4515c4c79322be7efea8f4","family_id":"qf-mod38-topbasis-subbasis-boundary","seed":202611073082,"semantic_template":"qf-mod38-topbasis-subbasis-boundary/{\"basis\":[[0,1],[1,2],[1,3],[2,3],[2,4],[3,4]],\"n\":5}"},"fields":{"title":"A Cover Alone Does Not Supply the Local Basis Axiom","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)","explanation":"Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)."},"skills":["A Cover Alone Does Not Supply the Local Basis Axiom"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0083","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8647ed244270828e2d1f55f81497dfb91ebfb882fac6540ed2fec18b679f4e52","family_id":"qf-mod38-topbasis-subbasis-boundary","seed":202611073083,"semantic_template":"qf-mod38-topbasis-subbasis-boundary/{\"basis\":[[0,1],[0,2],[1,2],[1,3],[2,3],[2,4],[3,4]],\"n\":5}"},"fields":{"title":"A Cover Alone Does Not Supply the Local Basis Axiom","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [0, 2], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)","explanation":"Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)."},"skills":["A Cover Alone Does Not Supply the Local Basis Axiom"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0084","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"057533640e99ecf6c6355301e266fcbdd3e48416a7126cdd5e3c62280870d219","family_id":"qf-mod38-topbasis-subbasis-boundary","seed":202611073084,"semantic_template":"qf-mod38-topbasis-subbasis-boundary/{\"basis\":[[0,1],[0,3],[1,2],[2,3],[3,4]],\"n\":5}"},"fields":{"title":"A Cover Alone Does Not Supply the Local Basis Axiom","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [0, 3], [1, 2], [2, 3], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)","explanation":"Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)."},"skills":["A Cover Alone Does Not Supply the Local Basis Axiom"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0085","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ace52ddc619cbc2bd70b37b729141aa348371209bae9a2cc812083cf449e1014","family_id":"qf-mod38-topbasis-subbasis-boundary","seed":202611073085,"semantic_template":"qf-mod38-topbasis-subbasis-boundary/{\"basis\":[[0,1],[0,2],[0,3],[1,2],[2,3],[3,4]],\"n\":5}"},"fields":{"title":"A Cover Alone Does Not Supply the Local Basis Axiom","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)","explanation":"Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)."},"skills":["A Cover Alone Does Not Supply the Local Basis Axiom"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0086","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1dba1af1eccfaa7599cc7a1452fb1ec9c0167116d87eac68a00e0901015bf18c","family_id":"qf-mod38-topbasis-subbasis-boundary","seed":202611073086,"semantic_template":"qf-mod38-topbasis-subbasis-boundary/{\"basis\":[[0,1],[0,3],[1,2],[1,3],[2,3],[3,4]],\"n\":5}"},"fields":{"title":"A Cover Alone Does Not Supply the Local Basis Axiom","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [0, 3], [1, 2], [1, 3], [2, 3], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)","explanation":"Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)."},"skills":["A Cover Alone Does Not Supply the Local Basis Axiom"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0087","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f23a4ccad79914aeab0a345a8dea12131e710645c9da52281dd687b70d990b64","family_id":"qf-mod38-topbasis-subbasis-boundary","seed":202611073087,"semantic_template":"qf-mod38-topbasis-subbasis-boundary/{\"basis\":[[0,1],[0,2],[0,3],[1,2],[1,3],[2,3],[3,4]],\"n\":5}"},"fields":{"title":"A Cover Alone Does Not Supply the Local Basis Axiom","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)","explanation":"Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)."},"skills":["A Cover Alone Does Not Supply the Local Basis Axiom"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0088","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"242e71dc3b172b3693d2036c1b3a0030364166129c5ff9cb55a782e73e514d82","family_id":"qf-mod38-topbasis-subbasis-boundary","seed":202611073088,"semantic_template":"qf-mod38-topbasis-subbasis-boundary/{\"basis\":[[0,1],[0,3],[1,2],[2,3],[2,4],[3,4]],\"n\":5}"},"fields":{"title":"A Cover Alone Does Not Supply the Local Basis Axiom","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [0, 3], [1, 2], [2, 3], [2, 4], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)","explanation":"Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)."},"skills":["A Cover Alone Does Not Supply the Local Basis Axiom"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0089","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa9529a9d411a59e384ae2da0bf2e90f09cf42b90be86957c3ee171de60b3ea1","family_id":"qf-mod38-topbasis-subbasis-boundary","seed":202611073089,"semantic_template":"qf-mod38-topbasis-subbasis-boundary/{\"basis\":[[0,1],[0,2],[0,3],[1,2],[2,3],[2,4],[3,4]],\"n\":5}"},"fields":{"title":"A Cover Alone Does Not Supply the Local Basis Axiom","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [0, 2], [0, 3], [1, 2], [2, 3], [2, 4], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)","explanation":"Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)."},"skills":["A Cover Alone Does Not Supply the Local Basis Axiom"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0090","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8be2df024c0d254ff71219f37a84003a59680d88796fd98d9b17a1549e7dc52c","family_id":"qf-mod38-topbasis-subbasis-boundary","seed":202611073090,"semantic_template":"qf-mod38-topbasis-subbasis-boundary/{\"basis\":[[0,1],[0,3],[1,2],[1,3],[2,3],[2,4],[3,4]],\"n\":5}"},"fields":{"title":"A Cover Alone Does Not Supply the Local Basis Axiom","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [0, 3], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)","explanation":"Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)."},"skills":["A Cover Alone Does Not Supply the Local Basis Axiom"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0091","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f753ac7a4bd5e183eb96b66504279123529cbfddf6c1dddecf7bd5067d156f2e","family_id":"qf-mod38-topbasis-subbasis-boundary","seed":202611073091,"semantic_template":"qf-mod38-topbasis-subbasis-boundary/{\"basis\":[[0,1],[0,2],[0,3],[1,2],[1,3],[2,3],[2,4],[3,4]],\"n\":5}"},"fields":{"title":"A Cover Alone Does Not Supply the Local Basis Axiom","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3], [2, 4], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)","explanation":"Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)."},"skills":["A Cover Alone Does Not Supply the Local Basis Axiom"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0092","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6f56c4aeb6cbbc0e954b4c31031c8e5e8b15a4fd6a0b9614bbeee658829aa1f","family_id":"qf-mod38-topbasis-subbasis-boundary","seed":202611073092,"semantic_template":"qf-mod38-topbasis-subbasis-boundary/{\"basis\":[[0,1],[1,2],[1,4],[2,3],[3,4]],\"n\":5}"},"fields":{"title":"A Cover Alone Does Not Supply the Local Basis Axiom","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [1, 2], [1, 4], [2, 3], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)","explanation":"Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)."},"skills":["A Cover Alone Does Not Supply the Local Basis Axiom"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0093","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"563f74a6862be6e4abe51ac7ba26a2f7ea8224e0d9b40de72e0e66b7738d941d","family_id":"qf-mod38-topbasis-subbasis-boundary","seed":202611073093,"semantic_template":"qf-mod38-topbasis-subbasis-boundary/{\"basis\":[[0,1],[0,2],[1,2],[1,4],[2,3],[3,4]],\"n\":5}"},"fields":{"title":"A Cover Alone Does Not Supply the Local Basis Axiom","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [0, 2], [1, 2], [1, 4], [2, 3], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)","explanation":"Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)."},"skills":["A Cover Alone Does Not Supply the Local Basis Axiom"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0094","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b6baf67b86c9a336f62965177b339187ad6566a17b8896aa197849bbfeb3f90d","family_id":"qf-mod38-topbasis-subbasis-boundary","seed":202611073094,"semantic_template":"qf-mod38-topbasis-subbasis-boundary/{\"basis\":[[0,1],[1,2],[1,3],[1,4],[2,3],[3,4]],\"n\":5}"},"fields":{"title":"A Cover Alone Does Not Supply the Local Basis Axiom","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)","explanation":"Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)."},"skills":["A Cover Alone Does Not Supply the Local Basis Axiom"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0095","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ca3b85e81bdb2055bcbc2389ac8cb1e5e76a0f9874cae4d6ca838adcf641b43","family_id":"qf-mod38-topbasis-subbasis-boundary","seed":202611073095,"semantic_template":"qf-mod38-topbasis-subbasis-boundary/{\"basis\":[[0,1],[0,2],[1,2],[1,3],[1,4],[2,3],[3,4]],\"n\":5}"},"fields":{"title":"A Cover Alone Does Not Supply the Local Basis Axiom","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [0, 2], [1, 2], [1, 3], [1, 4], [2, 3], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)","explanation":"Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)."},"skills":["A Cover Alone Does Not Supply the Local Basis Axiom"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0096","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3eb9a06b2d6fc365dc8a06c4ac4450a5622180d3c17b9247856cad04eaa30b9","family_id":"qf-mod38-topbasis-subbasis-boundary","seed":202611073096,"semantic_template":"qf-mod38-topbasis-subbasis-boundary/{\"basis\":[[0,1],[1,2],[1,4],[2,3],[2,4],[3,4]],\"n\":5}"},"fields":{"title":"A Cover Alone Does Not Supply the Local Basis Axiom","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [1, 2], [1, 4], [2, 3], [2, 4], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)","explanation":"Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)."},"skills":["A Cover Alone Does Not Supply the Local Basis Axiom"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0097","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e99fa6847b6a44982e974cc50f6d6152fe364cd08e8c221404d4f3389a72bde0","family_id":"qf-mod38-topbasis-subbasis-boundary","seed":202611073097,"semantic_template":"qf-mod38-topbasis-subbasis-boundary/{\"basis\":[[0,1],[0,2],[1,2],[1,4],[2,3],[2,4],[3,4]],\"n\":5}"},"fields":{"title":"A Cover Alone Does Not Supply the Local Basis Axiom","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [0, 2], [1, 2], [1, 4], [2, 3], [2, 4], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)","explanation":"Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)."},"skills":["A Cover Alone Does Not Supply the Local Basis Axiom"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0098","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65b7e26110cddf65659d6eed796197dec92f712017a96cab339342257ae0262a","family_id":"qf-mod38-topbasis-subbasis-boundary","seed":202611073098,"semantic_template":"qf-mod38-topbasis-subbasis-boundary/{\"basis\":[[0,1],[1,2],[1,3],[1,4],[2,3],[2,4],[3,4]],\"n\":5}"},"fields":{"title":"A Cover Alone Does Not Supply the Local Basis Axiom","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)","explanation":"Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)."},"skills":["A Cover Alone Does Not Supply the Local Basis Axiom"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0099","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83c877888478b27766114aa4dc9f52fa227257c8dc3f5b43c5d8b9e28a88f95b","family_id":"qf-mod38-topbasis-subbasis-boundary","seed":202611073099,"semantic_template":"qf-mod38-topbasis-subbasis-boundary/{\"basis\":[[0,1],[0,2],[1,2],[1,3],[1,4],[2,3],[2,4],[3,4]],\"n\":5}"},"fields":{"title":"A Cover Alone Does Not Supply the Local Basis Axiom","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [0, 2], [1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)","explanation":"Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)."},"skills":["A Cover Alone Does Not Supply the Local Basis Axiom"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261107-r03-topological-basis-0100","batch_id":"batch-20261014-advance-20261107-r03-topological-basis-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"228164d042f735750f14d7f36306e61d66de9da46642780714016c0f8e64641e","family_id":"qf-mod38-topbasis-subbasis-boundary","seed":202611073100,"semantic_template":"qf-mod38-topbasis-subbasis-boundary/{\"basis\":[[0,1],[0,3],[1,2],[1,4],[2,3],[3,4]],\"n\":5}"},"fields":{"title":"A Cover Alone Does Not Supply the Local Basis Axiom","stem":"Use finite carrier Fin(5)={0,…,4} and the complete canonical candidate family basis=[[0, 1], [0, 3], [1, 2], [1, 4], [2, 3], [3, 4]]. Its union must cover the carrier; for every point in each pairwise intersection a candidate element containing that point must lie inside the intersection. Complete finite data, not sampled intersections, supply this audit. Give (covers_carrier, local_refinement, is_basis, cover_alone_suffices_general_claim, boundary).","answer":{"kind":"expression","value":"(Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)","explanation":"Retain coverage, refinement and actual basis validity, then No for the general claim that cover alone suffices. The final No does not negate an independently valid basis or test whether this family is a subbasis. Generating from a subbasis generally needs finite intersections before unions. The exact result, retaining the requested order, is (Yes, No, No, No, cover_without_local_intersection_refinement_is_not_a_basis_and_may_only_be_a_subbasis)."},"skills":["A Cover Alone Does Not Supply the Local Basis Axiom"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}