{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0001","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0dad47b29c729de7854e2256c564f65d69e4576ebfb3dd8410bfd0492801f4c6","family_id":"qf-mod33-boundary-exact","seed":202611024001,"semantic_template":"qf-mod33-boundary-exact/{\"n\":5,\"subset\":[1,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[0]]}"},"fields":{"title":"Boundary Is Closure Minus Interior in the Same Topology","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 2]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.","answer":{"kind":"expression","value":"([1, 2], [1, 2, 3, 4], [], [1, 2, 3, 4])","explanation":"Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([1, 2], [1, 2, 3, 4], [], [1, 2, 3, 4])."},"skills":["Boundary Is Closure Minus Interior in the Same 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-20261102-r04-topological-boundary-0002","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"80154e1f5f7ac0a2c0557225e9f8bc3dc580fda17d10f154b97fa7d682c8567f","family_id":"qf-mod33-boundary-exact","seed":202611024002,"semantic_template":"qf-mod33-boundary-exact/{\"n\":5,\"subset\":[1,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,1],[0]]}"},"fields":{"title":"Boundary Is Closure Minus Interior in the Same Topology","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 2]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.","answer":{"kind":"expression","value":"([1, 2], [1, 2, 3, 4], [], [1, 2, 3, 4])","explanation":"Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([1, 2], [1, 2, 3, 4], [], [1, 2, 3, 4])."},"skills":["Boundary Is Closure Minus Interior in the Same 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-20261102-r04-topological-boundary-0003","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1248145a97b3c8b7992484704b0ba5ae797e6a7ba21adca63c312c1b09f246db","family_id":"qf-mod33-boundary-exact","seed":202611024003,"semantic_template":"qf-mod33-boundary-exact/{\"n\":5,\"subset\":[1,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,1],[0]]}"},"fields":{"title":"Boundary Is Closure Minus Interior in the Same Topology","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 3]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.","answer":{"kind":"expression","value":"([1, 3], [1, 2, 3, 4], [], [1, 2, 3, 4])","explanation":"Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([1, 3], [1, 2, 3, 4], [], [1, 2, 3, 4])."},"skills":["Boundary Is Closure Minus Interior in the Same 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-20261102-r04-topological-boundary-0004","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3778942d412d7a041a03e8965cdd2e057a4354c4c232fb3a1d576b31b6e237b9","family_id":"qf-mod33-boundary-exact","seed":202611024004,"semantic_template":"qf-mod33-boundary-exact/{\"n\":5,\"subset\":[1,3],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,1],[0]]}"},"fields":{"title":"Boundary Is Closure Minus Interior in the Same Topology","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 3]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.","answer":{"kind":"expression","value":"([1, 3], [1, 2, 3, 4], [], [1, 2, 3, 4])","explanation":"Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([1, 3], [1, 2, 3, 4], [], [1, 2, 3, 4])."},"skills":["Boundary Is Closure Minus Interior in the Same 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-20261102-r04-topological-boundary-0005","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef62fe61f75c4afe4b0020d20a2eefb198a5292699b725f614b97a2277e0c3f3","family_id":"qf-mod33-boundary-exact","seed":202611024005,"semantic_template":"qf-mod33-boundary-exact/{\"n\":5,\"subset\":[1,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,1],[0]]}"},"fields":{"title":"Boundary Is Closure Minus Interior in the Same Topology","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 4]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.","answer":{"kind":"expression","value":"([1, 4], [1, 2, 3, 4], [], [1, 2, 3, 4])","explanation":"Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([1, 4], [1, 2, 3, 4], [], [1, 2, 3, 4])."},"skills":["Boundary Is Closure Minus Interior in the Same 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-20261102-r04-topological-boundary-0006","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7a6fdb81a937688b23e029d65d953f12041f7277cef4b80171df45d2ed2d2be","family_id":"qf-mod33-boundary-exact","seed":202611024006,"semantic_template":"qf-mod33-boundary-exact/{\"n\":5,\"subset\":[1,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,1],[0]]}"},"fields":{"title":"Boundary Is Closure Minus Interior in the Same Topology","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 4]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.","answer":{"kind":"expression","value":"([1, 4], [1, 2, 3, 4], [], [1, 2, 3, 4])","explanation":"Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([1, 4], [1, 2, 3, 4], [], [1, 2, 3, 4])."},"skills":["Boundary Is Closure Minus Interior in the Same 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-20261102-r04-topological-boundary-0007","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12648898a0011627e15dd1878482420a39e9d9e67c25dccab4a4e5b6dc9be48d","family_id":"qf-mod33-boundary-exact","seed":202611024007,"semantic_template":"qf-mod33-boundary-exact/{\"n\":5,\"subset\":[1,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,2],[0]]}"},"fields":{"title":"Boundary Is Closure Minus Interior in the Same Topology","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 2]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.","answer":{"kind":"expression","value":"([1, 2], [1, 2, 3, 4], [], [1, 2, 3, 4])","explanation":"Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([1, 2], [1, 2, 3, 4], [], [1, 2, 3, 4])."},"skills":["Boundary Is Closure Minus Interior in the Same 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-20261102-r04-topological-boundary-0008","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea8b799ffbf9cb7dacb47f0b89a209100a798195529e4bf74141cea898a027b4","family_id":"qf-mod33-boundary-exact","seed":202611024008,"semantic_template":"qf-mod33-boundary-exact/{\"n\":5,\"subset\":[1,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,2],[0]]}"},"fields":{"title":"Boundary Is Closure Minus Interior in the Same Topology","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 2]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.","answer":{"kind":"expression","value":"([1, 2], [1, 2, 3, 4], [], [1, 2, 3, 4])","explanation":"Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([1, 2], [1, 2, 3, 4], [], [1, 2, 3, 4])."},"skills":["Boundary Is Closure Minus Interior in the Same 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-20261102-r04-topological-boundary-0009","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53d9cd6d1cd140fee37b136811f99b601e4d9e663867850f94ba31a6251c0708","family_id":"qf-mod33-boundary-exact","seed":202611024009,"semantic_template":"qf-mod33-boundary-exact/{\"n\":5,\"subset\":[2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,2],[0]]}"},"fields":{"title":"Boundary Is Closure Minus Interior in the Same Topology","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 3]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.","answer":{"kind":"expression","value":"([2, 3], [1, 2, 3, 4], [], [1, 2, 3, 4])","explanation":"Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([2, 3], [1, 2, 3, 4], [], [1, 2, 3, 4])."},"skills":["Boundary Is Closure Minus Interior in the Same 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-20261102-r04-topological-boundary-0010","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7701e59f3dfcf6b4397a98488c5bd4e96d8ea7994c8405cd08e41c39a3b5b790","family_id":"qf-mod33-boundary-exact","seed":202611024010,"semantic_template":"qf-mod33-boundary-exact/{\"n\":5,\"subset\":[2,3],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,2],[0]]}"},"fields":{"title":"Boundary Is Closure Minus Interior in the Same Topology","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 3]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.","answer":{"kind":"expression","value":"([2, 3], [1, 2, 3, 4], [], [1, 2, 3, 4])","explanation":"Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([2, 3], [1, 2, 3, 4], [], [1, 2, 3, 4])."},"skills":["Boundary Is Closure Minus Interior in the Same 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-20261102-r04-topological-boundary-0011","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d3755295f10b6b0b289e9c63fa65cfc71b8a6b38a037deca86849af2f4b1fa3","family_id":"qf-mod33-boundary-exact","seed":202611024011,"semantic_template":"qf-mod33-boundary-exact/{\"n\":5,\"subset\":[2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,2],[0]]}"},"fields":{"title":"Boundary Is Closure Minus Interior in the Same Topology","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 4]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.","answer":{"kind":"expression","value":"([2, 4], [1, 2, 3, 4], [], [1, 2, 3, 4])","explanation":"Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([2, 4], [1, 2, 3, 4], [], [1, 2, 3, 4])."},"skills":["Boundary Is Closure Minus Interior in the Same 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-20261102-r04-topological-boundary-0012","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c3d3e102bd3980c1da6ee1ec2bc6bd4561c222f1cb1226b79477f93cb363ef5","family_id":"qf-mod33-boundary-exact","seed":202611024012,"semantic_template":"qf-mod33-boundary-exact/{\"n\":5,\"subset\":[2,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,2],[0]]}"},"fields":{"title":"Boundary Is Closure Minus Interior in the Same Topology","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 4]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.","answer":{"kind":"expression","value":"([2, 4], [1, 2, 3, 4], [], [1, 2, 3, 4])","explanation":"Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([2, 4], [1, 2, 3, 4], [], [1, 2, 3, 4])."},"skills":["Boundary Is Closure Minus Interior in the Same 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-20261102-r04-topological-boundary-0013","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d0f8b45aa200dd9d21e5c18cce9f762c1893804bb6ce2ce049cc1ec41592df2","family_id":"qf-mod33-boundary-exact","seed":202611024013,"semantic_template":"qf-mod33-boundary-exact/{\"n\":5,\"subset\":[1,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,3],[0]]}"},"fields":{"title":"Boundary Is Closure Minus Interior in the Same Topology","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 3]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.","answer":{"kind":"expression","value":"([1, 3], [1, 2, 3, 4], [], [1, 2, 3, 4])","explanation":"Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([1, 3], [1, 2, 3, 4], [], [1, 2, 3, 4])."},"skills":["Boundary Is Closure Minus Interior in the Same 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-20261102-r04-topological-boundary-0014","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a12ae689aa73a7400c64ed78cb99294aba90384ce7cdc27a272c139ce64dc29e","family_id":"qf-mod33-boundary-exact","seed":202611024014,"semantic_template":"qf-mod33-boundary-exact/{\"n\":5,\"subset\":[1,3],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,3],[0]]}"},"fields":{"title":"Boundary Is Closure Minus Interior in the Same Topology","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 3]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.","answer":{"kind":"expression","value":"([1, 3], [1, 2, 3, 4], [], [1, 2, 3, 4])","explanation":"Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([1, 3], [1, 2, 3, 4], [], [1, 2, 3, 4])."},"skills":["Boundary Is Closure Minus Interior in the Same 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-20261102-r04-topological-boundary-0015","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ba457754fd5938f3eebe822a0a96262390be18a36c231197e92b50c78d420a3","family_id":"qf-mod33-boundary-exact","seed":202611024015,"semantic_template":"qf-mod33-boundary-exact/{\"n\":5,\"subset\":[2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,3],[0]]}"},"fields":{"title":"Boundary Is Closure Minus Interior in the Same Topology","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 3]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.","answer":{"kind":"expression","value":"([2, 3], [1, 2, 3, 4], [], [1, 2, 3, 4])","explanation":"Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([2, 3], [1, 2, 3, 4], [], [1, 2, 3, 4])."},"skills":["Boundary Is Closure Minus Interior in the Same 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-20261102-r04-topological-boundary-0016","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c869a9e24729b4f4f6e7c931be21a3c4a31707e1d749ee91ab7de6dcdc6c8196","family_id":"qf-mod33-boundary-exact","seed":202611024016,"semantic_template":"qf-mod33-boundary-exact/{\"n\":5,\"subset\":[2,3],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,3],[0]]}"},"fields":{"title":"Boundary Is Closure Minus Interior in the Same Topology","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 3]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.","answer":{"kind":"expression","value":"([2, 3], [1, 2, 3, 4], [], [1, 2, 3, 4])","explanation":"Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([2, 3], [1, 2, 3, 4], [], [1, 2, 3, 4])."},"skills":["Boundary Is Closure Minus Interior in the Same 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-20261102-r04-topological-boundary-0017","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3b0bdbebb56dec1d6d1aceae171c9e1d6c4dbfc37365420404b4fb4efd0e232","family_id":"qf-mod33-boundary-exact","seed":202611024017,"semantic_template":"qf-mod33-boundary-exact/{\"n\":5,\"subset\":[3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,3],[0]]}"},"fields":{"title":"Boundary Is Closure Minus Interior in the Same Topology","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[3, 4]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.","answer":{"kind":"expression","value":"([3, 4], [1, 2, 3, 4], [], [1, 2, 3, 4])","explanation":"Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([3, 4], [1, 2, 3, 4], [], [1, 2, 3, 4])."},"skills":["Boundary Is Closure Minus Interior in the Same 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-20261102-r04-topological-boundary-0018","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6cf3849c438fdbd73bd0ed23459b3c6f436026a89bf697a54558098f25a21106","family_id":"qf-mod33-boundary-exact","seed":202611024018,"semantic_template":"qf-mod33-boundary-exact/{\"n\":5,\"subset\":[3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,3],[0]]}"},"fields":{"title":"Boundary Is Closure Minus Interior in the Same Topology","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[3, 4]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.","answer":{"kind":"expression","value":"([3, 4], [1, 2, 3, 4], [], [1, 2, 3, 4])","explanation":"Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([3, 4], [1, 2, 3, 4], [], [1, 2, 3, 4])."},"skills":["Boundary Is Closure Minus Interior in the Same 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-20261102-r04-topological-boundary-0019","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"808205ce7d31ae15897d62850792cfc10920ba2f35a020d98d9f7dd4c827d91a","family_id":"qf-mod33-boundary-exact","seed":202611024019,"semantic_template":"qf-mod33-boundary-exact/{\"n\":5,\"subset\":[1,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,4],[0]]}"},"fields":{"title":"Boundary Is Closure Minus Interior in the Same Topology","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 4]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.","answer":{"kind":"expression","value":"([1, 4], [1, 2, 3, 4], [], [1, 2, 3, 4])","explanation":"Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([1, 4], [1, 2, 3, 4], [], [1, 2, 3, 4])."},"skills":["Boundary Is Closure Minus Interior in the Same 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-20261102-r04-topological-boundary-0020","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"518067c6813ce788a37d50d8e8eddad8625fbd8530a8a1c439bd99ae48a12bac","family_id":"qf-mod33-boundary-exact","seed":202611024020,"semantic_template":"qf-mod33-boundary-exact/{\"n\":5,\"subset\":[1,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,4],[0]]}"},"fields":{"title":"Boundary Is Closure Minus Interior in the Same Topology","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 4]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.","answer":{"kind":"expression","value":"([1, 4], [1, 2, 3, 4], [], [1, 2, 3, 4])","explanation":"Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([1, 4], [1, 2, 3, 4], [], [1, 2, 3, 4])."},"skills":["Boundary Is Closure Minus Interior in the Same 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-20261102-r04-topological-boundary-0021","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2f4a6870a9e1cfe1407c02f9f9eeeca951196b4251bec0ca1e7485fa4992f32","family_id":"qf-mod33-boundary-exact","seed":202611024021,"semantic_template":"qf-mod33-boundary-exact/{\"n\":5,\"subset\":[2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,4],[0]]}"},"fields":{"title":"Boundary Is Closure Minus Interior in the Same Topology","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 4]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.","answer":{"kind":"expression","value":"([2, 4], [1, 2, 3, 4], [], [1, 2, 3, 4])","explanation":"Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([2, 4], [1, 2, 3, 4], [], [1, 2, 3, 4])."},"skills":["Boundary Is Closure Minus Interior in the Same 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-20261102-r04-topological-boundary-0022","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0699cd0d6582ff4cbd08c73750d142bccb3a3ee5a0db51e4b0576286a6934858","family_id":"qf-mod33-boundary-exact","seed":202611024022,"semantic_template":"qf-mod33-boundary-exact/{\"n\":5,\"subset\":[2,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,4],[0]]}"},"fields":{"title":"Boundary Is Closure Minus Interior in the Same Topology","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 4]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.","answer":{"kind":"expression","value":"([2, 4], [1, 2, 3, 4], [], [1, 2, 3, 4])","explanation":"Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([2, 4], [1, 2, 3, 4], [], [1, 2, 3, 4])."},"skills":["Boundary Is Closure Minus Interior in the Same 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-20261102-r04-topological-boundary-0023","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7d70976f7e5ea1f744fce79670674169cc8913f95bce2e334eae0bde803a202","family_id":"qf-mod33-boundary-exact","seed":202611024023,"semantic_template":"qf-mod33-boundary-exact/{\"n\":5,\"subset\":[3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,4],[0]]}"},"fields":{"title":"Boundary Is Closure Minus Interior in the Same Topology","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[3, 4]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.","answer":{"kind":"expression","value":"([3, 4], [1, 2, 3, 4], [], [1, 2, 3, 4])","explanation":"Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([3, 4], [1, 2, 3, 4], [], [1, 2, 3, 4])."},"skills":["Boundary Is Closure Minus Interior in the Same 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-20261102-r04-topological-boundary-0024","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d595dbb065bfeb052ac09087bfaf5f0eda5d1b52304ba1e29f88692aba6321ee","family_id":"qf-mod33-boundary-exact","seed":202611024024,"semantic_template":"qf-mod33-boundary-exact/{\"n\":5,\"subset\":[3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,4],[0]]}"},"fields":{"title":"Boundary Is Closure Minus Interior in the Same Topology","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[3, 4]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.","answer":{"kind":"expression","value":"([3, 4], [1, 2, 3, 4], [], [1, 2, 3, 4])","explanation":"Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([3, 4], [1, 2, 3, 4], [], [1, 2, 3, 4])."},"skills":["Boundary Is Closure Minus Interior in the Same 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-20261102-r04-topological-boundary-0025","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2809c49e4cb433bd3b3f3a45f3bbc853c1fdefa96bb62e9caabc2df7f59470e","family_id":"qf-mod33-boundary-exact","seed":202611024025,"semantic_template":"qf-mod33-boundary-exact/{\"n\":5,\"subset\":[0,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[1]]}"},"fields":{"title":"Boundary Is Closure Minus Interior in the Same Topology","stem":"Use the complete finite topology=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[0, 2]. Give (subset,closure,interior,boundary), with resulting subsets in increasing carrier order.","answer":{"kind":"expression","value":"([0, 2], [0, 2, 3, 4], [], [0, 2, 3, 4])","explanation":"Intersect all closed supersets to get closure and unite all open subsets to get interior, then subtract the interior from the closure. A picture cannot replace these topology-relative constructions. The exact result, retaining the requested order, is ([0, 2], [0, 2, 3, 4], [], [0, 2, 3, 4])."},"skills":["Boundary Is Closure Minus Interior in the Same 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-20261102-r04-topological-boundary-0026","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7cd21a763f07a22ba4c8c812238d298d303791d4ba5cedbeded2e2da610d07b","family_id":"qf-mod33-boundary-membership","seed":202611024026,"semantic_template":"qf-mod33-boundary-membership/{\"n\":5,\"point\":0,\"subset\":[1,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[0]]}"},"fields":{"title":"Boundary Membership Is Not Just Membership in the Original Set","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 2], point=0. Give (point,boundary,in_boundary?,topology_relative_boundary).","answer":{"kind":"expression","value":"(0, [1, 2, 3, 4], No, topology_relative_boundary)","explanation":"Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (0, [1, 2, 3, 4], No, topology_relative_boundary)."},"skills":["Boundary Membership Is Not Just Membership in the Original Set"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0027","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1fb98c799a153d535cb960ca0d71d16657a080a2a1944e582340f6277ee19c77","family_id":"qf-mod33-boundary-membership","seed":202611024027,"semantic_template":"qf-mod33-boundary-membership/{\"n\":5,\"point\":1,\"subset\":[1,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,1],[0]]}"},"fields":{"title":"Boundary Membership Is Not Just Membership in the Original Set","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 2], point=1. Give (point,boundary,in_boundary?,topology_relative_boundary).","answer":{"kind":"expression","value":"(1, [1, 2, 3, 4], Yes, topology_relative_boundary)","explanation":"Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (1, [1, 2, 3, 4], Yes, topology_relative_boundary)."},"skills":["Boundary Membership Is Not Just Membership in the Original Set"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0028","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"286755cce578a5ca08400e1814baaf246d64b39e4db445a26e3224d8a78ad8ed","family_id":"qf-mod33-boundary-membership","seed":202611024028,"semantic_template":"qf-mod33-boundary-membership/{\"n\":5,\"point\":3,\"subset\":[1,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,1],[0]]}"},"fields":{"title":"Boundary Membership Is Not Just Membership in the Original Set","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 3], point=3. Give (point,boundary,in_boundary?,topology_relative_boundary).","answer":{"kind":"expression","value":"(3, [1, 2, 3, 4], Yes, topology_relative_boundary)","explanation":"Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (3, [1, 2, 3, 4], Yes, topology_relative_boundary)."},"skills":["Boundary Membership Is Not Just Membership in the Original Set"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0029","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68d1fd24b332a70154f90639d73bd654385238d351d889a0d53105bef3a4577b","family_id":"qf-mod33-boundary-membership","seed":202611024029,"semantic_template":"qf-mod33-boundary-membership/{\"n\":5,\"point\":4,\"subset\":[1,3],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,1],[0]]}"},"fields":{"title":"Boundary Membership Is Not Just Membership in the Original Set","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 3], point=4. Give (point,boundary,in_boundary?,topology_relative_boundary).","answer":{"kind":"expression","value":"(4, [1, 2, 3, 4], Yes, topology_relative_boundary)","explanation":"Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (4, [1, 2, 3, 4], Yes, topology_relative_boundary)."},"skills":["Boundary Membership Is Not Just Membership in the Original Set"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0030","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2eaebb5b8146c35de9acd98012665227cc0c14d08c1b058a2962ce0cca21364e","family_id":"qf-mod33-boundary-membership","seed":202611024030,"semantic_template":"qf-mod33-boundary-membership/{\"n\":5,\"point\":3,\"subset\":[1,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,1],[0]]}"},"fields":{"title":"Boundary Membership Is Not Just Membership in the Original Set","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 4], point=3. Give (point,boundary,in_boundary?,topology_relative_boundary).","answer":{"kind":"expression","value":"(3, [1, 2, 3, 4], Yes, topology_relative_boundary)","explanation":"Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (3, [1, 2, 3, 4], Yes, topology_relative_boundary)."},"skills":["Boundary Membership Is Not Just Membership in the Original Set"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0031","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"688274bed7ef2d618e1cd77bec4834cda4c6a051e4aeba85ae75240b04fc05a9","family_id":"qf-mod33-boundary-membership","seed":202611024031,"semantic_template":"qf-mod33-boundary-membership/{\"n\":5,\"point\":0,\"subset\":[1,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,1],[0]]}"},"fields":{"title":"Boundary Membership Is Not Just Membership in the Original Set","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 4], point=0. Give (point,boundary,in_boundary?,topology_relative_boundary).","answer":{"kind":"expression","value":"(0, [1, 2, 3, 4], No, topology_relative_boundary)","explanation":"Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (0, [1, 2, 3, 4], No, topology_relative_boundary)."},"skills":["Boundary Membership Is Not Just Membership in the Original Set"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0032","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7a2963f60dfc1ef3c89c811f4814aea0620a79489f13557dd7bb2bd537d618e","family_id":"qf-mod33-boundary-membership","seed":202611024032,"semantic_template":"qf-mod33-boundary-membership/{\"n\":5,\"point\":2,\"subset\":[1,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,2],[0]]}"},"fields":{"title":"Boundary Membership Is Not Just Membership in the Original Set","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 2], point=2. Give (point,boundary,in_boundary?,topology_relative_boundary).","answer":{"kind":"expression","value":"(2, [1, 2, 3, 4], Yes, topology_relative_boundary)","explanation":"Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (2, [1, 2, 3, 4], Yes, topology_relative_boundary)."},"skills":["Boundary Membership Is Not Just Membership in the Original Set"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0033","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4cb3110741ef2459175a961746fa4902857a11384a0d3fe1d50d65a9c0ea48bf","family_id":"qf-mod33-boundary-membership","seed":202611024033,"semantic_template":"qf-mod33-boundary-membership/{\"n\":5,\"point\":1,\"subset\":[1,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,2],[0]]}"},"fields":{"title":"Boundary Membership Is Not Just Membership in the Original Set","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 2], point=1. Give (point,boundary,in_boundary?,topology_relative_boundary).","answer":{"kind":"expression","value":"(1, [1, 2, 3, 4], Yes, topology_relative_boundary)","explanation":"Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (1, [1, 2, 3, 4], Yes, topology_relative_boundary)."},"skills":["Boundary Membership Is Not Just Membership in the Original Set"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0034","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d98ae5a3287184057c2d4cd810bd8a18d12f3c448ab0026bbf87267ac524a6e6","family_id":"qf-mod33-boundary-membership","seed":202611024034,"semantic_template":"qf-mod33-boundary-membership/{\"n\":5,\"point\":1,\"subset\":[2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,2],[0]]}"},"fields":{"title":"Boundary Membership Is Not Just Membership in the Original Set","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 3], point=1. Give (point,boundary,in_boundary?,topology_relative_boundary).","answer":{"kind":"expression","value":"(1, [1, 2, 3, 4], Yes, topology_relative_boundary)","explanation":"Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (1, [1, 2, 3, 4], Yes, topology_relative_boundary)."},"skills":["Boundary Membership Is Not Just Membership in the Original Set"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0035","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6baf5d5281a098b3f5222f3207825d7d01cb3d2ba75bdfc62580e5c996f39b6f","family_id":"qf-mod33-boundary-membership","seed":202611024035,"semantic_template":"qf-mod33-boundary-membership/{\"n\":5,\"point\":1,\"subset\":[2,3],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,2],[0]]}"},"fields":{"title":"Boundary Membership Is Not Just Membership in the Original Set","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 3], point=1. Give (point,boundary,in_boundary?,topology_relative_boundary).","answer":{"kind":"expression","value":"(1, [1, 2, 3, 4], Yes, topology_relative_boundary)","explanation":"Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (1, [1, 2, 3, 4], Yes, topology_relative_boundary)."},"skills":["Boundary Membership Is Not Just Membership in the Original Set"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0036","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"160bb232368f6348a7d5d91e7db4dab13f3bd45da7e8f5cd8e803895ce466aaf","family_id":"qf-mod33-boundary-membership","seed":202611024036,"semantic_template":"qf-mod33-boundary-membership/{\"n\":5,\"point\":0,\"subset\":[2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,2],[0]]}"},"fields":{"title":"Boundary Membership Is Not Just Membership in the Original Set","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 4], point=0. Give (point,boundary,in_boundary?,topology_relative_boundary).","answer":{"kind":"expression","value":"(0, [1, 2, 3, 4], No, topology_relative_boundary)","explanation":"Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (0, [1, 2, 3, 4], No, topology_relative_boundary)."},"skills":["Boundary Membership Is Not Just Membership in the Original Set"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0037","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06871b2dc8492dac5e1b3e751b483f6e5a9ff829fb4976308dce32a9183977b2","family_id":"qf-mod33-boundary-membership","seed":202611024037,"semantic_template":"qf-mod33-boundary-membership/{\"n\":5,\"point\":2,\"subset\":[2,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,2],[0]]}"},"fields":{"title":"Boundary Membership Is Not Just Membership in the Original Set","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 4], point=2. Give (point,boundary,in_boundary?,topology_relative_boundary).","answer":{"kind":"expression","value":"(2, [1, 2, 3, 4], Yes, topology_relative_boundary)","explanation":"Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (2, [1, 2, 3, 4], Yes, topology_relative_boundary)."},"skills":["Boundary Membership Is Not Just Membership in the Original Set"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0038","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e85205dde3f67c4dbe3812e56fe97e91c575f4ae0890bdbff7bad800587edaa","family_id":"qf-mod33-boundary-membership","seed":202611024038,"semantic_template":"qf-mod33-boundary-membership/{\"n\":5,\"point\":1,\"subset\":[1,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,3],[0]]}"},"fields":{"title":"Boundary Membership Is Not Just Membership in the Original Set","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 3], point=1. Give (point,boundary,in_boundary?,topology_relative_boundary).","answer":{"kind":"expression","value":"(1, [1, 2, 3, 4], Yes, topology_relative_boundary)","explanation":"Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (1, [1, 2, 3, 4], Yes, topology_relative_boundary)."},"skills":["Boundary Membership Is Not Just Membership in the Original Set"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0039","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0ead1c8f64745be0933deffd63ec8433ebda59176bd42fe111184892939f422","family_id":"qf-mod33-boundary-membership","seed":202611024039,"semantic_template":"qf-mod33-boundary-membership/{\"n\":5,\"point\":4,\"subset\":[1,3],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,3],[0]]}"},"fields":{"title":"Boundary Membership Is Not Just Membership in the Original Set","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 3], point=4. Give (point,boundary,in_boundary?,topology_relative_boundary).","answer":{"kind":"expression","value":"(4, [1, 2, 3, 4], Yes, topology_relative_boundary)","explanation":"Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (4, [1, 2, 3, 4], Yes, topology_relative_boundary)."},"skills":["Boundary Membership Is Not Just Membership in the Original Set"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0040","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f17b62914ff54304d3b5b7c4bfe726fef4e9854bd8a1694424ea38838a6a15e8","family_id":"qf-mod33-boundary-membership","seed":202611024040,"semantic_template":"qf-mod33-boundary-membership/{\"n\":5,\"point\":4,\"subset\":[2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,3],[0]]}"},"fields":{"title":"Boundary Membership Is Not Just Membership in the Original Set","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 3], point=4. Give (point,boundary,in_boundary?,topology_relative_boundary).","answer":{"kind":"expression","value":"(4, [1, 2, 3, 4], Yes, topology_relative_boundary)","explanation":"Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (4, [1, 2, 3, 4], Yes, topology_relative_boundary)."},"skills":["Boundary Membership Is Not Just Membership in the Original Set"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0041","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f09bb013f11f20ef2a2010de62fbfc0071c169016b992e4bd90bcef7f8e5eee1","family_id":"qf-mod33-boundary-membership","seed":202611024041,"semantic_template":"qf-mod33-boundary-membership/{\"n\":5,\"point\":0,\"subset\":[2,3],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,3],[0]]}"},"fields":{"title":"Boundary Membership Is Not Just Membership in the Original Set","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 3], point=0. Give (point,boundary,in_boundary?,topology_relative_boundary).","answer":{"kind":"expression","value":"(0, [1, 2, 3, 4], No, topology_relative_boundary)","explanation":"Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (0, [1, 2, 3, 4], No, topology_relative_boundary)."},"skills":["Boundary Membership Is Not Just Membership in the Original Set"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0042","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e84bfcb1c244118d0c6b4a2f33a8660bbefd0ede518256b22d984fd6a42c474","family_id":"qf-mod33-boundary-membership","seed":202611024042,"semantic_template":"qf-mod33-boundary-membership/{\"n\":5,\"point\":3,\"subset\":[3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,3],[0]]}"},"fields":{"title":"Boundary Membership Is Not Just Membership in the Original Set","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[3, 4], point=3. Give (point,boundary,in_boundary?,topology_relative_boundary).","answer":{"kind":"expression","value":"(3, [1, 2, 3, 4], Yes, topology_relative_boundary)","explanation":"Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (3, [1, 2, 3, 4], Yes, topology_relative_boundary)."},"skills":["Boundary Membership Is Not Just Membership in the Original Set"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0043","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4dc8ef60685cc22274f146526f16d9b098ecdb7b373f4cb7d102ef83868ea0b","family_id":"qf-mod33-boundary-membership","seed":202611024043,"semantic_template":"qf-mod33-boundary-membership/{\"n\":5,\"point\":4,\"subset\":[3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,3],[0]]}"},"fields":{"title":"Boundary Membership Is Not Just Membership in the Original Set","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[3, 4], point=4. Give (point,boundary,in_boundary?,topology_relative_boundary).","answer":{"kind":"expression","value":"(4, [1, 2, 3, 4], Yes, topology_relative_boundary)","explanation":"Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (4, [1, 2, 3, 4], Yes, topology_relative_boundary)."},"skills":["Boundary Membership Is Not Just Membership in the Original Set"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0044","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac172b5c3c1e960b3134c74e4741a310e1a421912e18558764d13501b7c43bdb","family_id":"qf-mod33-boundary-membership","seed":202611024044,"semantic_template":"qf-mod33-boundary-membership/{\"n\":5,\"point\":2,\"subset\":[1,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,4],[0]]}"},"fields":{"title":"Boundary Membership Is Not Just Membership in the Original Set","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 4], point=2. Give (point,boundary,in_boundary?,topology_relative_boundary).","answer":{"kind":"expression","value":"(2, [1, 2, 3, 4], Yes, topology_relative_boundary)","explanation":"Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (2, [1, 2, 3, 4], Yes, topology_relative_boundary)."},"skills":["Boundary Membership Is Not Just Membership in the Original Set"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0045","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3266c1d215052f0c1cd8b13e47e20d4a53074543ca8f4516992733249f6f3749","family_id":"qf-mod33-boundary-membership","seed":202611024045,"semantic_template":"qf-mod33-boundary-membership/{\"n\":5,\"point\":2,\"subset\":[1,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,4],[0]]}"},"fields":{"title":"Boundary Membership Is Not Just Membership in the Original Set","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 4], point=2. Give (point,boundary,in_boundary?,topology_relative_boundary).","answer":{"kind":"expression","value":"(2, [1, 2, 3, 4], Yes, topology_relative_boundary)","explanation":"Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (2, [1, 2, 3, 4], Yes, topology_relative_boundary)."},"skills":["Boundary Membership Is Not Just Membership in the Original Set"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0046","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1a3ef93581d7fe88c16349b3360f4cbc1566fc86f3bc1e3c149806247cf2c0b","family_id":"qf-mod33-boundary-membership","seed":202611024046,"semantic_template":"qf-mod33-boundary-membership/{\"n\":5,\"point\":0,\"subset\":[2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,4],[0]]}"},"fields":{"title":"Boundary Membership Is Not Just Membership in the Original Set","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 4], point=0. Give (point,boundary,in_boundary?,topology_relative_boundary).","answer":{"kind":"expression","value":"(0, [1, 2, 3, 4], No, topology_relative_boundary)","explanation":"Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (0, [1, 2, 3, 4], No, topology_relative_boundary)."},"skills":["Boundary Membership Is Not Just Membership in the Original Set"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0047","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee8168c1cdbdedc82751cedbd4baa0354c5e2efcbd58ca617b00e2b57bd99aee","family_id":"qf-mod33-boundary-membership","seed":202611024047,"semantic_template":"qf-mod33-boundary-membership/{\"n\":5,\"point\":4,\"subset\":[2,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,4],[0]]}"},"fields":{"title":"Boundary Membership Is Not Just Membership in the Original Set","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 4], point=4. Give (point,boundary,in_boundary?,topology_relative_boundary).","answer":{"kind":"expression","value":"(4, [1, 2, 3, 4], Yes, topology_relative_boundary)","explanation":"Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (4, [1, 2, 3, 4], Yes, topology_relative_boundary)."},"skills":["Boundary Membership Is Not Just Membership in the Original Set"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0048","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"531cddf975d324a1f94dc9329806f318ac4359bdce236ffe0d338c458a7a43fa","family_id":"qf-mod33-boundary-membership","seed":202611024048,"semantic_template":"qf-mod33-boundary-membership/{\"n\":5,\"point\":3,\"subset\":[3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,4],[0]]}"},"fields":{"title":"Boundary Membership Is Not Just Membership in the Original Set","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[3, 4], point=3. Give (point,boundary,in_boundary?,topology_relative_boundary).","answer":{"kind":"expression","value":"(3, [1, 2, 3, 4], Yes, topology_relative_boundary)","explanation":"Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (3, [1, 2, 3, 4], Yes, topology_relative_boundary)."},"skills":["Boundary Membership Is Not Just Membership in the Original Set"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0049","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17f334c39ea66a516dc91ead258ded1b21ae27940435f564e1136755f1505f48","family_id":"qf-mod33-boundary-membership","seed":202611024049,"semantic_template":"qf-mod33-boundary-membership/{\"n\":5,\"point\":2,\"subset\":[3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,4],[0]]}"},"fields":{"title":"Boundary Membership Is Not Just Membership in the Original Set","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[3, 4], point=2. Give (point,boundary,in_boundary?,topology_relative_boundary).","answer":{"kind":"expression","value":"(2, [1, 2, 3, 4], Yes, topology_relative_boundary)","explanation":"Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (2, [1, 2, 3, 4], Yes, topology_relative_boundary)."},"skills":["Boundary Membership Is Not Just Membership in the Original Set"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0050","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d358789a44ed7dc5e59a6e9dd4823e1155cd922583a08c2441f04c36fb2a8ca","family_id":"qf-mod33-boundary-membership","seed":202611024050,"semantic_template":"qf-mod33-boundary-membership/{\"n\":5,\"point\":4,\"subset\":[0,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[1]]}"},"fields":{"title":"Boundary Membership Is Not Just Membership in the Original Set","stem":"Use the complete finite topology=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[0, 2], point=4. Give (point,boundary,in_boundary?,topology_relative_boundary).","answer":{"kind":"expression","value":"(4, [0, 2, 3, 4], Yes, topology_relative_boundary)","explanation":"Compute the complete topology-relative boundary first, then test the point. Do not infer boundary status from an unstated metric or a drawing. The exact result, retaining the requested order, is (4, [0, 2, 3, 4], Yes, topology_relative_boundary)."},"skills":["Boundary Membership Is Not Just Membership in the Original Set"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0051","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ddd1feffe1c2c48d4aa487f44de50114ef9e995866a66954800f4868901fb782","family_id":"qf-mod33-boundary-complement-invariance","seed":202611024051,"semantic_template":"qf-mod33-boundary-complement-invariance/{\"n\":5,\"subset\":[1,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[0]]}"},"fields":{"title":"A Set and Its Complement Have the Same Topological Boundary","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 2]. Give (subset,complement,boundary_a,boundary_complement,equal?).","answer":{"kind":"expression","value":"([1, 2], [0, 3, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes)","explanation":"Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([1, 2], [0, 3, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes)."},"skills":["A Set and Its Complement Have the Same Topological Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0052","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"587d694a046c7302f722a7f8a5493603b22b3b8c574b1c60abe982e9b1c4c56c","family_id":"qf-mod33-boundary-complement-invariance","seed":202611024052,"semantic_template":"qf-mod33-boundary-complement-invariance/{\"n\":5,\"subset\":[1,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,1],[0]]}"},"fields":{"title":"A Set and Its Complement Have the Same Topological Boundary","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 2]. Give (subset,complement,boundary_a,boundary_complement,equal?).","answer":{"kind":"expression","value":"([1, 2], [0, 3, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes)","explanation":"Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([1, 2], [0, 3, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes)."},"skills":["A Set and Its Complement Have the Same Topological Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0053","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e509170d20f365f5f7d40a2d24632277c5f3386a5f074cd38eb4d7a62148967","family_id":"qf-mod33-boundary-complement-invariance","seed":202611024053,"semantic_template":"qf-mod33-boundary-complement-invariance/{\"n\":5,\"subset\":[1,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,1],[0]]}"},"fields":{"title":"A Set and Its Complement Have the Same Topological Boundary","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 3]. Give (subset,complement,boundary_a,boundary_complement,equal?).","answer":{"kind":"expression","value":"([1, 3], [0, 2, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes)","explanation":"Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([1, 3], [0, 2, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes)."},"skills":["A Set and Its Complement Have the Same Topological Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0054","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d0f28673e706113133e4e4b0a852ea9ac9f62b50a5cabb028bab07c23cad3d4","family_id":"qf-mod33-boundary-complement-invariance","seed":202611024054,"semantic_template":"qf-mod33-boundary-complement-invariance/{\"n\":5,\"subset\":[1,3],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,1],[0]]}"},"fields":{"title":"A Set and Its Complement Have the Same Topological Boundary","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 3]. Give (subset,complement,boundary_a,boundary_complement,equal?).","answer":{"kind":"expression","value":"([1, 3], [0, 2, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes)","explanation":"Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([1, 3], [0, 2, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes)."},"skills":["A Set and Its Complement Have the Same Topological Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0055","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"846088f1845665d09ca108df15471afca446787871e86caaff18106e4f767dc6","family_id":"qf-mod33-boundary-complement-invariance","seed":202611024055,"semantic_template":"qf-mod33-boundary-complement-invariance/{\"n\":5,\"subset\":[1,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,1],[0]]}"},"fields":{"title":"A Set and Its Complement Have the Same Topological Boundary","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 4]. Give (subset,complement,boundary_a,boundary_complement,equal?).","answer":{"kind":"expression","value":"([1, 4], [0, 2, 3], [1, 2, 3, 4], [1, 2, 3, 4], Yes)","explanation":"Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([1, 4], [0, 2, 3], [1, 2, 3, 4], [1, 2, 3, 4], Yes)."},"skills":["A Set and Its Complement Have the Same Topological Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0056","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6d9160b59612d37a8ed339b8a79df32eb3686d1c1cb224ab569676316d6849d","family_id":"qf-mod33-boundary-complement-invariance","seed":202611024056,"semantic_template":"qf-mod33-boundary-complement-invariance/{\"n\":5,\"subset\":[1,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,1],[0]]}"},"fields":{"title":"A Set and Its Complement Have the Same Topological Boundary","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 4]. Give (subset,complement,boundary_a,boundary_complement,equal?).","answer":{"kind":"expression","value":"([1, 4], [0, 2, 3], [1, 2, 3, 4], [1, 2, 3, 4], Yes)","explanation":"Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([1, 4], [0, 2, 3], [1, 2, 3, 4], [1, 2, 3, 4], Yes)."},"skills":["A Set and Its Complement Have the Same Topological Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0057","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2bb6d29cf00f4b4d16d8edebe37cfc023f2f2e76a77e828540159a225344444","family_id":"qf-mod33-boundary-complement-invariance","seed":202611024057,"semantic_template":"qf-mod33-boundary-complement-invariance/{\"n\":5,\"subset\":[1,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,2],[0]]}"},"fields":{"title":"A Set and Its Complement Have the Same Topological Boundary","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 2]. Give (subset,complement,boundary_a,boundary_complement,equal?).","answer":{"kind":"expression","value":"([1, 2], [0, 3, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes)","explanation":"Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([1, 2], [0, 3, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes)."},"skills":["A Set and Its Complement Have the Same Topological Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0058","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4bb29e03b9c723e3395ca2fe6f7544195e4f260901e0985aad29011117f345c4","family_id":"qf-mod33-boundary-complement-invariance","seed":202611024058,"semantic_template":"qf-mod33-boundary-complement-invariance/{\"n\":5,\"subset\":[1,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,2],[0]]}"},"fields":{"title":"A Set and Its Complement Have the Same Topological Boundary","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 2]. Give (subset,complement,boundary_a,boundary_complement,equal?).","answer":{"kind":"expression","value":"([1, 2], [0, 3, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes)","explanation":"Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([1, 2], [0, 3, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes)."},"skills":["A Set and Its Complement Have the Same Topological Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0059","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"957193fc73dbc1c180bef7c479718ce1dae1454494826ab538192fae3011dc2d","family_id":"qf-mod33-boundary-complement-invariance","seed":202611024059,"semantic_template":"qf-mod33-boundary-complement-invariance/{\"n\":5,\"subset\":[2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,2],[0]]}"},"fields":{"title":"A Set and Its Complement Have the Same Topological Boundary","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 3]. Give (subset,complement,boundary_a,boundary_complement,equal?).","answer":{"kind":"expression","value":"([2, 3], [0, 1, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes)","explanation":"Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([2, 3], [0, 1, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes)."},"skills":["A Set and Its Complement Have the Same Topological Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0060","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ba64fd964ddeff33311447ada5568d2baf2e10ed941cbd0717607b4b6ee0522","family_id":"qf-mod33-boundary-complement-invariance","seed":202611024060,"semantic_template":"qf-mod33-boundary-complement-invariance/{\"n\":5,\"subset\":[2,3],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,2],[0]]}"},"fields":{"title":"A Set and Its Complement Have the Same Topological Boundary","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 3]. Give (subset,complement,boundary_a,boundary_complement,equal?).","answer":{"kind":"expression","value":"([2, 3], [0, 1, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes)","explanation":"Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([2, 3], [0, 1, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes)."},"skills":["A Set and Its Complement Have the Same Topological Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0061","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"26280ad5df9036508890f938876e5f1f8942413dae8323a2d7d0a50c8d624bd0","family_id":"qf-mod33-boundary-complement-invariance","seed":202611024061,"semantic_template":"qf-mod33-boundary-complement-invariance/{\"n\":5,\"subset\":[2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,2],[0]]}"},"fields":{"title":"A Set and Its Complement Have the Same Topological Boundary","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 4]. Give (subset,complement,boundary_a,boundary_complement,equal?).","answer":{"kind":"expression","value":"([2, 4], [0, 1, 3], [1, 2, 3, 4], [1, 2, 3, 4], Yes)","explanation":"Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([2, 4], [0, 1, 3], [1, 2, 3, 4], [1, 2, 3, 4], Yes)."},"skills":["A Set and Its Complement Have the Same Topological Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0062","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0232fb952d52720a1cb443d42089de8f451b8f92be0aa7eb584f4cfce4e1adf","family_id":"qf-mod33-boundary-complement-invariance","seed":202611024062,"semantic_template":"qf-mod33-boundary-complement-invariance/{\"n\":5,\"subset\":[2,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,2],[0]]}"},"fields":{"title":"A Set and Its Complement Have the Same Topological Boundary","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 4]. Give (subset,complement,boundary_a,boundary_complement,equal?).","answer":{"kind":"expression","value":"([2, 4], [0, 1, 3], [1, 2, 3, 4], [1, 2, 3, 4], Yes)","explanation":"Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([2, 4], [0, 1, 3], [1, 2, 3, 4], [1, 2, 3, 4], Yes)."},"skills":["A Set and Its Complement Have the Same Topological Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0063","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e3c48ce15094fc3081b7e9457c96552b91dc8dc53d27e872849d4ac20c59a75f","family_id":"qf-mod33-boundary-complement-invariance","seed":202611024063,"semantic_template":"qf-mod33-boundary-complement-invariance/{\"n\":5,\"subset\":[1,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,3],[0]]}"},"fields":{"title":"A Set and Its Complement Have the Same Topological Boundary","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 3]. Give (subset,complement,boundary_a,boundary_complement,equal?).","answer":{"kind":"expression","value":"([1, 3], [0, 2, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes)","explanation":"Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([1, 3], [0, 2, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes)."},"skills":["A Set and Its Complement Have the Same Topological Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0064","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d72649d41b54cebdc3cc3592465d98474135efa3f2c59d55c6277b873b45eccd","family_id":"qf-mod33-boundary-complement-invariance","seed":202611024064,"semantic_template":"qf-mod33-boundary-complement-invariance/{\"n\":5,\"subset\":[1,3],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,3],[0]]}"},"fields":{"title":"A Set and Its Complement Have the Same Topological Boundary","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 3]. Give (subset,complement,boundary_a,boundary_complement,equal?).","answer":{"kind":"expression","value":"([1, 3], [0, 2, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes)","explanation":"Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([1, 3], [0, 2, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes)."},"skills":["A Set and Its Complement Have the Same Topological Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0065","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"450f11ac21d35b1d7c22303d0c138e63f20ec50e6f9eae9304bb9ee4e48f7182","family_id":"qf-mod33-boundary-complement-invariance","seed":202611024065,"semantic_template":"qf-mod33-boundary-complement-invariance/{\"n\":5,\"subset\":[2,3],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,3],[0]]}"},"fields":{"title":"A Set and Its Complement Have the Same Topological Boundary","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 3]. Give (subset,complement,boundary_a,boundary_complement,equal?).","answer":{"kind":"expression","value":"([2, 3], [0, 1, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes)","explanation":"Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([2, 3], [0, 1, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes)."},"skills":["A Set and Its Complement Have the Same Topological Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0066","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b92502b94e8e0bcd95d28fa6908cb64f358b5ea2d952e799a1b07057be1c16ff","family_id":"qf-mod33-boundary-complement-invariance","seed":202611024066,"semantic_template":"qf-mod33-boundary-complement-invariance/{\"n\":5,\"subset\":[2,3],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,3],[0]]}"},"fields":{"title":"A Set and Its Complement Have the Same Topological Boundary","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 3]. Give (subset,complement,boundary_a,boundary_complement,equal?).","answer":{"kind":"expression","value":"([2, 3], [0, 1, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes)","explanation":"Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([2, 3], [0, 1, 4], [1, 2, 3, 4], [1, 2, 3, 4], Yes)."},"skills":["A Set and Its Complement Have the Same Topological Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0067","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9050d8fd8f024af70f75c9966f8b63cc22c9297b50a2a5d752879bc578c716e7","family_id":"qf-mod33-boundary-complement-invariance","seed":202611024067,"semantic_template":"qf-mod33-boundary-complement-invariance/{\"n\":5,\"subset\":[3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,3],[0]]}"},"fields":{"title":"A Set and Its Complement Have the Same Topological Boundary","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[3, 4]. Give (subset,complement,boundary_a,boundary_complement,equal?).","answer":{"kind":"expression","value":"([3, 4], [0, 1, 2], [1, 2, 3, 4], [1, 2, 3, 4], Yes)","explanation":"Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([3, 4], [0, 1, 2], [1, 2, 3, 4], [1, 2, 3, 4], Yes)."},"skills":["A Set and Its Complement Have the Same Topological Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0068","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a89daef5061eb67490aead37b58093d0447281734aa9b609d1fdd355dcd8f677","family_id":"qf-mod33-boundary-complement-invariance","seed":202611024068,"semantic_template":"qf-mod33-boundary-complement-invariance/{\"n\":5,\"subset\":[3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,3],[0]]}"},"fields":{"title":"A Set and Its Complement Have the Same Topological Boundary","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[3, 4]. Give (subset,complement,boundary_a,boundary_complement,equal?).","answer":{"kind":"expression","value":"([3, 4], [0, 1, 2], [1, 2, 3, 4], [1, 2, 3, 4], Yes)","explanation":"Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([3, 4], [0, 1, 2], [1, 2, 3, 4], [1, 2, 3, 4], Yes)."},"skills":["A Set and Its Complement Have the Same Topological Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0069","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71dc3d1ec71bcd9f25571c0141c7ee6bf9ed4d0a823c9b274b0c37b6311a4363","family_id":"qf-mod33-boundary-complement-invariance","seed":202611024069,"semantic_template":"qf-mod33-boundary-complement-invariance/{\"n\":5,\"subset\":[1,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,4],[0]]}"},"fields":{"title":"A Set and Its Complement Have the Same Topological Boundary","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 4]. Give (subset,complement,boundary_a,boundary_complement,equal?).","answer":{"kind":"expression","value":"([1, 4], [0, 2, 3], [1, 2, 3, 4], [1, 2, 3, 4], Yes)","explanation":"Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([1, 4], [0, 2, 3], [1, 2, 3, 4], [1, 2, 3, 4], Yes)."},"skills":["A Set and Its Complement Have the Same Topological Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0070","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57b325518cc53da3700be2f2a4db312edc9e3abd1792ede90882df953bcdd152","family_id":"qf-mod33-boundary-complement-invariance","seed":202611024070,"semantic_template":"qf-mod33-boundary-complement-invariance/{\"n\":5,\"subset\":[1,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,4],[0]]}"},"fields":{"title":"A Set and Its Complement Have the Same Topological Boundary","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[1, 4]. Give (subset,complement,boundary_a,boundary_complement,equal?).","answer":{"kind":"expression","value":"([1, 4], [0, 2, 3], [1, 2, 3, 4], [1, 2, 3, 4], Yes)","explanation":"Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([1, 4], [0, 2, 3], [1, 2, 3, 4], [1, 2, 3, 4], Yes)."},"skills":["A Set and Its Complement Have the Same Topological Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0071","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"025e179478f01464770be4723dec1b486088744710d3da6df093063bd68bfbcf","family_id":"qf-mod33-boundary-complement-invariance","seed":202611024071,"semantic_template":"qf-mod33-boundary-complement-invariance/{\"n\":5,\"subset\":[2,4],\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,4],[0]]}"},"fields":{"title":"A Set and Its Complement Have the Same Topological Boundary","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 4]. Give (subset,complement,boundary_a,boundary_complement,equal?).","answer":{"kind":"expression","value":"([2, 4], [0, 1, 3], [1, 2, 3, 4], [1, 2, 3, 4], Yes)","explanation":"Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([2, 4], [0, 1, 3], [1, 2, 3, 4], [1, 2, 3, 4], Yes)."},"skills":["A Set and Its Complement Have the Same Topological Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0072","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"298705e5aa1e3ad24be5a75c2dc0fe2a51f1cc441b462844e66b2f30ed6345b6","family_id":"qf-mod33-boundary-complement-invariance","seed":202611024072,"semantic_template":"qf-mod33-boundary-complement-invariance/{\"n\":5,\"subset\":[2,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,4],[0]]}"},"fields":{"title":"A Set and Its Complement Have the Same Topological Boundary","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[2, 4]. Give (subset,complement,boundary_a,boundary_complement,equal?).","answer":{"kind":"expression","value":"([2, 4], [0, 1, 3], [1, 2, 3, 4], [1, 2, 3, 4], Yes)","explanation":"Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([2, 4], [0, 1, 3], [1, 2, 3, 4], [1, 2, 3, 4], Yes)."},"skills":["A Set and Its Complement Have the Same Topological Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0073","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0c528f851e6ca0cde6ac5d1bfe250de2ecbb59b357798028f10a7e48b239298","family_id":"qf-mod33-boundary-complement-invariance","seed":202611024073,"semantic_template":"qf-mod33-boundary-complement-invariance/{\"n\":5,\"subset\":[3,4],\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,4],[0]]}"},"fields":{"title":"A Set and Its Complement Have the Same Topological Boundary","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[3, 4]. Give (subset,complement,boundary_a,boundary_complement,equal?).","answer":{"kind":"expression","value":"([3, 4], [0, 1, 2], [1, 2, 3, 4], [1, 2, 3, 4], Yes)","explanation":"Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([3, 4], [0, 1, 2], [1, 2, 3, 4], [1, 2, 3, 4], Yes)."},"skills":["A Set and Its Complement Have the Same Topological Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0074","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"80b3de4ab748223b63c43e0058f455e1d2609175bb1d9f52c234a5fd6f4ea9b7","family_id":"qf-mod33-boundary-complement-invariance","seed":202611024074,"semantic_template":"qf-mod33-boundary-complement-invariance/{\"n\":5,\"subset\":[3,4],\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,4],[0]]}"},"fields":{"title":"A Set and Its Complement Have the Same Topological Boundary","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[3, 4]. Give (subset,complement,boundary_a,boundary_complement,equal?).","answer":{"kind":"expression","value":"([3, 4], [0, 1, 2], [1, 2, 3, 4], [1, 2, 3, 4], Yes)","explanation":"Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([3, 4], [0, 1, 2], [1, 2, 3, 4], [1, 2, 3, 4], Yes)."},"skills":["A Set and Its Complement Have the Same Topological Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0075","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c93e7dddaeb10d1498866dddce0615da749e5e7be77b2a9fa30ed1262d9e7806","family_id":"qf-mod33-boundary-complement-invariance","seed":202611024075,"semantic_template":"qf-mod33-boundary-complement-invariance/{\"n\":5,\"subset\":[0,2],\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[1]]}"},"fields":{"title":"A Set and Its Complement Have the Same Topological Boundary","stem":"Use the complete finite topology=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. Subset=[0, 2]. Give (subset,complement,boundary_a,boundary_complement,equal?).","answer":{"kind":"expression","value":"([0, 2], [1, 3, 4], [0, 2, 3, 4], [0, 2, 3, 4], Yes)","explanation":"Form the complement in the same carrier and independently compute both closure-minus-interior boundaries. Their equality concerns this same topology, not two different ambient spaces. The exact result, retaining the requested order, is ([0, 2], [1, 3, 4], [0, 2, 3, 4], [0, 2, 3, 4], Yes)."},"skills":["A Set and Its Complement Have the Same Topological Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0076","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db3145dacffbf157e97be751fd158cf4a23b0f87b83758583271b30c508fb971","family_id":"qf-mod33-boundary-empty-whole","seed":202611024076,"semantic_template":"qf-mod33-boundary-empty-whole/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[0]],\"which\":\"empty\"}"},"fields":{"title":"Empty and Whole Sets Both Have Empty Boundary","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=empty, restricted to empty or whole. Give (which,subset,boundary,empty?).","answer":{"kind":"expression","value":"(empty, [], [], Yes)","explanation":"Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (empty, [], [], Yes)."},"skills":["Empty and Whole Sets Both Have Empty Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0077","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c8af96ebed6815bdc01c44e336794c8616076632271cfa60bfa9989c03aae33","family_id":"qf-mod33-boundary-empty-whole","seed":202611024077,"semantic_template":"qf-mod33-boundary-empty-whole/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,1],[0]],\"which\":\"whole\"}"},"fields":{"title":"Empty and Whole Sets Both Have Empty Boundary","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=whole, restricted to empty or whole. Give (which,subset,boundary,empty?).","answer":{"kind":"expression","value":"(whole, [0, 1, 2, 3, 4], [], Yes)","explanation":"Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (whole, [0, 1, 2, 3, 4], [], Yes)."},"skills":["Empty and Whole Sets Both Have Empty Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0078","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a37db8e4273932a782451f7275406270ccec1b74421f520e147e212d64801e0a","family_id":"qf-mod33-boundary-empty-whole","seed":202611024078,"semantic_template":"qf-mod33-boundary-empty-whole/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,1],[0]],\"which\":\"empty\"}"},"fields":{"title":"Empty and Whole Sets Both Have Empty Boundary","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=empty, restricted to empty or whole. Give (which,subset,boundary,empty?).","answer":{"kind":"expression","value":"(empty, [], [], Yes)","explanation":"Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (empty, [], [], Yes)."},"skills":["Empty and Whole Sets Both Have Empty Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0079","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5899de26640fd06dc8dce375eb8a9de79c99bbec609848d9375459060d50f9e7","family_id":"qf-mod33-boundary-empty-whole","seed":202611024079,"semantic_template":"qf-mod33-boundary-empty-whole/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,1],[0]],\"which\":\"whole\"}"},"fields":{"title":"Empty and Whole Sets Both Have Empty Boundary","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=whole, restricted to empty or whole. Give (which,subset,boundary,empty?).","answer":{"kind":"expression","value":"(whole, [0, 1, 2, 3, 4], [], Yes)","explanation":"Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (whole, [0, 1, 2, 3, 4], [], Yes)."},"skills":["Empty and Whole Sets Both Have Empty Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0080","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22893f496adaaa875de55d58ff442cef0dc3aa6683fa0fec9bfe4912e482b246","family_id":"qf-mod33-boundary-empty-whole","seed":202611024080,"semantic_template":"qf-mod33-boundary-empty-whole/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,1],[0]],\"which\":\"empty\"}"},"fields":{"title":"Empty and Whole Sets Both Have Empty Boundary","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=empty, restricted to empty or whole. Give (which,subset,boundary,empty?).","answer":{"kind":"expression","value":"(empty, [], [], Yes)","explanation":"Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (empty, [], [], Yes)."},"skills":["Empty and Whole Sets Both Have Empty Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0081","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0596edf8cd873fbc5b9816a8a956801edf11483c4b835efe1f9d1b451ce543ce","family_id":"qf-mod33-boundary-empty-whole","seed":202611024081,"semantic_template":"qf-mod33-boundary-empty-whole/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,1],[0]],\"which\":\"whole\"}"},"fields":{"title":"Empty and Whole Sets Both Have Empty Boundary","stem":"Use the complete finite topology=[[], [0], [0, 1], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=whole, restricted to empty or whole. Give (which,subset,boundary,empty?).","answer":{"kind":"expression","value":"(whole, [0, 1, 2, 3, 4], [], Yes)","explanation":"Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (whole, [0, 1, 2, 3, 4], [], Yes)."},"skills":["Empty and Whole Sets Both Have Empty Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0082","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eaa2f07cab2be040b87ac6c5b8ec33837fdab6eac439aebfbc8d9df2db5937a1","family_id":"qf-mod33-boundary-empty-whole","seed":202611024082,"semantic_template":"qf-mod33-boundary-empty-whole/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,2],[0]],\"which\":\"empty\"}"},"fields":{"title":"Empty and Whole Sets Both Have Empty Boundary","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=empty, restricted to empty or whole. Give (which,subset,boundary,empty?).","answer":{"kind":"expression","value":"(empty, [], [], Yes)","explanation":"Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (empty, [], [], Yes)."},"skills":["Empty and Whole Sets Both Have Empty Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0083","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"caf0e6fd6b62c7340b16e675142dd8ffd8dab4b0ea0a22cce551e4fa05c0d7e3","family_id":"qf-mod33-boundary-empty-whole","seed":202611024083,"semantic_template":"qf-mod33-boundary-empty-whole/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,2],[0]],\"which\":\"whole\"}"},"fields":{"title":"Empty and Whole Sets Both Have Empty Boundary","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=whole, restricted to empty or whole. Give (which,subset,boundary,empty?).","answer":{"kind":"expression","value":"(whole, [0, 1, 2, 3, 4], [], Yes)","explanation":"Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (whole, [0, 1, 2, 3, 4], [], Yes)."},"skills":["Empty and Whole Sets Both Have Empty Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0084","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dda9d04fd6347f1482691144e936a2c0d064be3d683c74283e701940b3dd3d24","family_id":"qf-mod33-boundary-empty-whole","seed":202611024084,"semantic_template":"qf-mod33-boundary-empty-whole/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,2],[0]],\"which\":\"empty\"}"},"fields":{"title":"Empty and Whole Sets Both Have Empty Boundary","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=empty, restricted to empty or whole. Give (which,subset,boundary,empty?).","answer":{"kind":"expression","value":"(empty, [], [], Yes)","explanation":"Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (empty, [], [], Yes)."},"skills":["Empty and Whole Sets Both Have Empty Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0085","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"909cc235e69aa3828e80cbafdf6f6670f7d699ddc0d29e1c69ef5a13e7976bf4","family_id":"qf-mod33-boundary-empty-whole","seed":202611024085,"semantic_template":"qf-mod33-boundary-empty-whole/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,2],[0]],\"which\":\"whole\"}"},"fields":{"title":"Empty and Whole Sets Both Have Empty Boundary","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=whole, restricted to empty or whole. Give (which,subset,boundary,empty?).","answer":{"kind":"expression","value":"(whole, [0, 1, 2, 3, 4], [], Yes)","explanation":"Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (whole, [0, 1, 2, 3, 4], [], Yes)."},"skills":["Empty and Whole Sets Both Have Empty Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0086","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69d9a5dd398ac525976470392a67c6849d6f13fb3a3c2198d2cf499a03856d53","family_id":"qf-mod33-boundary-empty-whole","seed":202611024086,"semantic_template":"qf-mod33-boundary-empty-whole/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,2],[0]],\"which\":\"empty\"}"},"fields":{"title":"Empty and Whole Sets Both Have Empty Boundary","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=empty, restricted to empty or whole. Give (which,subset,boundary,empty?).","answer":{"kind":"expression","value":"(empty, [], [], Yes)","explanation":"Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (empty, [], [], Yes)."},"skills":["Empty and Whole Sets Both Have Empty Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0087","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9e4054ad147fed69671a0f26788c309d0c0e3e4f8162af90fa1e2a049dae757","family_id":"qf-mod33-boundary-empty-whole","seed":202611024087,"semantic_template":"qf-mod33-boundary-empty-whole/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,2],[0]],\"which\":\"whole\"}"},"fields":{"title":"Empty and Whole Sets Both Have Empty Boundary","stem":"Use the complete finite topology=[[], [0], [0, 2], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=whole, restricted to empty or whole. Give (which,subset,boundary,empty?).","answer":{"kind":"expression","value":"(whole, [0, 1, 2, 3, 4], [], Yes)","explanation":"Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (whole, [0, 1, 2, 3, 4], [], Yes)."},"skills":["Empty and Whole Sets Both Have Empty Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0088","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d19992a29b91e26cdee5688c669fe232d0d50c731602d7d54512790ef4dd9e5","family_id":"qf-mod33-boundary-empty-whole","seed":202611024088,"semantic_template":"qf-mod33-boundary-empty-whole/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,3],[0]],\"which\":\"empty\"}"},"fields":{"title":"Empty and Whole Sets Both Have Empty Boundary","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=empty, restricted to empty or whole. Give (which,subset,boundary,empty?).","answer":{"kind":"expression","value":"(empty, [], [], Yes)","explanation":"Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (empty, [], [], Yes)."},"skills":["Empty and Whole Sets Both Have Empty Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0089","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8a53ded5c1c5fbe7710aaca5a89a7013abc7734d50b3c79d33fdd15d972050a","family_id":"qf-mod33-boundary-empty-whole","seed":202611024089,"semantic_template":"qf-mod33-boundary-empty-whole/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,3],[0]],\"which\":\"whole\"}"},"fields":{"title":"Empty and Whole Sets Both Have Empty Boundary","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 1, 3], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=whole, restricted to empty or whole. Give (which,subset,boundary,empty?).","answer":{"kind":"expression","value":"(whole, [0, 1, 2, 3, 4], [], Yes)","explanation":"Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (whole, [0, 1, 2, 3, 4], [], Yes)."},"skills":["Empty and Whole Sets Both Have Empty Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0090","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fbd0ffd904875e9ec1eacaf26b67ce604f80586b8e57cc9ee953120aae7a440e","family_id":"qf-mod33-boundary-empty-whole","seed":202611024090,"semantic_template":"qf-mod33-boundary-empty-whole/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,3],[0]],\"which\":\"empty\"}"},"fields":{"title":"Empty and Whole Sets Both Have Empty Boundary","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=empty, restricted to empty or whole. Give (which,subset,boundary,empty?).","answer":{"kind":"expression","value":"(empty, [], [], Yes)","explanation":"Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (empty, [], [], Yes)."},"skills":["Empty and Whole Sets Both Have Empty Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0091","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72117924f5a977e53d097e88ab0c2f9b466ebd19aced355b122c2a48ea76529b","family_id":"qf-mod33-boundary-empty-whole","seed":202611024091,"semantic_template":"qf-mod33-boundary-empty-whole/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,3],[0]],\"which\":\"whole\"}"},"fields":{"title":"Empty and Whole Sets Both Have Empty Boundary","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 2, 3], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=whole, restricted to empty or whole. Give (which,subset,boundary,empty?).","answer":{"kind":"expression","value":"(whole, [0, 1, 2, 3, 4], [], Yes)","explanation":"Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (whole, [0, 1, 2, 3, 4], [], Yes)."},"skills":["Empty and Whole Sets Both Have Empty Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0092","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70977eca46e652f7e083be7bb47d650b48f6750a9823fab96e6ad9c4291c5eb0","family_id":"qf-mod33-boundary-empty-whole","seed":202611024092,"semantic_template":"qf-mod33-boundary-empty-whole/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,3],[0]],\"which\":\"empty\"}"},"fields":{"title":"Empty and Whole Sets Both Have Empty Boundary","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=empty, restricted to empty or whole. Give (which,subset,boundary,empty?).","answer":{"kind":"expression","value":"(empty, [], [], Yes)","explanation":"Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (empty, [], [], Yes)."},"skills":["Empty and Whole Sets Both Have Empty Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0093","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"540776c6c43e116379b4b3f8e308bc08da76c1af7fcd37da5a9c79ca90ff99f3","family_id":"qf-mod33-boundary-empty-whole","seed":202611024093,"semantic_template":"qf-mod33-boundary-empty-whole/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,3],[0]],\"which\":\"whole\"}"},"fields":{"title":"Empty and Whole Sets Both Have Empty Boundary","stem":"Use the complete finite topology=[[], [0], [0, 3], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=whole, restricted to empty or whole. Give (which,subset,boundary,empty?).","answer":{"kind":"expression","value":"(whole, [0, 1, 2, 3, 4], [], Yes)","explanation":"Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (whole, [0, 1, 2, 3, 4], [], Yes)."},"skills":["Empty and Whole Sets Both Have Empty Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0094","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe2f65cb95614fb4d44c4c5e5053be05322db6a7336778b6a97a2a5c7f064e79","family_id":"qf-mod33-boundary-empty-whole","seed":202611024094,"semantic_template":"qf-mod33-boundary-empty-whole/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,4],[0]],\"which\":\"empty\"}"},"fields":{"title":"Empty and Whole Sets Both Have Empty Boundary","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=empty, restricted to empty or whole. Give (which,subset,boundary,empty?).","answer":{"kind":"expression","value":"(empty, [], [], Yes)","explanation":"Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (empty, [], [], Yes)."},"skills":["Empty and Whole Sets Both Have Empty Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0095","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fce2a6e18d47678ac6d484894aff3e14ab479f3786fde4778c8170dcc68f39c5","family_id":"qf-mod33-boundary-empty-whole","seed":202611024095,"semantic_template":"qf-mod33-boundary-empty-whole/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,4],[0]],\"which\":\"whole\"}"},"fields":{"title":"Empty and Whole Sets Both Have Empty Boundary","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 1, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=whole, restricted to empty or whole. Give (which,subset,boundary,empty?).","answer":{"kind":"expression","value":"(whole, [0, 1, 2, 3, 4], [], Yes)","explanation":"Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (whole, [0, 1, 2, 3, 4], [], Yes)."},"skills":["Empty and Whole Sets Both Have Empty Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0096","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f30d420011102e465d581d31d229eddc311b12e78a7f93e1a2152cf8d683e37c","family_id":"qf-mod33-boundary-empty-whole","seed":202611024096,"semantic_template":"qf-mod33-boundary-empty-whole/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,4],[0]],\"which\":\"empty\"}"},"fields":{"title":"Empty and Whole Sets Both Have Empty Boundary","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 1, 2, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=empty, restricted to empty or whole. Give (which,subset,boundary,empty?).","answer":{"kind":"expression","value":"(empty, [], [], Yes)","explanation":"Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (empty, [], [], Yes)."},"skills":["Empty and Whole Sets Both Have Empty Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0097","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c44789f6a11e461754643f1f91a41674960556dfaa85977188edb4f7ace55454","family_id":"qf-mod33-boundary-empty-whole","seed":202611024097,"semantic_template":"qf-mod33-boundary-empty-whole/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,4],[0]],\"which\":\"whole\"}"},"fields":{"title":"Empty and Whole Sets Both Have Empty Boundary","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 2, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=whole, restricted to empty or whole. Give (which,subset,boundary,empty?).","answer":{"kind":"expression","value":"(whole, [0, 1, 2, 3, 4], [], Yes)","explanation":"Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (whole, [0, 1, 2, 3, 4], [], Yes)."},"skills":["Empty and Whole Sets Both Have Empty Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0098","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"683a5398b4eb3b774ff4732049a1f56bb236e685d7479c6396e028cf24759976","family_id":"qf-mod33-boundary-empty-whole","seed":202611024098,"semantic_template":"qf-mod33-boundary-empty-whole/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,4],[0]],\"which\":\"empty\"}"},"fields":{"title":"Empty and Whole Sets Both Have Empty Boundary","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 1, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=empty, restricted to empty or whole. Give (which,subset,boundary,empty?).","answer":{"kind":"expression","value":"(empty, [], [], Yes)","explanation":"Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (empty, [], [], Yes)."},"skills":["Empty and Whole Sets Both Have Empty Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0099","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c87a28017bf3338d873725ef6cbcd014e007b10ffd50b2da561946c72fd074c6","family_id":"qf-mod33-boundary-empty-whole","seed":202611024099,"semantic_template":"qf-mod33-boundary-empty-whole/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,4],[0]],\"which\":\"whole\"}"},"fields":{"title":"Empty and Whole Sets Both Have Empty Boundary","stem":"Use the complete finite topology=[[], [0], [0, 4], [0, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=whole, restricted to empty or whole. Give (which,subset,boundary,empty?).","answer":{"kind":"expression","value":"(whole, [0, 1, 2, 3, 4], [], Yes)","explanation":"Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (whole, [0, 1, 2, 3, 4], [], Yes)."},"skills":["Empty and Whole Sets Both Have Empty Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261102-r04-topological-boundary-0100","batch_id":"batch-20261014-advance-20261102-r04-topological-boundary-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6679386a54f7dc61ad918e107554fbe19147d4a4ffca09824235f56ed9bb6fc2","family_id":"qf-mod33-boundary-empty-whole","seed":202611024100,"semantic_template":"qf-mod33-boundary-empty-whole/{\"n\":5,\"topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[1]],\"which\":\"empty\"}"},"fields":{"title":"Empty and Whole Sets Both Have Empty Boundary","stem":"Use the complete finite topology=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]] on the same carrier Fin(5)={0,…,4}. Each subset is an increasing duplicate-free carrier list. Validate the unique empty/whole opens and all finite union/intersection closure obligations first. All complements, closed sets, closures, interiors and boundaries refer to this same carrier/topology; no metric or visual boundary is silently supplied. which=empty, restricted to empty or whole. Give (which,subset,boundary,empty?).","answer":{"kind":"expression","value":"(empty, [], [], Yes)","explanation":"Use the indicated empty subset or complete carrier. Both are open and closed in this topology, so closure minus interior is empty. The exact result, retaining the requested order, is (empty, [], [], Yes)."},"skills":["Empty and Whole Sets Both Have Empty Boundary"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}