{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0001","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b885906adec6aac099ba1c8492b55bb31390bee2f75bf01f67d6c3977e98c9e","family_id":"qf-ext19-neighborhood-witness","seed":202610192001,"semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[0,1],\"n\":4,\"opens\":[[],[0],[0,1],[0,1,2],[0,1,2,3]],\"x\":0}"},"fields":{"title":"Point Guard and Every Open Neighborhood Witness","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=0, candidate N=[0, 1]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.","answer":{"kind":"expression","value":"(4, 0, Yes, [0, 1], [[0], [0, 1]], Yes)","explanation":"Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, Yes, [0, 1], [[0], [0, 1]], Yes)."},"skills":["Point Guard and Every Open Neighborhood Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0002","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2235743a966b2ba8d12a0836ad243d7804d1a3f84f651951946481cb776399a","family_id":"qf-ext19-neighborhood-witness","seed":202610192002,"semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[2],\"n\":4,\"opens\":[[],[1],[0,1],[0,1,2],[0,1,2,3]],\"x\":1}"},"fields":{"title":"Point Guard and Every Open Neighborhood Witness","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=1, candidate N=[2]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.","answer":{"kind":"expression","value":"(4, 1, Yes, [2], [], No)","explanation":"Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 1, Yes, [2], [], No)."},"skills":["Point Guard and Every Open Neighborhood Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0003","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e069fa3de497bc06ce129f49e17f1d3b6d979b3f8294b226d3d3f2496174665d","family_id":"qf-ext19-neighborhood-witness","seed":202610192003,"semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[0,2],\"n\":4,\"opens\":[[],[2],[0,2],[0,1,2],[0,1,2,3]],\"x\":2}"},"fields":{"title":"Point Guard and Every Open Neighborhood Witness","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [0, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=2, candidate N=[0, 2]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.","answer":{"kind":"expression","value":"(4, 2, Yes, [0, 2], [[2], [0, 2]], Yes)","explanation":"Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, Yes, [0, 2], [[2], [0, 2]], Yes)."},"skills":["Point Guard and Every Open Neighborhood Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0004","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f6d0c6141c421e3511d7e197b7abede403af7668e6417c36f12b0fb295cdb2b","family_id":"qf-ext19-neighborhood-witness","seed":202610192004,"semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[1,2],\"n\":4,\"opens\":[[],[3],[0,3],[0,1,3],[0,1,2,3]],\"x\":3}"},"fields":{"title":"Point Guard and Every Open Neighborhood Witness","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=3, candidate N=[1, 2]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.","answer":{"kind":"expression","value":"(4, 3, Yes, [1, 2], [], No)","explanation":"Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, Yes, [1, 2], [], No)."},"skills":["Point Guard and Every Open Neighborhood Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0005","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bedd419945dd88dd1ae1f862a6d59b9b753e2888d273fe59d26bbd89ebf37125","family_id":"qf-ext19-neighborhood-witness","seed":202610192005,"semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[0,1,2],\"n\":4,\"opens\":[[],[0],[0,2],[0,1,2],[0,1,2,3]],\"x\":-1}"},"fields":{"title":"Point Guard and Every Open Neighborhood Witness","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=-1, candidate N=[0, 1, 2]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.","answer":{"kind":"expression","value":"(4, -1, No, [0, 1, 2], not-evaluated, not-evaluated)","explanation":"Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, -1, No, [0, 1, 2], not-evaluated, not-evaluated)."},"skills":["Point Guard and Every Open Neighborhood Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0006","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da0de90d2920f2e0e477e9abc463ca65ae89dbcbab979b544115e3f967139897","family_id":"qf-ext19-neighborhood-witness","seed":202610192006,"semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[3],\"n\":4,\"opens\":[[],[1],[1,2],[0,1,2],[0,1,2,3]],\"x\":1}"},"fields":{"title":"Point Guard and Every Open Neighborhood Witness","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=1, candidate N=[3]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.","answer":{"kind":"expression","value":"(4, 1, Yes, [3], [], No)","explanation":"Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 1, Yes, [3], [], No)."},"skills":["Point Guard and Every Open Neighborhood Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0007","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7fdb633048b0b6d4cd5144f68f236e172c4d23bcf201517bad21f2afc4907986","family_id":"qf-ext19-neighborhood-witness","seed":202610192007,"semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[0,3],\"n\":4,\"opens\":[[],[2],[1,2],[0,1,2],[0,1,2,3]],\"x\":2}"},"fields":{"title":"Point Guard and Every Open Neighborhood Witness","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [1, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=2, candidate N=[0, 3]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.","answer":{"kind":"expression","value":"(4, 2, Yes, [0, 3], [], No)","explanation":"Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, Yes, [0, 3], [], No)."},"skills":["Point Guard and Every Open Neighborhood Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0008","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3fad248459c5341c6a857165e735700b5d845a923c79270e3bef4791a051b03","family_id":"qf-ext19-neighborhood-witness","seed":202610192008,"semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[1,3],\"n\":4,\"opens\":[[],[3],[1,3],[0,1,3],[0,1,2,3]],\"x\":3}"},"fields":{"title":"Point Guard and Every Open Neighborhood Witness","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=3, candidate N=[1, 3]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.","answer":{"kind":"expression","value":"(4, 3, Yes, [1, 3], [[3], [1, 3]], Yes)","explanation":"Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, Yes, [1, 3], [[3], [1, 3]], Yes)."},"skills":["Point Guard and Every Open Neighborhood Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0009","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8689660280c87216b19ffc8593971abf2cb47b5e9aa72435d48a1caae43e468","family_id":"qf-ext19-neighborhood-witness","seed":202610192009,"semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[0,1,3],\"n\":4,\"opens\":[[],[0],[0,3],[0,1,3],[0,1,2,3]],\"x\":0}"},"fields":{"title":"Point Guard and Every Open Neighborhood Witness","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=0, candidate N=[0, 1, 3]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.","answer":{"kind":"expression","value":"(4, 0, Yes, [0, 1, 3], [[0], [0, 3], [0, 1, 3]], Yes)","explanation":"Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, Yes, [0, 1, 3], [[0], [0, 3], [0, 1, 3]], Yes)."},"skills":["Point Guard and Every Open Neighborhood Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0010","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c266cc8a931de214d9263a49eb6c5523585eab9a41c086253785b15999b8f85d","family_id":"qf-ext19-neighborhood-witness","seed":202610192010,"semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[2,3],\"n\":4,\"opens\":[[],[1],[1,3],[0,1,3],[0,1,2,3]],\"x\":-1}"},"fields":{"title":"Point Guard and Every Open Neighborhood Witness","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=-1, candidate N=[2, 3]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.","answer":{"kind":"expression","value":"(4, -1, No, [2, 3], not-evaluated, not-evaluated)","explanation":"Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, -1, No, [2, 3], not-evaluated, not-evaluated)."},"skills":["Point Guard and Every Open Neighborhood Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0011","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7b1fd73730c138f18b18ba36bd9909e42da8f7934c816aa6e1fe72404120e9c","family_id":"qf-ext19-neighborhood-witness","seed":202610192011,"semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[0,2,3],\"n\":4,\"opens\":[[],[2],[2,3],[0,2,3],[0,1,2,3]],\"x\":2}"},"fields":{"title":"Point Guard and Every Open Neighborhood Witness","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=2, candidate N=[0, 2, 3]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.","answer":{"kind":"expression","value":"(4, 2, Yes, [0, 2, 3], [[2], [2, 3], [0, 2, 3]], Yes)","explanation":"Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, Yes, [0, 2, 3], [[2], [2, 3], [0, 2, 3]], Yes)."},"skills":["Point Guard and Every Open Neighborhood Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0012","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4db64a8f2fcb2f505ddc08680bb4244b9951b3776cea376b1a460ba42cfc601","family_id":"qf-ext19-neighborhood-witness","seed":202610192012,"semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[1,2,3],\"n\":4,\"opens\":[[],[3],[2,3],[0,2,3],[0,1,2,3]],\"x\":3}"},"fields":{"title":"Point Guard and Every Open Neighborhood Witness","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=3, candidate N=[1, 2, 3]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.","answer":{"kind":"expression","value":"(4, 3, Yes, [1, 2, 3], [[3], [2, 3]], Yes)","explanation":"Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, Yes, [1, 2, 3], [[3], [2, 3]], Yes)."},"skills":["Point Guard and Every Open Neighborhood Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0013","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60bf25a2ab79e01622d6795c76e529c7f1e968bb728daf01c173f348ec50e3c9","family_id":"qf-ext19-neighborhood-witness","seed":202610192013,"semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[0,1,2,3],\"n\":4,\"opens\":[[],[0],[0,1],[0,1,3],[0,1,2,3]],\"x\":0}"},"fields":{"title":"Point Guard and Every Open Neighborhood Witness","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=0, candidate N=[0, 1, 2, 3]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.","answer":{"kind":"expression","value":"(4, 0, Yes, [0, 1, 2, 3], [[0], [0, 1], [0, 1, 3], [0, 1, 2, 3]], Yes)","explanation":"Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, Yes, [0, 1, 2, 3], [[0], [0, 1], [0, 1, 3], [0, 1, 2, 3]], Yes)."},"skills":["Point Guard and Every Open Neighborhood Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0014","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5edf74942dd0651f0ecd6af3ac5c0c0020ddcb190bef26fb33eb1c1732e35ac9","family_id":"qf-ext19-neighborhood-witness","seed":202610192014,"semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[],\"n\":4,\"opens\":[[],[1],[0,1],[0,1,3],[0,1,2,3]],\"x\":1}"},"fields":{"title":"Point Guard and Every Open Neighborhood Witness","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=1, candidate N=[]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.","answer":{"kind":"expression","value":"(4, 1, Yes, [], [], No)","explanation":"Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 1, Yes, [], [], No)."},"skills":["Point Guard and Every Open Neighborhood Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0015","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6155508d2cbd7b5070661b7dafb97306aa92e19374330c9028d6090cf8316237","family_id":"qf-ext19-neighborhood-witness","seed":202610192015,"semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[0],\"n\":4,\"opens\":[[],[2],[0,2],[0,2,3],[0,1,2,3]],\"x\":-1}"},"fields":{"title":"Point Guard and Every Open Neighborhood Witness","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=-1, candidate N=[0]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.","answer":{"kind":"expression","value":"(4, -1, No, [0], not-evaluated, not-evaluated)","explanation":"Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, -1, No, [0], not-evaluated, not-evaluated)."},"skills":["Point Guard and Every Open Neighborhood Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0016","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6721224bbe074f42688e440231a61229c1e894826044a3b6104087657df1655","family_id":"qf-ext19-neighborhood-witness","seed":202610192016,"semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[1],\"n\":4,\"opens\":[[],[3],[0,3],[0,2,3],[0,1,2,3]],\"x\":3}"},"fields":{"title":"Point Guard and Every Open Neighborhood Witness","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=3, candidate N=[1]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.","answer":{"kind":"expression","value":"(4, 3, Yes, [1], [], No)","explanation":"Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, Yes, [1], [], No)."},"skills":["Point Guard and Every Open Neighborhood Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0017","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00739c6ece6767dbf07992580e655c9e7d98553155fa51276bc028f33b54b7cd","family_id":"qf-ext19-neighborhood-witness","seed":202610192017,"semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[0,1],\"n\":4,\"opens\":[[],[0],[0,2],[0,2,3],[0,1,2,3]],\"x\":0}"},"fields":{"title":"Point Guard and Every Open Neighborhood Witness","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=0, candidate N=[0, 1]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.","answer":{"kind":"expression","value":"(4, 0, Yes, [0, 1], [[0]], Yes)","explanation":"Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, Yes, [0, 1], [[0]], Yes)."},"skills":["Point Guard and Every Open Neighborhood Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0018","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f64280cce5305ca2605ef1d9c8782ab5730c0e26556d6e96d8d3dd6952b48703","family_id":"qf-ext19-neighborhood-witness","seed":202610192018,"semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[2],\"n\":4,\"opens\":[[],[1],[1,2],[1,2,3],[0,1,2,3]],\"x\":1}"},"fields":{"title":"Point Guard and Every Open Neighborhood Witness","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=1, candidate N=[2]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.","answer":{"kind":"expression","value":"(4, 1, Yes, [2], [], No)","explanation":"Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 1, Yes, [2], [], No)."},"skills":["Point Guard and Every Open Neighborhood Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0019","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28c7ecc3a0b2c930918a73d1dc1085dda6b2979415677d25a40efe421f3f39e2","family_id":"qf-ext19-neighborhood-witness","seed":202610192019,"semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[0,2],\"n\":4,\"opens\":[[],[2],[1,2],[1,2,3],[0,1,2,3]],\"x\":2}"},"fields":{"title":"Point Guard and Every Open Neighborhood Witness","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=2, candidate N=[0, 2]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.","answer":{"kind":"expression","value":"(4, 2, Yes, [0, 2], [[2]], Yes)","explanation":"Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, Yes, [0, 2], [[2]], Yes)."},"skills":["Point Guard and Every Open Neighborhood Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0020","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13b071392d63907412a4f8f4c53b0da33e759b920536d1b6db955bccb0b21504","family_id":"qf-ext19-neighborhood-witness","seed":202610192020,"semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[1,2],\"n\":4,\"opens\":[[],[3],[1,3],[1,2,3],[0,1,2,3]],\"x\":-1}"},"fields":{"title":"Point Guard and Every Open Neighborhood Witness","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=-1, candidate N=[1, 2]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.","answer":{"kind":"expression","value":"(4, -1, No, [1, 2], not-evaluated, not-evaluated)","explanation":"Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, -1, No, [1, 2], not-evaluated, not-evaluated)."},"skills":["Point Guard and Every Open Neighborhood Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0021","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e68695af41cba75adb4c9c67d043b486c59af4757969a9d40f35273def299af8","family_id":"qf-ext19-neighborhood-witness","seed":202610192021,"semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[0,1,2],\"n\":4,\"opens\":[[],[0],[0,3],[0,2,3],[0,1,2,3]],\"x\":0}"},"fields":{"title":"Point Guard and Every Open Neighborhood Witness","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=0, candidate N=[0, 1, 2]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.","answer":{"kind":"expression","value":"(4, 0, Yes, [0, 1, 2], [[0]], Yes)","explanation":"Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, Yes, [0, 1, 2], [[0]], Yes)."},"skills":["Point Guard and Every Open Neighborhood Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0022","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"553176071d4a8212dbb6411b39f6fa41ad444e7c4adac09a4dd36e404ec16750","family_id":"qf-ext19-neighborhood-witness","seed":202610192022,"semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[3],\"n\":4,\"opens\":[[],[1],[1,3],[1,2,3],[0,1,2,3]],\"x\":1}"},"fields":{"title":"Point Guard and Every Open Neighborhood Witness","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=1, candidate N=[3]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.","answer":{"kind":"expression","value":"(4, 1, Yes, [3], [], No)","explanation":"Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 1, Yes, [3], [], No)."},"skills":["Point Guard and Every Open Neighborhood Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0023","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a287a9686b6f536d17250fe1101d7ea6b426345180d9c24b59a939476e41d002","family_id":"qf-ext19-neighborhood-witness","seed":202610192023,"semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[0,3],\"n\":4,\"opens\":[[],[2],[2,3],[1,2,3],[0,1,2,3]],\"x\":2}"},"fields":{"title":"Point Guard and Every Open Neighborhood Witness","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=2, candidate N=[0, 3]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.","answer":{"kind":"expression","value":"(4, 2, Yes, [0, 3], [], No)","explanation":"Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, Yes, [0, 3], [], No)."},"skills":["Point Guard and Every Open Neighborhood Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0024","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4b32f7859b22093e4f90869c44a6d2319fc778801f119d9d827afae6ba4a08e","family_id":"qf-ext19-neighborhood-witness","seed":202610192024,"semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[1,3],\"n\":4,\"opens\":[[],[3],[2,3],[1,2,3],[0,1,2,3]],\"x\":3}"},"fields":{"title":"Point Guard and Every Open Neighborhood Witness","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=3, candidate N=[1, 3]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.","answer":{"kind":"expression","value":"(4, 3, Yes, [1, 3], [[3]], Yes)","explanation":"Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, Yes, [1, 3], [[3]], Yes)."},"skills":["Point Guard and Every Open Neighborhood Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0025","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a9179f09af24647abb95bc63fcaac9138d5e15b2437f4a4005465350ae8109c","family_id":"qf-ext19-neighborhood-witness","seed":202610192025,"semantic_template":"qf-ext19-neighborhood-witness/{\"candidate\":[0,1],\"n\":3,\"opens\":[[],[0],[0,1],[0,1,2]],\"x\":-1}"},"fields":{"title":"Point Guard and Every Open Neighborhood Witness","stem":"Use the complete finite topology on X=Fin(3)={0,…,2} with opens=[[], [0], [0, 1], [0, 1, 2]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=-1, candidate N=[0, 1]. Give (n, x, point in X?, N, all open U with x∈U⊆N or not-evaluated, neighborhood? or not-evaluated). N itself need not be open.","answer":{"kind":"expression","value":"(3, -1, No, [0, 1], not-evaluated, not-evaluated)","explanation":"Type-check the point before evaluating neighborhood claims, then enumerate every open witness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, -1, No, [0, 1], not-evaluated, not-evaluated)."},"skills":["Point Guard and Every Open Neighborhood Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0026","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c35195b7b21730cbaf73d72ae8bac2138d75b83baaebe1d9023cba63975737f","family_id":"qf-ext19-neighborhood-qualification","seed":202610192026,"semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[0,1],\"n\":4,\"opens\":[[],[0],[0,1],[0,1,2],[0,1,2,3]],\"x\":0}"},"fields":{"title":"Containment, Openness, and Neighborhood Status Are Different","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=0, candidate N=[0, 1]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.","answer":{"kind":"expression","value":"(4, 0, Yes, [0, 1], Yes, Yes, Yes)","explanation":"Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, Yes, [0, 1], Yes, Yes, Yes)."},"skills":["Containment, Openness, and Neighborhood Status Are Different"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0027","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"944e60b3b554fe926641cc72f7fa882e294c3787c13ade32d3833859b167d8f2","family_id":"qf-ext19-neighborhood-qualification","seed":202610192027,"semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[2],\"n\":4,\"opens\":[[],[1],[0,1],[0,1,2],[0,1,2,3]],\"x\":1}"},"fields":{"title":"Containment, Openness, and Neighborhood Status Are Different","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=1, candidate N=[2]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.","answer":{"kind":"expression","value":"(4, 1, Yes, [2], No, No, No)","explanation":"Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 1, Yes, [2], No, No, No)."},"skills":["Containment, Openness, and Neighborhood Status Are Different"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0028","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ccebceb0e84422f3de911187569749e2a7810cb34c9c3d007f260651bf65636","family_id":"qf-ext19-neighborhood-qualification","seed":202610192028,"semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[0,2],\"n\":4,\"opens\":[[],[2],[0,2],[0,1,2],[0,1,2,3]],\"x\":2}"},"fields":{"title":"Containment, Openness, and Neighborhood Status Are Different","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [0, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=2, candidate N=[0, 2]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.","answer":{"kind":"expression","value":"(4, 2, Yes, [0, 2], Yes, Yes, Yes)","explanation":"Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, Yes, [0, 2], Yes, Yes, Yes)."},"skills":["Containment, Openness, and Neighborhood Status Are Different"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0029","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71b84534d81d47785919d9bd51354f7467801c0ddddbe207b99d1196ab32bc36","family_id":"qf-ext19-neighborhood-qualification","seed":202610192029,"semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[1,2],\"n\":4,\"opens\":[[],[3],[0,3],[0,1,3],[0,1,2,3]],\"x\":3}"},"fields":{"title":"Containment, Openness, and Neighborhood Status Are Different","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=3, candidate N=[1, 2]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.","answer":{"kind":"expression","value":"(4, 3, Yes, [1, 2], No, No, No)","explanation":"Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, Yes, [1, 2], No, No, No)."},"skills":["Containment, Openness, and Neighborhood Status Are Different"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0030","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"886fa78fb746c8e42ac922171f42f6f66ddf51c1c9d4802ddcccc3d207b8658b","family_id":"qf-ext19-neighborhood-qualification","seed":202610192030,"semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[0,1,2],\"n\":4,\"opens\":[[],[0],[0,2],[0,1,2],[0,1,2,3]],\"x\":-1}"},"fields":{"title":"Containment, Openness, and Neighborhood Status Are Different","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=-1, candidate N=[0, 1, 2]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.","answer":{"kind":"expression","value":"(4, -1, No, [0, 1, 2], not-evaluated, not-evaluated, not-evaluated)","explanation":"Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, -1, No, [0, 1, 2], not-evaluated, not-evaluated, not-evaluated)."},"skills":["Containment, Openness, and Neighborhood Status Are Different"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0031","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78b0d2b6d647b36062dc4f02d997d784c959882f10d077c16d2377239b630e23","family_id":"qf-ext19-neighborhood-qualification","seed":202610192031,"semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[3],\"n\":4,\"opens\":[[],[1],[1,2],[0,1,2],[0,1,2,3]],\"x\":1}"},"fields":{"title":"Containment, Openness, and Neighborhood Status Are Different","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=1, candidate N=[3]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.","answer":{"kind":"expression","value":"(4, 1, Yes, [3], No, No, No)","explanation":"Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 1, Yes, [3], No, No, No)."},"skills":["Containment, Openness, and Neighborhood Status Are Different"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0032","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5ecb3921ad144035693bd9de98424afe38a90eca844e2ed9a038c989a929666","family_id":"qf-ext19-neighborhood-qualification","seed":202610192032,"semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[0,3],\"n\":4,\"opens\":[[],[2],[1,2],[0,1,2],[0,1,2,3]],\"x\":2}"},"fields":{"title":"Containment, Openness, and Neighborhood Status Are Different","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [1, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=2, candidate N=[0, 3]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.","answer":{"kind":"expression","value":"(4, 2, Yes, [0, 3], No, No, No)","explanation":"Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, Yes, [0, 3], No, No, No)."},"skills":["Containment, Openness, and Neighborhood Status Are Different"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0033","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0ce70ec64884602b86755f98cf64130d6d0a7b8b7b0e29e5ff9de0f0db8c379","family_id":"qf-ext19-neighborhood-qualification","seed":202610192033,"semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[1,3],\"n\":4,\"opens\":[[],[3],[1,3],[0,1,3],[0,1,2,3]],\"x\":3}"},"fields":{"title":"Containment, Openness, and Neighborhood Status Are Different","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=3, candidate N=[1, 3]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.","answer":{"kind":"expression","value":"(4, 3, Yes, [1, 3], Yes, Yes, Yes)","explanation":"Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, Yes, [1, 3], Yes, Yes, Yes)."},"skills":["Containment, Openness, and Neighborhood Status Are Different"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0034","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"639b8cce3cd36845aee681ffc827ec6bfa0a0f07e73fa3a1f9d1a97d3e0932a2","family_id":"qf-ext19-neighborhood-qualification","seed":202610192034,"semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[0,1,3],\"n\":4,\"opens\":[[],[0],[0,3],[0,1,3],[0,1,2,3]],\"x\":0}"},"fields":{"title":"Containment, Openness, and Neighborhood Status Are Different","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=0, candidate N=[0, 1, 3]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.","answer":{"kind":"expression","value":"(4, 0, Yes, [0, 1, 3], Yes, Yes, Yes)","explanation":"Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, Yes, [0, 1, 3], Yes, Yes, Yes)."},"skills":["Containment, Openness, and Neighborhood Status Are Different"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0035","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"727d1ef277f1f26d54b021f1e4ec6be4b555f9228996102ce976d8f830cb24b5","family_id":"qf-ext19-neighborhood-qualification","seed":202610192035,"semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[2,3],\"n\":4,\"opens\":[[],[1],[1,3],[0,1,3],[0,1,2,3]],\"x\":-1}"},"fields":{"title":"Containment, Openness, and Neighborhood Status Are Different","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=-1, candidate N=[2, 3]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.","answer":{"kind":"expression","value":"(4, -1, No, [2, 3], not-evaluated, not-evaluated, not-evaluated)","explanation":"Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, -1, No, [2, 3], not-evaluated, not-evaluated, not-evaluated)."},"skills":["Containment, Openness, and Neighborhood Status Are Different"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0036","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b87b434f591be152457a74b24c38edcaecba1d84ae210ea986f5fc71ed845c1","family_id":"qf-ext19-neighborhood-qualification","seed":202610192036,"semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[0,2,3],\"n\":4,\"opens\":[[],[2],[2,3],[0,2,3],[0,1,2,3]],\"x\":2}"},"fields":{"title":"Containment, Openness, and Neighborhood Status Are Different","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=2, candidate N=[0, 2, 3]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.","answer":{"kind":"expression","value":"(4, 2, Yes, [0, 2, 3], Yes, Yes, Yes)","explanation":"Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, Yes, [0, 2, 3], Yes, Yes, Yes)."},"skills":["Containment, Openness, and Neighborhood Status Are Different"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0037","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d9bc9d2b909357a7f3e8743db802e1ea7ae1e71a003c91d6843d811a7746e92","family_id":"qf-ext19-neighborhood-qualification","seed":202610192037,"semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[1,2,3],\"n\":4,\"opens\":[[],[3],[2,3],[0,2,3],[0,1,2,3]],\"x\":3}"},"fields":{"title":"Containment, Openness, and Neighborhood Status Are Different","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=3, candidate N=[1, 2, 3]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.","answer":{"kind":"expression","value":"(4, 3, Yes, [1, 2, 3], Yes, No, Yes)","explanation":"Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, Yes, [1, 2, 3], Yes, No, Yes)."},"skills":["Containment, Openness, and Neighborhood Status Are Different"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0038","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"279a0927301f9f91eb2f73bdec97d1c582a6689b122c80cd4ab1940bf36ab45c","family_id":"qf-ext19-neighborhood-qualification","seed":202610192038,"semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[0,1,2,3],\"n\":4,\"opens\":[[],[0],[0,1],[0,1,3],[0,1,2,3]],\"x\":0}"},"fields":{"title":"Containment, Openness, and Neighborhood Status Are Different","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=0, candidate N=[0, 1, 2, 3]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.","answer":{"kind":"expression","value":"(4, 0, Yes, [0, 1, 2, 3], Yes, Yes, Yes)","explanation":"Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, Yes, [0, 1, 2, 3], Yes, Yes, Yes)."},"skills":["Containment, Openness, and Neighborhood Status Are Different"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0039","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ff0cafcaf1782b13b1feedd04c3d46e3dda6fdd0c9b343f14da39afb84c5b0a","family_id":"qf-ext19-neighborhood-qualification","seed":202610192039,"semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[],\"n\":4,\"opens\":[[],[1],[0,1],[0,1,3],[0,1,2,3]],\"x\":1}"},"fields":{"title":"Containment, Openness, and Neighborhood Status Are Different","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=1, candidate N=[]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.","answer":{"kind":"expression","value":"(4, 1, Yes, [], No, Yes, No)","explanation":"Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 1, Yes, [], No, Yes, No)."},"skills":["Containment, Openness, and Neighborhood Status Are Different"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0040","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d1264828d9dbd9c281ce3aac80d114a5f67d910806f26d909d294c879b2c5a4","family_id":"qf-ext19-neighborhood-qualification","seed":202610192040,"semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[0],\"n\":4,\"opens\":[[],[2],[0,2],[0,2,3],[0,1,2,3]],\"x\":-1}"},"fields":{"title":"Containment, Openness, and Neighborhood Status Are Different","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=-1, candidate N=[0]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.","answer":{"kind":"expression","value":"(4, -1, No, [0], not-evaluated, not-evaluated, not-evaluated)","explanation":"Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, -1, No, [0], not-evaluated, not-evaluated, not-evaluated)."},"skills":["Containment, Openness, and Neighborhood Status Are Different"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0041","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29510637ca67b559ba45e8200beee5c00bc7c083187af345b0ff4851b2c62c9b","family_id":"qf-ext19-neighborhood-qualification","seed":202610192041,"semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[1],\"n\":4,\"opens\":[[],[3],[0,3],[0,2,3],[0,1,2,3]],\"x\":3}"},"fields":{"title":"Containment, Openness, and Neighborhood Status Are Different","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=3, candidate N=[1]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.","answer":{"kind":"expression","value":"(4, 3, Yes, [1], No, No, No)","explanation":"Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, Yes, [1], No, No, No)."},"skills":["Containment, Openness, and Neighborhood Status Are Different"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0042","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7025c7d134ea911d2d0bba2e8959eeb3539df0be71489db6c523163cf7e715da","family_id":"qf-ext19-neighborhood-qualification","seed":202610192042,"semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[0,1],\"n\":4,\"opens\":[[],[0],[0,2],[0,2,3],[0,1,2,3]],\"x\":0}"},"fields":{"title":"Containment, Openness, and Neighborhood Status Are Different","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=0, candidate N=[0, 1]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.","answer":{"kind":"expression","value":"(4, 0, Yes, [0, 1], Yes, No, Yes)","explanation":"Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, Yes, [0, 1], Yes, No, Yes)."},"skills":["Containment, Openness, and Neighborhood Status Are Different"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0043","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4cb129191b20ff300f91509af114e9729c4b10ff7dc250c934a6313bdfcd65b","family_id":"qf-ext19-neighborhood-qualification","seed":202610192043,"semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[2],\"n\":4,\"opens\":[[],[1],[1,2],[1,2,3],[0,1,2,3]],\"x\":1}"},"fields":{"title":"Containment, Openness, and Neighborhood Status Are Different","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=1, candidate N=[2]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.","answer":{"kind":"expression","value":"(4, 1, Yes, [2], No, No, No)","explanation":"Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 1, Yes, [2], No, No, No)."},"skills":["Containment, Openness, and Neighborhood Status Are Different"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0044","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c083067a8de472a3f9f8e145fff0e7d511c605bae07e74d66d1a2c062cb8f7c4","family_id":"qf-ext19-neighborhood-qualification","seed":202610192044,"semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[0,2],\"n\":4,\"opens\":[[],[2],[1,2],[1,2,3],[0,1,2,3]],\"x\":2}"},"fields":{"title":"Containment, Openness, and Neighborhood Status Are Different","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=2, candidate N=[0, 2]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.","answer":{"kind":"expression","value":"(4, 2, Yes, [0, 2], Yes, No, Yes)","explanation":"Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, Yes, [0, 2], Yes, No, Yes)."},"skills":["Containment, Openness, and Neighborhood Status Are Different"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0045","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6877475a7e2240e510fb5c0a038998b245fbd571defdf88772ca6f8c7c017f36","family_id":"qf-ext19-neighborhood-qualification","seed":202610192045,"semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[1,2],\"n\":4,\"opens\":[[],[3],[1,3],[1,2,3],[0,1,2,3]],\"x\":-1}"},"fields":{"title":"Containment, Openness, and Neighborhood Status Are Different","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=-1, candidate N=[1, 2]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.","answer":{"kind":"expression","value":"(4, -1, No, [1, 2], not-evaluated, not-evaluated, not-evaluated)","explanation":"Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, -1, No, [1, 2], not-evaluated, not-evaluated, not-evaluated)."},"skills":["Containment, Openness, and Neighborhood Status Are Different"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0046","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0bcde5af787c01fd8180fcc7541cce3a3561a6b3a7006c08ef1680e24b2194a5","family_id":"qf-ext19-neighborhood-qualification","seed":202610192046,"semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[0,1,2],\"n\":4,\"opens\":[[],[0],[0,3],[0,2,3],[0,1,2,3]],\"x\":0}"},"fields":{"title":"Containment, Openness, and Neighborhood Status Are Different","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=0, candidate N=[0, 1, 2]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.","answer":{"kind":"expression","value":"(4, 0, Yes, [0, 1, 2], Yes, No, Yes)","explanation":"Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, Yes, [0, 1, 2], Yes, No, Yes)."},"skills":["Containment, Openness, and Neighborhood Status Are Different"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0047","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c323af9cc4253122c95f0fc34303341803ee8159550e00319e56217877ab788b","family_id":"qf-ext19-neighborhood-qualification","seed":202610192047,"semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[3],\"n\":4,\"opens\":[[],[1],[1,3],[1,2,3],[0,1,2,3]],\"x\":1}"},"fields":{"title":"Containment, Openness, and Neighborhood Status Are Different","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=1, candidate N=[3]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.","answer":{"kind":"expression","value":"(4, 1, Yes, [3], No, No, No)","explanation":"Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 1, Yes, [3], No, No, No)."},"skills":["Containment, Openness, and Neighborhood Status Are Different"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0048","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b699127c08a93555530a1633fdb20e1cbf93ef113819ad247b6268f34b81e437","family_id":"qf-ext19-neighborhood-qualification","seed":202610192048,"semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[0,3],\"n\":4,\"opens\":[[],[2],[2,3],[1,2,3],[0,1,2,3]],\"x\":2}"},"fields":{"title":"Containment, Openness, and Neighborhood Status Are Different","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=2, candidate N=[0, 3]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.","answer":{"kind":"expression","value":"(4, 2, Yes, [0, 3], No, No, No)","explanation":"Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, Yes, [0, 3], No, No, No)."},"skills":["Containment, Openness, and Neighborhood Status Are Different"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0049","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1fd069393e89031c262d790ef8e3c1adf368aa84bf7ce6923fd2fa9cf8547d1","family_id":"qf-ext19-neighborhood-qualification","seed":202610192049,"semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[1,3],\"n\":4,\"opens\":[[],[3],[2,3],[1,2,3],[0,1,2,3]],\"x\":3}"},"fields":{"title":"Containment, Openness, and Neighborhood Status Are Different","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=3, candidate N=[1, 3]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.","answer":{"kind":"expression","value":"(4, 3, Yes, [1, 3], Yes, No, Yes)","explanation":"Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, Yes, [1, 3], Yes, No, Yes)."},"skills":["Containment, Openness, and Neighborhood Status Are Different"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0050","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b1ad455f7730fb33724a240d77a187ab26c94e8bfc5e711c8dd19d823100075","family_id":"qf-ext19-neighborhood-qualification","seed":202610192050,"semantic_template":"qf-ext19-neighborhood-qualification/{\"candidate\":[0,1],\"n\":3,\"opens\":[[],[0],[0,1],[0,1,2]],\"x\":-1}"},"fields":{"title":"Containment, Openness, and Neighborhood Status Are Different","stem":"Use the complete finite topology on X=Fin(3)={0,…,2} with opens=[[], [0], [0, 1], [0, 1, 2]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Point x=-1, candidate N=[0, 1]. Give (n, x, point in X?, N, x∈N?, N open?, N a neighborhood?). If x is outside X, use not-evaluated for the final three claims.","answer":{"kind":"expression","value":"(3, -1, No, [0, 1], not-evaluated, not-evaluated, not-evaluated)","explanation":"Membership alone is insufficient; a neighborhood contains a point-containing open set and may itself be nonopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, -1, No, [0, 1], not-evaluated, not-evaluated, not-evaluated)."},"skills":["Containment, Openness, and Neighborhood Status Are Different"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0051","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24ad60e771f1ac90ea9c29b5cf71f871d0eaf27fecb483a939e77678a790cbcc","family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192051,"semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[0],[0,1],[0,1,2],[0,1,2,3]],\"small\":[0],\"x\":0}"},"fields":{"title":"Neighborhood Upward-Closure Guards","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=0, small=[0], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).","answer":{"kind":"expression","value":"(4, 0, Yes, [0], [0, 1, 2, 3], Yes, Yes, Yes)","explanation":"Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, Yes, [0], [0, 1, 2, 3], Yes, Yes, Yes)."},"skills":["Neighborhood Upward-Closure Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0052","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90595db4a0b3f2ca8189254754d10fd3848b7a2892fd740b2812def15850506a","family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192052,"semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[1],[0,1],[0,1,2],[0,1,2,3]],\"small\":[1],\"x\":1}"},"fields":{"title":"Neighborhood Upward-Closure Guards","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=1, small=[1], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).","answer":{"kind":"expression","value":"(4, 1, Yes, [1], [0, 1, 2, 3], Yes, Yes, Yes)","explanation":"Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 1, Yes, [1], [0, 1, 2, 3], Yes, Yes, Yes)."},"skills":["Neighborhood Upward-Closure Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0053","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb5bf64458d05953921a0511c7f7a93a7f6dd7efacf3f74a20c0c8f86c317789","family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192053,"semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[2],[0,2],[0,1,2],[0,1,2,3]],\"small\":[2],\"x\":2}"},"fields":{"title":"Neighborhood Upward-Closure Guards","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [0, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=2, small=[2], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).","answer":{"kind":"expression","value":"(4, 2, Yes, [2], [0, 1, 2, 3], Yes, Yes, Yes)","explanation":"Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, Yes, [2], [0, 1, 2, 3], Yes, Yes, Yes)."},"skills":["Neighborhood Upward-Closure Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0054","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d95268dc7ac996f65bad06757145526903c0efd519214ac422fe205298741cc9","family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192054,"semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[3],[0,3],[0,1,3],[0,1,2,3]],\"small\":[3],\"x\":3}"},"fields":{"title":"Neighborhood Upward-Closure Guards","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=3, small=[3], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).","answer":{"kind":"expression","value":"(4, 3, Yes, [3], [0, 1, 2, 3], Yes, Yes, Yes)","explanation":"Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, Yes, [3], [0, 1, 2, 3], Yes, Yes, Yes)."},"skills":["Neighborhood Upward-Closure Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0055","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4027696bc8a9053e22ea6ca729b4f60a44ba6e7c6b362f867d0474c93d532618","family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192055,"semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[0],[0,2],[0,1,2],[0,1,2,3]],\"small\":[0],\"x\":-1}"},"fields":{"title":"Neighborhood Upward-Closure Guards","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=-1, small=[0], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).","answer":{"kind":"expression","value":"(4, -1, No, [0], [0, 1, 2, 3], Yes, not-evaluated, not-evaluated)","explanation":"Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, -1, No, [0], [0, 1, 2, 3], Yes, not-evaluated, not-evaluated)."},"skills":["Neighborhood Upward-Closure Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0056","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08dff28747dd381c6e0c938d33e4e204295d4a8e2705695c78363edd4a77f7ba","family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192056,"semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[1],[1,2],[0,1,2],[0,1,2,3]],\"small\":[1],\"x\":1}"},"fields":{"title":"Neighborhood Upward-Closure Guards","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=1, small=[1], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).","answer":{"kind":"expression","value":"(4, 1, Yes, [1], [0, 1, 2, 3], Yes, Yes, Yes)","explanation":"Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 1, Yes, [1], [0, 1, 2, 3], Yes, Yes, Yes)."},"skills":["Neighborhood Upward-Closure Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0057","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9969196d89dd2cd74a71c841061adbadc7aa46f17a5749e8383799020a14cbc","family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192057,"semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[2],[1,2],[0,1,2],[0,1,2,3]],\"small\":[2],\"x\":2}"},"fields":{"title":"Neighborhood Upward-Closure Guards","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [1, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=2, small=[2], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).","answer":{"kind":"expression","value":"(4, 2, Yes, [2], [0, 1, 2, 3], Yes, Yes, Yes)","explanation":"Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, Yes, [2], [0, 1, 2, 3], Yes, Yes, Yes)."},"skills":["Neighborhood Upward-Closure Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0058","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c831fd0e32c3a2345eef5fbd833de4e6be55de3a08839a06b5316113a981cdb","family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192058,"semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[3],[1,3],[0,1,3],[0,1,2,3]],\"small\":[3],\"x\":3}"},"fields":{"title":"Neighborhood Upward-Closure Guards","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=3, small=[3], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).","answer":{"kind":"expression","value":"(4, 3, Yes, [3], [0, 1, 2, 3], Yes, Yes, Yes)","explanation":"Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, Yes, [3], [0, 1, 2, 3], Yes, Yes, Yes)."},"skills":["Neighborhood Upward-Closure Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0059","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79669d95f895ce1a855f87350c7bd8f4a2067d90e8be1782c6fe8073ab9238d3","family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192059,"semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[0],[0,3],[0,1,3],[0,1,2,3]],\"small\":[0],\"x\":0}"},"fields":{"title":"Neighborhood Upward-Closure Guards","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=0, small=[0], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).","answer":{"kind":"expression","value":"(4, 0, Yes, [0], [0, 1, 2, 3], Yes, Yes, Yes)","explanation":"Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, Yes, [0], [0, 1, 2, 3], Yes, Yes, Yes)."},"skills":["Neighborhood Upward-Closure Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0060","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa79ae106b2fba1c3d50834c3da538d33b59525ccf47ea0aea721ac65a3c5a96","family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192060,"semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[1],[1,3],[0,1,3],[0,1,2,3]],\"small\":[0,1,3],\"x\":-1}"},"fields":{"title":"Neighborhood Upward-Closure Guards","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=-1, small=[0, 1, 3], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).","answer":{"kind":"expression","value":"(4, -1, No, [0, 1, 3], [0, 1, 2, 3], Yes, not-evaluated, not-evaluated)","explanation":"Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, -1, No, [0, 1, 3], [0, 1, 2, 3], Yes, not-evaluated, not-evaluated)."},"skills":["Neighborhood Upward-Closure Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0061","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2a6b4a1323aeb6083d4449376ddc4a9ce243ff0c4261afea566ce91034c9c4c","family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192061,"semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[2],[2,3],[0,2,3],[0,1,2,3]],\"small\":[2],\"x\":2}"},"fields":{"title":"Neighborhood Upward-Closure Guards","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=2, small=[2], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).","answer":{"kind":"expression","value":"(4, 2, Yes, [2], [0, 1, 2, 3], Yes, Yes, Yes)","explanation":"Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, Yes, [2], [0, 1, 2, 3], Yes, Yes, Yes)."},"skills":["Neighborhood Upward-Closure Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0062","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b5f27eb5e0f4ba057c3854e73df295fe74e093a08139e5380fb50728c39cc52","family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192062,"semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[3],[2,3],[0,2,3],[0,1,2,3]],\"small\":[3],\"x\":3}"},"fields":{"title":"Neighborhood Upward-Closure Guards","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=3, small=[3], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).","answer":{"kind":"expression","value":"(4, 3, Yes, [3], [0, 1, 2, 3], Yes, Yes, Yes)","explanation":"Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, Yes, [3], [0, 1, 2, 3], Yes, Yes, Yes)."},"skills":["Neighborhood Upward-Closure Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0063","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb389c7cc13a25f91e47a40d53d525defaec82a29ea2257363fb97387afacbdc","family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192063,"semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[0],[0,1],[0,1,3],[0,1,2,3]],\"small\":[0],\"x\":0}"},"fields":{"title":"Neighborhood Upward-Closure Guards","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=0, small=[0], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).","answer":{"kind":"expression","value":"(4, 0, Yes, [0], [0, 1, 2, 3], Yes, Yes, Yes)","explanation":"Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, Yes, [0], [0, 1, 2, 3], Yes, Yes, Yes)."},"skills":["Neighborhood Upward-Closure Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0064","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11edf6dbcc489e55dabc4ac61c21048b97ddf328f8d2ab1363ac26260a15b78e","family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192064,"semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[1],[0,1],[0,1,3],[0,1,2,3]],\"small\":[1],\"x\":1}"},"fields":{"title":"Neighborhood Upward-Closure Guards","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=1, small=[1], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).","answer":{"kind":"expression","value":"(4, 1, Yes, [1], [0, 1, 2, 3], Yes, Yes, Yes)","explanation":"Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 1, Yes, [1], [0, 1, 2, 3], Yes, Yes, Yes)."},"skills":["Neighborhood Upward-Closure Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0065","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c36cf36b40910188e094276f73d7a9314f358306523c6a052e8233f57caede17","family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192065,"semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[2],[0,2],[0,2,3],[0,1,2,3]],\"small\":[0,2],\"x\":-1}"},"fields":{"title":"Neighborhood Upward-Closure Guards","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=-1, small=[0, 2], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).","answer":{"kind":"expression","value":"(4, -1, No, [0, 2], [0, 1, 2, 3], Yes, not-evaluated, not-evaluated)","explanation":"Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, -1, No, [0, 2], [0, 1, 2, 3], Yes, not-evaluated, not-evaluated)."},"skills":["Neighborhood Upward-Closure Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0066","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"09cde8a03e2bb45a00d6b986aec3b935226ab39b07b7547f087ead405e168c4a","family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192066,"semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[3],[0,3],[0,2,3],[0,1,2,3]],\"small\":[3],\"x\":3}"},"fields":{"title":"Neighborhood Upward-Closure Guards","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=3, small=[3], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).","answer":{"kind":"expression","value":"(4, 3, Yes, [3], [0, 1, 2, 3], Yes, Yes, Yes)","explanation":"Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, Yes, [3], [0, 1, 2, 3], Yes, Yes, Yes)."},"skills":["Neighborhood Upward-Closure Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0067","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc1fd1ee447623e77b1335bc192fedd020af6d8c55aece80623ad0c29e54be7d","family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192067,"semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[0],[0,2],[0,2,3],[0,1,2,3]],\"small\":[0],\"x\":0}"},"fields":{"title":"Neighborhood Upward-Closure Guards","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=0, small=[0], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).","answer":{"kind":"expression","value":"(4, 0, Yes, [0], [0, 1, 2, 3], Yes, Yes, Yes)","explanation":"Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, Yes, [0], [0, 1, 2, 3], Yes, Yes, Yes)."},"skills":["Neighborhood Upward-Closure Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0068","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e97898a6ba7aacec0f60870708366ee00d151dc5ae80ef94039c0c3e6c7ad5f","family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192068,"semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[1],[1,2],[1,2,3],[0,1,2,3]],\"small\":[1],\"x\":1}"},"fields":{"title":"Neighborhood Upward-Closure Guards","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=1, small=[1], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).","answer":{"kind":"expression","value":"(4, 1, Yes, [1], [0, 1, 2, 3], Yes, Yes, Yes)","explanation":"Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 1, Yes, [1], [0, 1, 2, 3], Yes, Yes, Yes)."},"skills":["Neighborhood Upward-Closure Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0069","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0fa148a33793a001fae0935bca728ad0ad83a54aa26f289e0b01ecfcd1992c46","family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192069,"semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[2],[1,2],[1,2,3],[0,1,2,3]],\"small\":[2],\"x\":2}"},"fields":{"title":"Neighborhood Upward-Closure Guards","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=2, small=[2], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).","answer":{"kind":"expression","value":"(4, 2, Yes, [2], [0, 1, 2, 3], Yes, Yes, Yes)","explanation":"Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, Yes, [2], [0, 1, 2, 3], Yes, Yes, Yes)."},"skills":["Neighborhood Upward-Closure Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0070","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d89699cccd07a6c610ad6fe1141bf0834d26acf2b38ac1f2aa77c0c92d243ee","family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192070,"semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[3],[1,3],[1,2,3],[0,1,2,3]],\"small\":[0,1,2,3],\"x\":-1}"},"fields":{"title":"Neighborhood Upward-Closure Guards","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=-1, small=[0, 1, 2, 3], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).","answer":{"kind":"expression","value":"(4, -1, No, [0, 1, 2, 3], [0, 1, 2, 3], Yes, not-evaluated, not-evaluated)","explanation":"Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, -1, No, [0, 1, 2, 3], [0, 1, 2, 3], Yes, not-evaluated, not-evaluated)."},"skills":["Neighborhood Upward-Closure Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0071","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c54d7ee644230ee0528f71bf499cc521cab7fb0409eb2b7fb4c2ad6c6435ffb","family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192071,"semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[0],[0,3],[0,2,3],[0,1,2,3]],\"small\":[0],\"x\":0}"},"fields":{"title":"Neighborhood Upward-Closure Guards","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=0, small=[0], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).","answer":{"kind":"expression","value":"(4, 0, Yes, [0], [0, 1, 2, 3], Yes, Yes, Yes)","explanation":"Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, Yes, [0], [0, 1, 2, 3], Yes, Yes, Yes)."},"skills":["Neighborhood Upward-Closure Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0072","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1afa461201d9cf3e12c814488fc5b1ac7840a15c6ada1e7ef13637865124e96","family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192072,"semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[1],[1,3],[1,2,3],[0,1,2,3]],\"small\":[1],\"x\":1}"},"fields":{"title":"Neighborhood Upward-Closure Guards","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=1, small=[1], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).","answer":{"kind":"expression","value":"(4, 1, Yes, [1], [0, 1, 2, 3], Yes, Yes, Yes)","explanation":"Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 1, Yes, [1], [0, 1, 2, 3], Yes, Yes, Yes)."},"skills":["Neighborhood Upward-Closure Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0073","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f778731638b5f810602701de955b9307147201131cdf0f26aa5263329c84fc68","family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192073,"semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[2],[2,3],[1,2,3],[0,1,2,3]],\"small\":[2],\"x\":2}"},"fields":{"title":"Neighborhood Upward-Closure Guards","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=2, small=[2], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).","answer":{"kind":"expression","value":"(4, 2, Yes, [2], [0, 1, 2, 3], Yes, Yes, Yes)","explanation":"Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, Yes, [2], [0, 1, 2, 3], Yes, Yes, Yes)."},"skills":["Neighborhood Upward-Closure Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0074","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"201b617470d7f14ec672579076b3b1a8a9dd64ce73e6d9bd295fb901d0581275","family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192074,"semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2,3],\"n\":4,\"opens\":[[],[3],[2,3],[1,2,3],[0,1,2,3]],\"small\":[3],\"x\":3}"},"fields":{"title":"Neighborhood Upward-Closure Guards","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=3, small=[3], large=[0, 1, 2, 3]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).","answer":{"kind":"expression","value":"(4, 3, Yes, [3], [0, 1, 2, 3], Yes, Yes, Yes)","explanation":"Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, Yes, [3], [0, 1, 2, 3], Yes, Yes, Yes)."},"skills":["Neighborhood Upward-Closure Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0075","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3707b094714bbfd005aacd553712090806382402e25b616f8adb205a65d4b823","family_id":"qf-ext19-neighborhood-upward-closure","seed":202610192075,"semantic_template":"qf-ext19-neighborhood-upward-closure/{\"large\":[0,1,2],\"n\":3,\"opens\":[[],[0],[0,1],[0,1,2]],\"small\":[0],\"x\":-1}"},"fields":{"title":"Neighborhood Upward-Closure Guards","stem":"Use the complete finite topology on X=Fin(3)={0,…,2} with opens=[[], [0], [0, 1], [0, 1, 2]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=-1, small=[0], large=[0, 1, 2]. Give (n, x, point typed?, small, large, small⊆large?, small neighborhood? or not-evaluated, large neighborhood? only if the point is typed, inclusion holds and small is a neighborhood; otherwise not-evaluated).","answer":{"kind":"expression","value":"(3, -1, No, [0], [0, 1, 2], Yes, not-evaluated, not-evaluated)","explanation":"Do not evaluate the final conditional obligation before all its stated premises qualify. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, -1, No, [0], [0, 1, 2], Yes, not-evaluated, not-evaluated)."},"skills":["Neighborhood Upward-Closure Guards"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0076","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f4872b9568ae70e5dd3c64f5491c068549f21da7291cd89e0607ffda56ffeb0","family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192076,"semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[0],[0,1],[0,1,2],[0,1,2,3]],\"x\":0}"},"fields":{"title":"All Neighborhoods of a Qualified Finite Point","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=0 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.","answer":{"kind":"expression","value":"(4, 0, [[0], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]])","explanation":"The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, [[0], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]])."},"skills":["All Neighborhoods of a Qualified Finite Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0077","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c66b324b8aa64117e807d73e415defabe150db05d5db9121f51208c5d8118679","family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192077,"semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[1],[0,1],[0,1,2],[0,1,2,3]],\"x\":1}"},"fields":{"title":"All Neighborhoods of a Qualified Finite Point","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=1 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.","answer":{"kind":"expression","value":"(4, 1, [[1], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[1], [0, 1], [1, 2], [0, 1, 2], [1, 3], [0, 1, 3], [1, 2, 3], [0, 1, 2, 3]])","explanation":"The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 1, [[1], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[1], [0, 1], [1, 2], [0, 1, 2], [1, 3], [0, 1, 3], [1, 2, 3], [0, 1, 2, 3]])."},"skills":["All Neighborhoods of a Qualified Finite Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0078","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c6d91eb0b9bd69a7df88e5dacc7142f11712be546e67b7dea07e707f6f8d604","family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192078,"semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[2],[0,2],[0,1,2],[0,1,2,3]],\"x\":2}"},"fields":{"title":"All Neighborhoods of a Qualified Finite Point","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [0, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=2 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.","answer":{"kind":"expression","value":"(4, 2, [[2], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[2], [0, 2], [1, 2], [0, 1, 2], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])","explanation":"The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, [[2], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[2], [0, 2], [1, 2], [0, 1, 2], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])."},"skills":["All Neighborhoods of a Qualified Finite Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0079","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba95d01edccf5dc197c2bc3ea661899f34bdc2e3adfa47573b0bd332b88a3227","family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192079,"semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[3],[0,3],[0,1,3],[0,1,2,3]],\"x\":3}"},"fields":{"title":"All Neighborhoods of a Qualified Finite Point","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=3 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.","answer":{"kind":"expression","value":"(4, 3, [[3], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])","explanation":"The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, [[3], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])."},"skills":["All Neighborhoods of a Qualified Finite Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0080","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"26bed98b748dcde4ba8625fde25092689453b50e780c2a0f2b4f310507b6fc05","family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192080,"semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[0],[0,2],[0,1,2],[0,1,2,3]],\"x\":0}"},"fields":{"title":"All Neighborhoods of a Qualified Finite Point","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=0 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.","answer":{"kind":"expression","value":"(4, 0, [[0], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]])","explanation":"The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, [[0], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]])."},"skills":["All Neighborhoods of a Qualified Finite Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0081","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c7004f4d7775aa2323c7666590f11f897f038614af9638204db12a78836da07","family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192081,"semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[1],[1,2],[0,1,2],[0,1,2,3]],\"x\":1}"},"fields":{"title":"All Neighborhoods of a Qualified Finite Point","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=1 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.","answer":{"kind":"expression","value":"(4, 1, [[1], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[1], [0, 1], [1, 2], [0, 1, 2], [1, 3], [0, 1, 3], [1, 2, 3], [0, 1, 2, 3]])","explanation":"The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 1, [[1], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[1], [0, 1], [1, 2], [0, 1, 2], [1, 3], [0, 1, 3], [1, 2, 3], [0, 1, 2, 3]])."},"skills":["All Neighborhoods of a Qualified Finite Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0082","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"809f598ca18eb064499b3c22a3840ba6d2ece487a7e6bdfb7f2801d4c6303d5b","family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192082,"semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[2],[1,2],[0,1,2],[0,1,2,3]],\"x\":2}"},"fields":{"title":"All Neighborhoods of a Qualified Finite Point","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [1, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=2 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.","answer":{"kind":"expression","value":"(4, 2, [[2], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[2], [0, 2], [1, 2], [0, 1, 2], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])","explanation":"The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, [[2], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[2], [0, 2], [1, 2], [0, 1, 2], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])."},"skills":["All Neighborhoods of a Qualified Finite Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0083","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8413c8babb22ca50b0a1ac260a8c1dc496caf2355e3f018e661574c465c03c1f","family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192083,"semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[3],[1,3],[0,1,3],[0,1,2,3]],\"x\":3}"},"fields":{"title":"All Neighborhoods of a Qualified Finite Point","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=3 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.","answer":{"kind":"expression","value":"(4, 3, [[3], [1, 3], [0, 1, 3], [0, 1, 2, 3]], [[3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])","explanation":"The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, [[3], [1, 3], [0, 1, 3], [0, 1, 2, 3]], [[3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])."},"skills":["All Neighborhoods of a Qualified Finite Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0084","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59fee72fee46bf24e4f4ecc59b3a331337960afaef2437ef0ea237a431c81e54","family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192084,"semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[0],[0,3],[0,1,3],[0,1,2,3]],\"x\":0}"},"fields":{"title":"All Neighborhoods of a Qualified Finite Point","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=0 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.","answer":{"kind":"expression","value":"(4, 0, [[0], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]])","explanation":"The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, [[0], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]])."},"skills":["All Neighborhoods of a Qualified Finite Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0085","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2bb37f339be23b78934bce6a24ecb0c349c8252a73275551e25c134fbdbf0a93","family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192085,"semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[1],[1,3],[0,1,3],[0,1,2,3]],\"x\":0}"},"fields":{"title":"All Neighborhoods of a Qualified Finite Point","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=0 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.","answer":{"kind":"expression","value":"(4, 0, [[0, 1, 3], [0, 1, 2, 3]], [[0, 1, 3], [0, 1, 2, 3]])","explanation":"The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, [[0, 1, 3], [0, 1, 2, 3]], [[0, 1, 3], [0, 1, 2, 3]])."},"skills":["All Neighborhoods of a Qualified Finite Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0086","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"98356afd60eebbd06c2a83ed0eee5b49e72bbdbad8bde9b8b0f4f8464b32f211","family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192086,"semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[2],[2,3],[0,2,3],[0,1,2,3]],\"x\":2}"},"fields":{"title":"All Neighborhoods of a Qualified Finite Point","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=2 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.","answer":{"kind":"expression","value":"(4, 2, [[2], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[2], [0, 2], [1, 2], [0, 1, 2], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])","explanation":"The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, [[2], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[2], [0, 2], [1, 2], [0, 1, 2], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])."},"skills":["All Neighborhoods of a Qualified Finite Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0087","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b79622d68b396537eb3f1598c525031e5b2a0233bdde2c50b020e1f257cb509","family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192087,"semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[3],[2,3],[0,2,3],[0,1,2,3]],\"x\":3}"},"fields":{"title":"All Neighborhoods of a Qualified Finite Point","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=3 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.","answer":{"kind":"expression","value":"(4, 3, [[3], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])","explanation":"The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, [[3], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])."},"skills":["All Neighborhoods of a Qualified Finite Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0088","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"261878b3256abb63bc9596747f62bae51797d209f7a1f7d4f5b373e9a1646de5","family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192088,"semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[0],[0,1],[0,1,3],[0,1,2,3]],\"x\":0}"},"fields":{"title":"All Neighborhoods of a Qualified Finite Point","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=0 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.","answer":{"kind":"expression","value":"(4, 0, [[0], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]])","explanation":"The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, [[0], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]])."},"skills":["All Neighborhoods of a Qualified Finite Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0089","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63fa889014558b6a469ae34f181c8de1ff90199656eacbca2b82d388e3761a92","family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192089,"semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[1],[0,1],[0,1,3],[0,1,2,3]],\"x\":1}"},"fields":{"title":"All Neighborhoods of a Qualified Finite Point","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=1 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.","answer":{"kind":"expression","value":"(4, 1, [[1], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[1], [0, 1], [1, 2], [0, 1, 2], [1, 3], [0, 1, 3], [1, 2, 3], [0, 1, 2, 3]])","explanation":"The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 1, [[1], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[1], [0, 1], [1, 2], [0, 1, 2], [1, 3], [0, 1, 3], [1, 2, 3], [0, 1, 2, 3]])."},"skills":["All Neighborhoods of a Qualified Finite Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0090","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7ee36d4110345a4f4f61f28e0b4eb17c77bc63ca448bd98d30b21fdfca04a9a","family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192090,"semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[2],[0,2],[0,2,3],[0,1,2,3]],\"x\":0}"},"fields":{"title":"All Neighborhoods of a Qualified Finite Point","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=0 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.","answer":{"kind":"expression","value":"(4, 0, [[0, 2], [0, 2, 3], [0, 1, 2, 3]], [[0, 2], [0, 1, 2], [0, 2, 3], [0, 1, 2, 3]])","explanation":"The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, [[0, 2], [0, 2, 3], [0, 1, 2, 3]], [[0, 2], [0, 1, 2], [0, 2, 3], [0, 1, 2, 3]])."},"skills":["All Neighborhoods of a Qualified Finite Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0091","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75f2f756c2ac0cb7e7b99961bfbf8c620723c4f2d530e49de9f24fde4239500f","family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192091,"semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[3],[0,3],[0,2,3],[0,1,2,3]],\"x\":3}"},"fields":{"title":"All Neighborhoods of a Qualified Finite Point","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=3 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.","answer":{"kind":"expression","value":"(4, 3, [[3], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])","explanation":"The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, [[3], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])."},"skills":["All Neighborhoods of a Qualified Finite Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0092","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd21ae411d84805876b40ec5bb02222615600efbd56eaa22a83461c70c9cb5cb","family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192092,"semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[0],[0,2],[0,2,3],[0,1,2,3]],\"x\":0}"},"fields":{"title":"All Neighborhoods of a Qualified Finite Point","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=0 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.","answer":{"kind":"expression","value":"(4, 0, [[0], [0, 2], [0, 2, 3], [0, 1, 2, 3]], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]])","explanation":"The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, [[0], [0, 2], [0, 2, 3], [0, 1, 2, 3]], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]])."},"skills":["All Neighborhoods of a Qualified Finite Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0093","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71480066e9fd5155a9a88b0e657abf3cdb3a6b4d088003b9ed383dc768686e7e","family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192093,"semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[1],[1,2],[1,2,3],[0,1,2,3]],\"x\":1}"},"fields":{"title":"All Neighborhoods of a Qualified Finite Point","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=1 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.","answer":{"kind":"expression","value":"(4, 1, [[1], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[1], [0, 1], [1, 2], [0, 1, 2], [1, 3], [0, 1, 3], [1, 2, 3], [0, 1, 2, 3]])","explanation":"The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 1, [[1], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[1], [0, 1], [1, 2], [0, 1, 2], [1, 3], [0, 1, 3], [1, 2, 3], [0, 1, 2, 3]])."},"skills":["All Neighborhoods of a Qualified Finite Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0094","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8293460fe83bd6dc5ae9f350a8427f96506ef838a6e0230730cd8d1902e2aa61","family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192094,"semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[2],[1,2],[1,2,3],[0,1,2,3]],\"x\":2}"},"fields":{"title":"All Neighborhoods of a Qualified Finite Point","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=2 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.","answer":{"kind":"expression","value":"(4, 2, [[2], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[2], [0, 2], [1, 2], [0, 1, 2], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])","explanation":"The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, [[2], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[2], [0, 2], [1, 2], [0, 1, 2], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])."},"skills":["All Neighborhoods of a Qualified Finite Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0095","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"928082f9479cfb9f6ed94aa5bf9271ff03637c0261663354f2645d6d152df3fe","family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192095,"semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[3],[1,3],[1,2,3],[0,1,2,3]],\"x\":0}"},"fields":{"title":"All Neighborhoods of a Qualified Finite Point","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=0 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.","answer":{"kind":"expression","value":"(4, 0, [[0, 1, 2, 3]], [[0, 1, 2, 3]])","explanation":"The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, [[0, 1, 2, 3]], [[0, 1, 2, 3]])."},"skills":["All Neighborhoods of a Qualified Finite Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0096","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6f219cd2d6fddb8f8f1ac1d9363822c74233465f3b04bd102e8e43cad88517f","family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192096,"semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[0],[0,3],[0,2,3],[0,1,2,3]],\"x\":0}"},"fields":{"title":"All Neighborhoods of a Qualified Finite Point","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=0 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.","answer":{"kind":"expression","value":"(4, 0, [[0], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]])","explanation":"The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 0, [[0], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]])."},"skills":["All Neighborhoods of a Qualified Finite Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0097","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58d04c777fa61a7f95ebf599dd7ecf6d94b1cdec8adc244c9b63e02859294abe","family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192097,"semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[1],[1,3],[1,2,3],[0,1,2,3]],\"x\":1}"},"fields":{"title":"All Neighborhoods of a Qualified Finite Point","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=1 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.","answer":{"kind":"expression","value":"(4, 1, [[1], [1, 3], [1, 2, 3], [0, 1, 2, 3]], [[1], [0, 1], [1, 2], [0, 1, 2], [1, 3], [0, 1, 3], [1, 2, 3], [0, 1, 2, 3]])","explanation":"The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 1, [[1], [1, 3], [1, 2, 3], [0, 1, 2, 3]], [[1], [0, 1], [1, 2], [0, 1, 2], [1, 3], [0, 1, 3], [1, 2, 3], [0, 1, 2, 3]])."},"skills":["All Neighborhoods of a Qualified Finite Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0098","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cbaa299b785c07a7662b972e0d18224d378a0d1ba78d2f28d0663fc0ebdc7255","family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192098,"semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[2],[2,3],[1,2,3],[0,1,2,3]],\"x\":2}"},"fields":{"title":"All Neighborhoods of a Qualified Finite Point","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=2 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.","answer":{"kind":"expression","value":"(4, 2, [[2], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[2], [0, 2], [1, 2], [0, 1, 2], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])","explanation":"The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 2, [[2], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[2], [0, 2], [1, 2], [0, 1, 2], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])."},"skills":["All Neighborhoods of a Qualified Finite Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0099","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"445bc9b258a67dee6cd25a2048a59d25be54865a3d534090b0a017f8760bdaba","family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192099,"semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":4,\"opens\":[[],[3],[2,3],[1,2,3],[0,1,2,3]],\"x\":3}"},"fields":{"title":"All Neighborhoods of a Qualified Finite Point","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=3 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.","answer":{"kind":"expression","value":"(4, 3, [[3], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])","explanation":"The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, 3, [[3], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]])."},"skills":["All Neighborhoods of a Qualified Finite Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r02-neighborhood-0100","batch_id":"batch-20261013-advance-20261019-r02-neighborhood-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca8505da6c347cf8da1fadafa789ad7a3aa14a9e33cab736990b42f5b8753c44","family_id":"qf-ext19-neighborhood-complete-inventory","seed":202610192100,"semantic_template":"qf-ext19-neighborhood-complete-inventory/{\"n\":3,\"opens\":[[],[0],[0,1],[0,1,2]],\"x\":0}"},"fields":{"title":"All Neighborhoods of a Qualified Finite Point","stem":"Use the complete finite topology on X=Fin(3)={0,…,2} with opens=[[], [0], [0, 1], [0, 1, 2]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. x=0 is required to lie in X. Give (n, x, all open sets containing x, all neighborhoods of x). Enumerate the entire powerset in subset-mask order.","answer":{"kind":"expression","value":"(3, 0, [[0], [0, 1], [0, 1, 2]], [[0], [0, 1], [0, 2], [0, 1, 2]])","explanation":"The open witnesses form an open neighborhood base, not a promised unique minimal neighborhood; retain nonopen neighborhoods too. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, 0, [[0], [0, 1], [0, 1, 2]], [[0], [0, 1], [0, 2], [0, 1, 2]])."},"skills":["All Neighborhoods of a Qualified Finite Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}