{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0001","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff6ab432dd741ba9122a02e921725d149966caed7938becd3a4e7b14086a6c79","family_id":"qf-mod48-component-connected-subsets","seed":202611177001,"semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1],[2,3,4,5]]}"},"fields":{"title":"The Union of All Connected Subsets through a Basepoint Is Its Component","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1], [2, 3, 4, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).","answer":{"kind":"expression","value":"(0, [[0], [0, 1]], [0, 1], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation":"Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 1]], [0, 1], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)."},"skills":["The Union of All Connected Subsets through a Basepoint Is Its Component"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0002","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57bb15c003a91014198ec8b13e832bb879271ba48b81cc33f03c59c728d1c31b","family_id":"qf-mod48-component-connected-subsets","seed":202611177002,"semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2],[1,3,4,5]]}"},"fields":{"title":"The Union of All Connected Subsets through a Basepoint Is Its Component","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2], [1, 3, 4, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).","answer":{"kind":"expression","value":"(0, [[0], [0, 2]], [0, 2], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation":"Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 2]], [0, 2], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)."},"skills":["The Union of All Connected Subsets through a Basepoint Is Its Component"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0003","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"edb415050ad48bfdc5ae29cac50249508286469fc0116dd4035df704c31d0db8","family_id":"qf-mod48-component-connected-subsets","seed":202611177003,"semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2],[3,4,5]]}"},"fields":{"title":"The Union of All Connected Subsets through a Basepoint Is Its Component","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2], [3, 4, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).","answer":{"kind":"expression","value":"(0, [[0], [0, 1], [0, 2], [0, 1, 2]], [0, 1, 2], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation":"Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 1], [0, 2], [0, 1, 2]], [0, 1, 2], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)."},"skills":["The Union of All Connected Subsets through a Basepoint Is Its Component"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0004","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e1c57b0fa225944f2fbb67e607cb74c3751e392e1b27d06a783f996c15824eb","family_id":"qf-mod48-component-connected-subsets","seed":202611177004,"semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,3],[1,2,4,5]]}"},"fields":{"title":"The Union of All Connected Subsets through a Basepoint Is Its Component","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 3], [1, 2, 4, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).","answer":{"kind":"expression","value":"(0, [[0], [0, 3]], [0, 3], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation":"Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 3]], [0, 3], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)."},"skills":["The Union of All Connected Subsets through a Basepoint Is Its Component"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0005","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f28273f0b2feb3943954beaf532fb62188369295f3b6b69b757467d57b01a061","family_id":"qf-mod48-component-connected-subsets","seed":202611177005,"semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,3],[2,4,5]]}"},"fields":{"title":"The Union of All Connected Subsets through a Basepoint Is Its Component","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 3], [2, 4, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).","answer":{"kind":"expression","value":"(0, [[0], [0, 1], [0, 3], [0, 1, 3]], [0, 1, 3], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation":"Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 1], [0, 3], [0, 1, 3]], [0, 1, 3], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)."},"skills":["The Union of All Connected Subsets through a Basepoint Is Its Component"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0006","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b16f0b660e7ff03786f22c84dcf11b5c41d94b31bfd2bad89c94c6e41f82609","family_id":"qf-mod48-component-connected-subsets","seed":202611177006,"semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2,3],[1,4,5]]}"},"fields":{"title":"The Union of All Connected Subsets through a Basepoint Is Its Component","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 3], [1, 4, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).","answer":{"kind":"expression","value":"(0, [[0], [0, 2], [0, 3], [0, 2, 3]], [0, 2, 3], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation":"Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 2], [0, 3], [0, 2, 3]], [0, 2, 3], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)."},"skills":["The Union of All Connected Subsets through a Basepoint Is Its Component"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0007","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01da8868b92b22ceea2c0d95873b3c581d25b26f31936473a298dc051cbddcd7","family_id":"qf-mod48-component-connected-subsets","seed":202611177007,"semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2,3],[4,5]]}"},"fields":{"title":"The Union of All Connected Subsets through a Basepoint Is Its Component","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3], [4, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).","answer":{"kind":"expression","value":"(0, [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]], [0, 1, 2, 3], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation":"Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]], [0, 1, 2, 3], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)."},"skills":["The Union of All Connected Subsets through a Basepoint Is Its Component"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0008","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"719dc90f5b85385021b8918adb3111b1b89b67fe52bdd6317374e26ff666ec28","family_id":"qf-mod48-component-connected-subsets","seed":202611177008,"semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,4],[1,2,3,5]]}"},"fields":{"title":"The Union of All Connected Subsets through a Basepoint Is Its Component","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 4], [1, 2, 3, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).","answer":{"kind":"expression","value":"(0, [[0], [0, 4]], [0, 4], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation":"Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 4]], [0, 4], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)."},"skills":["The Union of All Connected Subsets through a Basepoint Is Its Component"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0009","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c65c8a0cd9aa14a987243f4712d98a64097f48b7d74c012e639b5baa82930eee","family_id":"qf-mod48-component-connected-subsets","seed":202611177009,"semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,4],[2,3,5]]}"},"fields":{"title":"The Union of All Connected Subsets through a Basepoint Is Its Component","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 4], [2, 3, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).","answer":{"kind":"expression","value":"(0, [[0], [0, 1], [0, 4], [0, 1, 4]], [0, 1, 4], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation":"Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 1], [0, 4], [0, 1, 4]], [0, 1, 4], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)."},"skills":["The Union of All Connected Subsets through a Basepoint Is Its Component"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0010","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5e28e0420f163387e559346f77e255f98054ab968b322d7682c1ea9b905faf5","family_id":"qf-mod48-component-connected-subsets","seed":202611177010,"semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2,4],[1,3,5]]}"},"fields":{"title":"The Union of All Connected Subsets through a Basepoint Is Its Component","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 4], [1, 3, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).","answer":{"kind":"expression","value":"(0, [[0], [0, 2], [0, 4], [0, 2, 4]], [0, 2, 4], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation":"Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 2], [0, 4], [0, 2, 4]], [0, 2, 4], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)."},"skills":["The Union of All Connected Subsets through a Basepoint Is Its Component"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0011","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b75ef83ee6a4ff4a1e565ddc1fd22e7d0be0f80ba40c39fca37c4116db3a216","family_id":"qf-mod48-component-connected-subsets","seed":202611177011,"semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2,4],[3,5]]}"},"fields":{"title":"The Union of All Connected Subsets through a Basepoint Is Its Component","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 4], [3, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).","answer":{"kind":"expression","value":"(0, [[0], [0, 1], [0, 2], [0, 1, 2], [0, 4], [0, 1, 4], [0, 2, 4], [0, 1, 2, 4]], [0, 1, 2, 4], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation":"Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 1], [0, 2], [0, 1, 2], [0, 4], [0, 1, 4], [0, 2, 4], [0, 1, 2, 4]], [0, 1, 2, 4], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)."},"skills":["The Union of All Connected Subsets through a Basepoint Is Its Component"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0012","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7eabd0f4e2d7f31c61b220fb0b77e520c6688b01ced5a4db249bfb2c306a25bd","family_id":"qf-mod48-component-connected-subsets","seed":202611177012,"semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,3,4],[1,2,5]]}"},"fields":{"title":"The Union of All Connected Subsets through a Basepoint Is Its Component","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 3, 4], [1, 2, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).","answer":{"kind":"expression","value":"(0, [[0], [0, 3], [0, 4], [0, 3, 4]], [0, 3, 4], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation":"Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 3], [0, 4], [0, 3, 4]], [0, 3, 4], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)."},"skills":["The Union of All Connected Subsets through a Basepoint Is Its Component"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0013","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37de92e00aee6277308507f82e264f044115d581285a229d74cbd53e2f348bcf","family_id":"qf-mod48-component-connected-subsets","seed":202611177013,"semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,3,4],[2,5]]}"},"fields":{"title":"The Union of All Connected Subsets through a Basepoint Is Its Component","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 3, 4], [2, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).","answer":{"kind":"expression","value":"(0, [[0], [0, 1], [0, 3], [0, 1, 3], [0, 4], [0, 1, 4], [0, 3, 4], [0, 1, 3, 4]], [0, 1, 3, 4], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation":"Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 1], [0, 3], [0, 1, 3], [0, 4], [0, 1, 4], [0, 3, 4], [0, 1, 3, 4]], [0, 1, 3, 4], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)."},"skills":["The Union of All Connected Subsets through a Basepoint Is Its Component"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0014","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a36b03ecac03fe46d4439126af4928db9f776531c1eca77c60e6fa67d1b56804","family_id":"qf-mod48-component-connected-subsets","seed":202611177014,"semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2,3,4],[1,5]]}"},"fields":{"title":"The Union of All Connected Subsets through a Basepoint Is Its Component","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 3, 4], [1, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).","answer":{"kind":"expression","value":"(0, [[0], [0, 2], [0, 3], [0, 2, 3], [0, 4], [0, 2, 4], [0, 3, 4], [0, 2, 3, 4]], [0, 2, 3, 4], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation":"Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 2], [0, 3], [0, 2, 3], [0, 4], [0, 2, 4], [0, 3, 4], [0, 2, 3, 4]], [0, 2, 3, 4], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)."},"skills":["The Union of All Connected Subsets through a Basepoint Is Its Component"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0015","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b6418cbae608bd127403b252501046377c3fa7cdaa86c467a062dfa9bf38718","family_id":"qf-mod48-component-connected-subsets","seed":202611177015,"semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2,3,4],[5]]}"},"fields":{"title":"The Union of All Connected Subsets through a Basepoint Is Its Component","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4], [5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).","answer":{"kind":"expression","value":"(0, [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3], [0, 4], [0, 1, 4], [0, 2, 4], [0, 1, 2, 4], [0, 3, 4], [0, 1, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], [0, 1, 2, 3, 4], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation":"Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3], [0, 4], [0, 1, 4], [0, 2, 4], [0, 1, 2, 4], [0, 3, 4], [0, 1, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], [0, 1, 2, 3, 4], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)."},"skills":["The Union of All Connected Subsets through a Basepoint Is Its Component"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0016","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"94a4282d060d5e1475ca0e63fd79066eaee39ed5daac1ccc7eacc06d78aac872","family_id":"qf-mod48-component-connected-subsets","seed":202611177016,"semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,5],[1,2,3,4]]}"},"fields":{"title":"The Union of All Connected Subsets through a Basepoint Is Its Component","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 5], [1, 2, 3, 4]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).","answer":{"kind":"expression","value":"(0, [[0], [0, 5]], [0, 5], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation":"Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 5]], [0, 5], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)."},"skills":["The Union of All Connected Subsets through a Basepoint Is Its Component"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0017","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba33c09d4fa295fc97530b5d824a0be1298e0b35b6414f5234a94150b10e2808","family_id":"qf-mod48-component-connected-subsets","seed":202611177017,"semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,5],[2,3,4]]}"},"fields":{"title":"The Union of All Connected Subsets through a Basepoint Is Its Component","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 5], [2, 3, 4]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).","answer":{"kind":"expression","value":"(0, [[0], [0, 1], [0, 5], [0, 1, 5]], [0, 1, 5], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation":"Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 1], [0, 5], [0, 1, 5]], [0, 1, 5], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)."},"skills":["The Union of All Connected Subsets through a Basepoint Is Its Component"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0018","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"053e7a3618b605a117b5fdf19b6f1a620910653fe3f0bfda066c666fced4abf8","family_id":"qf-mod48-component-connected-subsets","seed":202611177018,"semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2,5],[1,3,4]]}"},"fields":{"title":"The Union of All Connected Subsets through a Basepoint Is Its Component","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 5], [1, 3, 4]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).","answer":{"kind":"expression","value":"(0, [[0], [0, 2], [0, 5], [0, 2, 5]], [0, 2, 5], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation":"Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 2], [0, 5], [0, 2, 5]], [0, 2, 5], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)."},"skills":["The Union of All Connected Subsets through a Basepoint Is Its Component"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0019","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f12c39c1fed64e3422267e54e9455a7a3ba2c50e7338e9a4863c29d793abc778","family_id":"qf-mod48-component-connected-subsets","seed":202611177019,"semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2,5],[3,4]]}"},"fields":{"title":"The Union of All Connected Subsets through a Basepoint Is Its Component","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 5], [3, 4]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).","answer":{"kind":"expression","value":"(0, [[0], [0, 1], [0, 2], [0, 1, 2], [0, 5], [0, 1, 5], [0, 2, 5], [0, 1, 2, 5]], [0, 1, 2, 5], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation":"Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 1], [0, 2], [0, 1, 2], [0, 5], [0, 1, 5], [0, 2, 5], [0, 1, 2, 5]], [0, 1, 2, 5], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)."},"skills":["The Union of All Connected Subsets through a Basepoint Is Its Component"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0020","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f497bbc6aba97f2590a67645cb7321c454c548e776dcc105fe378f843a01c3b5","family_id":"qf-mod48-component-connected-subsets","seed":202611177020,"semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,3,5],[1,2,4]]}"},"fields":{"title":"The Union of All Connected Subsets through a Basepoint Is Its Component","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 3, 5], [1, 2, 4]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).","answer":{"kind":"expression","value":"(0, [[0], [0, 3], [0, 5], [0, 3, 5]], [0, 3, 5], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation":"Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 3], [0, 5], [0, 3, 5]], [0, 3, 5], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)."},"skills":["The Union of All Connected Subsets through a Basepoint Is Its Component"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0021","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f6d47f579ef10f289815dd4030630c16382fa1394c02bc97b27c79acc37a959","family_id":"qf-mod48-component-connected-subsets","seed":202611177021,"semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,3,5],[2,4]]}"},"fields":{"title":"The Union of All Connected Subsets through a Basepoint Is Its Component","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 3, 5], [2, 4]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).","answer":{"kind":"expression","value":"(0, [[0], [0, 1], [0, 3], [0, 1, 3], [0, 5], [0, 1, 5], [0, 3, 5], [0, 1, 3, 5]], [0, 1, 3, 5], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation":"Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 1], [0, 3], [0, 1, 3], [0, 5], [0, 1, 5], [0, 3, 5], [0, 1, 3, 5]], [0, 1, 3, 5], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)."},"skills":["The Union of All Connected Subsets through a Basepoint Is Its Component"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0022","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3d3314f609304767a5519b6859262b478f28cb58dc95f30e8937b5904e9a947","family_id":"qf-mod48-component-connected-subsets","seed":202611177022,"semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2,3,5],[1,4]]}"},"fields":{"title":"The Union of All Connected Subsets through a Basepoint Is Its Component","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 3, 5], [1, 4]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).","answer":{"kind":"expression","value":"(0, [[0], [0, 2], [0, 3], [0, 2, 3], [0, 5], [0, 2, 5], [0, 3, 5], [0, 2, 3, 5]], [0, 2, 3, 5], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation":"Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 2], [0, 3], [0, 2, 3], [0, 5], [0, 2, 5], [0, 3, 5], [0, 2, 3, 5]], [0, 2, 3, 5], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)."},"skills":["The Union of All Connected Subsets through a Basepoint Is Its Component"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0023","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa4f5f9584dd9a892e1a48bd7dc0626a260a6cee34a5dece9c59ab0c75f631cd","family_id":"qf-mod48-component-connected-subsets","seed":202611177023,"semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2,3,5],[4]]}"},"fields":{"title":"The Union of All Connected Subsets through a Basepoint Is Its Component","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 5], [4]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).","answer":{"kind":"expression","value":"(0, [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3], [0, 5], [0, 1, 5], [0, 2, 5], [0, 1, 2, 5], [0, 3, 5], [0, 1, 3, 5], [0, 2, 3, 5], [0, 1, 2, 3, 5]], [0, 1, 2, 3, 5], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation":"Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3], [0, 5], [0, 1, 5], [0, 2, 5], [0, 1, 2, 5], [0, 3, 5], [0, 1, 3, 5], [0, 2, 3, 5], [0, 1, 2, 3, 5]], [0, 1, 2, 3, 5], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)."},"skills":["The Union of All Connected Subsets through a Basepoint Is Its Component"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0024","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4bb675234f09032e1df8e49f761f7829c7a79943329ebf56599742fd8d57d898","family_id":"qf-mod48-component-connected-subsets","seed":202611177024,"semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,4,5],[1,2,3]]}"},"fields":{"title":"The Union of All Connected Subsets through a Basepoint Is Its Component","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 4, 5], [1, 2, 3]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).","answer":{"kind":"expression","value":"(0, [[0], [0, 4], [0, 5], [0, 4, 5]], [0, 4, 5], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation":"Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 4], [0, 5], [0, 4, 5]], [0, 4, 5], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)."},"skills":["The Union of All Connected Subsets through a Basepoint Is Its Component"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0025","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5dfd08bf9fc8ba6647922fb7981d0715f2764a25f9b7608621aa36dced85b19","family_id":"qf-mod48-component-connected-subsets","seed":202611177025,"semantic_template":"component-connected-subsets|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,4,5],[2,3]]}"},"fields":{"title":"The Union of All Connected Subsets through a Basepoint Is Its Component","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 4, 5], [2, 3]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, all_containing_connected_subsets, their_union, union_connected, certificate).","answer":{"kind":"expression","value":"(0, [[0], [0, 1], [0, 4], [0, 1, 4], [0, 5], [0, 1, 5], [0, 4, 5], [0, 1, 4, 5]], [0, 1, 4, 5], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)","explanation":"Enumerate every nonempty connected subspace containing the basepoint, take their union and verify its connectedness. The common basepoint is essential; this is not a maximum-cardinality search or an arbitrary graph component. The exact result, retaining the requested order, is (0, [[0], [0, 1], [0, 4], [0, 1, 4], [0, 5], [0, 1, 5], [0, 4, 5], [0, 1, 4, 5]], [0, 1, 4, 5], Yes, union_of_connected_subsets_sharing_basepoint_is_connected)."},"skills":["The Union of All Connected Subsets through a Basepoint Is Its Component"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0026","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"957217c15568eb44da480d059f3aff410f9c0a418f2b8ee5ca04f6b0fc0b7fac","family_id":"qf-mod48-component-maximality","seed":202611177026,"semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1],[2,3,4,5]]}"},"fields":{"title":"The Basepoint Component Is Uniquely Inclusion-Maximal and Closed","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1], [2, 3, 4, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).","answer":{"kind":"expression","value":"(0, [0, 1], [[0], [0, 1]], Yes, Yes, unique_inclusion_maximal_connected_subset)","explanation":"Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 1], [[0], [0, 1]], Yes, Yes, unique_inclusion_maximal_connected_subset)."},"skills":["The Basepoint Component Is Uniquely Inclusion-Maximal and Closed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0027","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4cf63abacc790485c2ec65cd46f69c39d7433e159929c65848f8ac9af3060f36","family_id":"qf-mod48-component-maximality","seed":202611177027,"semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2],[1,3,4,5]]}"},"fields":{"title":"The Basepoint Component Is Uniquely Inclusion-Maximal and Closed","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2], [1, 3, 4, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).","answer":{"kind":"expression","value":"(0, [0, 2], [[0], [0, 2]], Yes, Yes, unique_inclusion_maximal_connected_subset)","explanation":"Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 2], [[0], [0, 2]], Yes, Yes, unique_inclusion_maximal_connected_subset)."},"skills":["The Basepoint Component Is Uniquely Inclusion-Maximal and Closed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0028","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab85142e420b05049020b69233ba760776caddc8203f5ad5a5a4cb2e73c4f31c","family_id":"qf-mod48-component-maximality","seed":202611177028,"semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2],[3,4,5]]}"},"fields":{"title":"The Basepoint Component Is Uniquely Inclusion-Maximal and Closed","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2], [3, 4, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).","answer":{"kind":"expression","value":"(0, [0, 1, 2], [[0], [0, 1], [0, 2], [0, 1, 2]], Yes, Yes, unique_inclusion_maximal_connected_subset)","explanation":"Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 1, 2], [[0], [0, 1], [0, 2], [0, 1, 2]], Yes, Yes, unique_inclusion_maximal_connected_subset)."},"skills":["The Basepoint Component Is Uniquely Inclusion-Maximal and Closed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0029","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6155433198b691724645d4c9c0ee868c861333a36fc8c2996d00bd930f67adc3","family_id":"qf-mod48-component-maximality","seed":202611177029,"semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,3],[1,2,4,5]]}"},"fields":{"title":"The Basepoint Component Is Uniquely Inclusion-Maximal and Closed","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 3], [1, 2, 4, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).","answer":{"kind":"expression","value":"(0, [0, 3], [[0], [0, 3]], Yes, Yes, unique_inclusion_maximal_connected_subset)","explanation":"Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 3], [[0], [0, 3]], Yes, Yes, unique_inclusion_maximal_connected_subset)."},"skills":["The Basepoint Component Is Uniquely Inclusion-Maximal and Closed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0030","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc9f091e3798aeb421346b126a217360a7080680f5b0023069c86e2303186a35","family_id":"qf-mod48-component-maximality","seed":202611177030,"semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,3],[2,4,5]]}"},"fields":{"title":"The Basepoint Component Is Uniquely Inclusion-Maximal and Closed","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 3], [2, 4, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).","answer":{"kind":"expression","value":"(0, [0, 1, 3], [[0], [0, 1], [0, 3], [0, 1, 3]], Yes, Yes, unique_inclusion_maximal_connected_subset)","explanation":"Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 1, 3], [[0], [0, 1], [0, 3], [0, 1, 3]], Yes, Yes, unique_inclusion_maximal_connected_subset)."},"skills":["The Basepoint Component Is Uniquely Inclusion-Maximal and Closed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0031","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c2c569c0bb2e8bb31ddb147ed3f6afc1fecf2e541ecfbbc8cd7d85db6d3387f","family_id":"qf-mod48-component-maximality","seed":202611177031,"semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2,3],[1,4,5]]}"},"fields":{"title":"The Basepoint Component Is Uniquely Inclusion-Maximal and Closed","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 3], [1, 4, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).","answer":{"kind":"expression","value":"(0, [0, 2, 3], [[0], [0, 2], [0, 3], [0, 2, 3]], Yes, Yes, unique_inclusion_maximal_connected_subset)","explanation":"Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 2, 3], [[0], [0, 2], [0, 3], [0, 2, 3]], Yes, Yes, unique_inclusion_maximal_connected_subset)."},"skills":["The Basepoint Component Is Uniquely Inclusion-Maximal and Closed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0032","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c48ec3a69ac1f3aada9b63bf38ff504e1af8bb65c9cb2f9052eb41e27d83028f","family_id":"qf-mod48-component-maximality","seed":202611177032,"semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2,3],[4,5]]}"},"fields":{"title":"The Basepoint Component Is Uniquely Inclusion-Maximal and Closed","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3], [4, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).","answer":{"kind":"expression","value":"(0, [0, 1, 2, 3], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]], Yes, Yes, unique_inclusion_maximal_connected_subset)","explanation":"Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 1, 2, 3], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3]], Yes, Yes, unique_inclusion_maximal_connected_subset)."},"skills":["The Basepoint Component Is Uniquely Inclusion-Maximal and Closed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0033","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd535f4a1be65a7d85b1a0a9d949c3008fa435ab7859e37577419bf56efe4096","family_id":"qf-mod48-component-maximality","seed":202611177033,"semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,4],[1,2,3,5]]}"},"fields":{"title":"The Basepoint Component Is Uniquely Inclusion-Maximal and Closed","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 4], [1, 2, 3, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).","answer":{"kind":"expression","value":"(0, [0, 4], [[0], [0, 4]], Yes, Yes, unique_inclusion_maximal_connected_subset)","explanation":"Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 4], [[0], [0, 4]], Yes, Yes, unique_inclusion_maximal_connected_subset)."},"skills":["The Basepoint Component Is Uniquely Inclusion-Maximal and Closed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0034","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5a0beaf05aeab6d9b24321ce3d256a12347b7fbce7dd13fccf3b44987f19647","family_id":"qf-mod48-component-maximality","seed":202611177034,"semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,4],[2,3,5]]}"},"fields":{"title":"The Basepoint Component Is Uniquely Inclusion-Maximal and Closed","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 4], [2, 3, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).","answer":{"kind":"expression","value":"(0, [0, 1, 4], [[0], [0, 1], [0, 4], [0, 1, 4]], Yes, Yes, unique_inclusion_maximal_connected_subset)","explanation":"Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 1, 4], [[0], [0, 1], [0, 4], [0, 1, 4]], Yes, Yes, unique_inclusion_maximal_connected_subset)."},"skills":["The Basepoint Component Is Uniquely Inclusion-Maximal and Closed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0035","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d1c1d4f365e76f709f58dd634d2ba91efe731c39862a0b803e542583cf0b571","family_id":"qf-mod48-component-maximality","seed":202611177035,"semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2,4],[1,3,5]]}"},"fields":{"title":"The Basepoint Component Is Uniquely Inclusion-Maximal and Closed","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 4], [1, 3, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).","answer":{"kind":"expression","value":"(0, [0, 2, 4], [[0], [0, 2], [0, 4], [0, 2, 4]], Yes, Yes, unique_inclusion_maximal_connected_subset)","explanation":"Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 2, 4], [[0], [0, 2], [0, 4], [0, 2, 4]], Yes, Yes, unique_inclusion_maximal_connected_subset)."},"skills":["The Basepoint Component Is Uniquely Inclusion-Maximal and Closed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0036","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0bcff2e2afe61d6639c57472e7d3f9c791235a74038c40b39aaf5434d4ca8095","family_id":"qf-mod48-component-maximality","seed":202611177036,"semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2,4],[3,5]]}"},"fields":{"title":"The Basepoint Component Is Uniquely Inclusion-Maximal and Closed","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 4], [3, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).","answer":{"kind":"expression","value":"(0, [0, 1, 2, 4], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 4], [0, 1, 4], [0, 2, 4], [0, 1, 2, 4]], Yes, Yes, unique_inclusion_maximal_connected_subset)","explanation":"Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 1, 2, 4], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 4], [0, 1, 4], [0, 2, 4], [0, 1, 2, 4]], Yes, Yes, unique_inclusion_maximal_connected_subset)."},"skills":["The Basepoint Component Is Uniquely Inclusion-Maximal and Closed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0037","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"514c6cc9fee37c62b6f0b05f2e119c89ec50937a43aae3d85c0aa4aac5aa7dbb","family_id":"qf-mod48-component-maximality","seed":202611177037,"semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,3,4],[1,2,5]]}"},"fields":{"title":"The Basepoint Component Is Uniquely Inclusion-Maximal and Closed","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 3, 4], [1, 2, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).","answer":{"kind":"expression","value":"(0, [0, 3, 4], [[0], [0, 3], [0, 4], [0, 3, 4]], Yes, Yes, unique_inclusion_maximal_connected_subset)","explanation":"Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 3, 4], [[0], [0, 3], [0, 4], [0, 3, 4]], Yes, Yes, unique_inclusion_maximal_connected_subset)."},"skills":["The Basepoint Component Is Uniquely Inclusion-Maximal and Closed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0038","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7f18db8da225c579f463becc30ab75538250cb7dddc07db98f0921df6a6d0b5","family_id":"qf-mod48-component-maximality","seed":202611177038,"semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,3,4],[2,5]]}"},"fields":{"title":"The Basepoint Component Is Uniquely Inclusion-Maximal and Closed","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 3, 4], [2, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).","answer":{"kind":"expression","value":"(0, [0, 1, 3, 4], [[0], [0, 1], [0, 3], [0, 1, 3], [0, 4], [0, 1, 4], [0, 3, 4], [0, 1, 3, 4]], Yes, Yes, unique_inclusion_maximal_connected_subset)","explanation":"Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 1, 3, 4], [[0], [0, 1], [0, 3], [0, 1, 3], [0, 4], [0, 1, 4], [0, 3, 4], [0, 1, 3, 4]], Yes, Yes, unique_inclusion_maximal_connected_subset)."},"skills":["The Basepoint Component Is Uniquely Inclusion-Maximal and Closed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0039","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d20091364c4a705943ede99c18db2e93eba2aafd2ccd771222d0019f3edcb6d6","family_id":"qf-mod48-component-maximality","seed":202611177039,"semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2,3,4],[1,5]]}"},"fields":{"title":"The Basepoint Component Is Uniquely Inclusion-Maximal and Closed","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 3, 4], [1, 5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).","answer":{"kind":"expression","value":"(0, [0, 2, 3, 4], [[0], [0, 2], [0, 3], [0, 2, 3], [0, 4], [0, 2, 4], [0, 3, 4], [0, 2, 3, 4]], Yes, Yes, unique_inclusion_maximal_connected_subset)","explanation":"Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 2, 3, 4], [[0], [0, 2], [0, 3], [0, 2, 3], [0, 4], [0, 2, 4], [0, 3, 4], [0, 2, 3, 4]], Yes, Yes, unique_inclusion_maximal_connected_subset)."},"skills":["The Basepoint Component Is Uniquely Inclusion-Maximal and Closed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0040","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3fbfa3024a3e4c82289ec21cf6d9940fb625c37767035db5036a598385eb5a87","family_id":"qf-mod48-component-maximality","seed":202611177040,"semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2,3,4],[5]]}"},"fields":{"title":"The Basepoint Component Is Uniquely Inclusion-Maximal and Closed","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4], [5]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).","answer":{"kind":"expression","value":"(0, [0, 1, 2, 3, 4], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3], [0, 4], [0, 1, 4], [0, 2, 4], [0, 1, 2, 4], [0, 3, 4], [0, 1, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], Yes, Yes, unique_inclusion_maximal_connected_subset)","explanation":"Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 1, 2, 3, 4], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3], [0, 4], [0, 1, 4], [0, 2, 4], [0, 1, 2, 4], [0, 3, 4], [0, 1, 3, 4], [0, 2, 3, 4], [0, 1, 2, 3, 4]], Yes, Yes, unique_inclusion_maximal_connected_subset)."},"skills":["The Basepoint Component Is Uniquely Inclusion-Maximal and Closed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0041","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"15b0e5dd9c080815048f0d0177c9f4ec4d3e479217a79762172cd4e4b4e05dd9","family_id":"qf-mod48-component-maximality","seed":202611177041,"semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,5],[1,2,3,4]]}"},"fields":{"title":"The Basepoint Component Is Uniquely Inclusion-Maximal and Closed","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 5], [1, 2, 3, 4]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).","answer":{"kind":"expression","value":"(0, [0, 5], [[0], [0, 5]], Yes, Yes, unique_inclusion_maximal_connected_subset)","explanation":"Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 5], [[0], [0, 5]], Yes, Yes, unique_inclusion_maximal_connected_subset)."},"skills":["The Basepoint Component Is Uniquely Inclusion-Maximal and Closed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0042","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6cb59299d0cfeeedb2ac70c532ccf73a69b46c729a5f713599853bb71d013911","family_id":"qf-mod48-component-maximality","seed":202611177042,"semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,5],[2,3,4]]}"},"fields":{"title":"The Basepoint Component Is Uniquely Inclusion-Maximal and Closed","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 5], [2, 3, 4]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).","answer":{"kind":"expression","value":"(0, [0, 1, 5], [[0], [0, 1], [0, 5], [0, 1, 5]], Yes, Yes, unique_inclusion_maximal_connected_subset)","explanation":"Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 1, 5], [[0], [0, 1], [0, 5], [0, 1, 5]], Yes, Yes, unique_inclusion_maximal_connected_subset)."},"skills":["The Basepoint Component Is Uniquely Inclusion-Maximal and Closed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0043","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b01faf3aa0aebe0317a8b8ff7ca9a843feeacca1843597928a6101ab4fb511f","family_id":"qf-mod48-component-maximality","seed":202611177043,"semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2,5],[1,3,4]]}"},"fields":{"title":"The Basepoint Component Is Uniquely Inclusion-Maximal and Closed","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 5], [1, 3, 4]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).","answer":{"kind":"expression","value":"(0, [0, 2, 5], [[0], [0, 2], [0, 5], [0, 2, 5]], Yes, Yes, unique_inclusion_maximal_connected_subset)","explanation":"Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 2, 5], [[0], [0, 2], [0, 5], [0, 2, 5]], Yes, Yes, unique_inclusion_maximal_connected_subset)."},"skills":["The Basepoint Component Is Uniquely Inclusion-Maximal and Closed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0044","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3631c59a45580cc364f1a863bcfe43c0422294a69cdab2a6a505879f9a2c485","family_id":"qf-mod48-component-maximality","seed":202611177044,"semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2,5],[3,4]]}"},"fields":{"title":"The Basepoint Component Is Uniquely Inclusion-Maximal and Closed","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 5], [3, 4]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).","answer":{"kind":"expression","value":"(0, [0, 1, 2, 5], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 5], [0, 1, 5], [0, 2, 5], [0, 1, 2, 5]], Yes, Yes, unique_inclusion_maximal_connected_subset)","explanation":"Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 1, 2, 5], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 5], [0, 1, 5], [0, 2, 5], [0, 1, 2, 5]], Yes, Yes, unique_inclusion_maximal_connected_subset)."},"skills":["The Basepoint Component Is Uniquely Inclusion-Maximal and Closed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0045","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0ef9d4ad909d826830166a22b1dbdbd839f30e569a7c70582ce7e88f03f03cb","family_id":"qf-mod48-component-maximality","seed":202611177045,"semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,3,5],[1,2,4]]}"},"fields":{"title":"The Basepoint Component Is Uniquely Inclusion-Maximal and Closed","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 3, 5], [1, 2, 4]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).","answer":{"kind":"expression","value":"(0, [0, 3, 5], [[0], [0, 3], [0, 5], [0, 3, 5]], Yes, Yes, unique_inclusion_maximal_connected_subset)","explanation":"Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 3, 5], [[0], [0, 3], [0, 5], [0, 3, 5]], Yes, Yes, unique_inclusion_maximal_connected_subset)."},"skills":["The Basepoint Component Is Uniquely Inclusion-Maximal and Closed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0046","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34231f32a5ccab5ad7469e5cbda6ad45ed3c2565d7654cc309520275e9f57b85","family_id":"qf-mod48-component-maximality","seed":202611177046,"semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,3,5],[2,4]]}"},"fields":{"title":"The Basepoint Component Is Uniquely Inclusion-Maximal and Closed","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 3, 5], [2, 4]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).","answer":{"kind":"expression","value":"(0, [0, 1, 3, 5], [[0], [0, 1], [0, 3], [0, 1, 3], [0, 5], [0, 1, 5], [0, 3, 5], [0, 1, 3, 5]], Yes, Yes, unique_inclusion_maximal_connected_subset)","explanation":"Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 1, 3, 5], [[0], [0, 1], [0, 3], [0, 1, 3], [0, 5], [0, 1, 5], [0, 3, 5], [0, 1, 3, 5]], Yes, Yes, unique_inclusion_maximal_connected_subset)."},"skills":["The Basepoint Component Is Uniquely Inclusion-Maximal and Closed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0047","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77efb015e7901b8ddfd3c42aacc1271129e4dc94b7fc362817170d489657682c","family_id":"qf-mod48-component-maximality","seed":202611177047,"semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2,3,5],[1,4]]}"},"fields":{"title":"The Basepoint Component Is Uniquely Inclusion-Maximal and Closed","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 3, 5], [1, 4]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).","answer":{"kind":"expression","value":"(0, [0, 2, 3, 5], [[0], [0, 2], [0, 3], [0, 2, 3], [0, 5], [0, 2, 5], [0, 3, 5], [0, 2, 3, 5]], Yes, Yes, unique_inclusion_maximal_connected_subset)","explanation":"Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 2, 3, 5], [[0], [0, 2], [0, 3], [0, 2, 3], [0, 5], [0, 2, 5], [0, 3, 5], [0, 2, 3, 5]], Yes, Yes, unique_inclusion_maximal_connected_subset)."},"skills":["The Basepoint Component Is Uniquely Inclusion-Maximal and Closed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0048","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a6e55bf97470b619b62d0fce0a411fae0670a170ed3e4a95ce9d98baa4e41af","family_id":"qf-mod48-component-maximality","seed":202611177048,"semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2,3,5],[4]]}"},"fields":{"title":"The Basepoint Component Is Uniquely Inclusion-Maximal and Closed","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 5], [4]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).","answer":{"kind":"expression","value":"(0, [0, 1, 2, 3, 5], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3], [0, 5], [0, 1, 5], [0, 2, 5], [0, 1, 2, 5], [0, 3, 5], [0, 1, 3, 5], [0, 2, 3, 5], [0, 1, 2, 3, 5]], Yes, Yes, unique_inclusion_maximal_connected_subset)","explanation":"Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 1, 2, 3, 5], [[0], [0, 1], [0, 2], [0, 1, 2], [0, 3], [0, 1, 3], [0, 2, 3], [0, 1, 2, 3], [0, 5], [0, 1, 5], [0, 2, 5], [0, 1, 2, 5], [0, 3, 5], [0, 1, 3, 5], [0, 2, 3, 5], [0, 1, 2, 3, 5]], Yes, Yes, unique_inclusion_maximal_connected_subset)."},"skills":["The Basepoint Component Is Uniquely Inclusion-Maximal and Closed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0049","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85bb156a6974401fa16f9f62af909d336b9a90504850516aab60d8ff5866dc96","family_id":"qf-mod48-component-maximality","seed":202611177049,"semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,4,5],[1,2,3]]}"},"fields":{"title":"The Basepoint Component Is Uniquely Inclusion-Maximal and Closed","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 4, 5], [1, 2, 3]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).","answer":{"kind":"expression","value":"(0, [0, 4, 5], [[0], [0, 4], [0, 5], [0, 4, 5]], Yes, Yes, unique_inclusion_maximal_connected_subset)","explanation":"Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 4, 5], [[0], [0, 4], [0, 5], [0, 4, 5]], Yes, Yes, unique_inclusion_maximal_connected_subset)."},"skills":["The Basepoint Component Is Uniquely Inclusion-Maximal and Closed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0050","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39aa7d091e9bdb29fca403ffcf0ddf438581eceaaacf907009dc3395a427466f","family_id":"qf-mod48-component-maximality","seed":202611177050,"semantic_template":"component-maximality|{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,4,5],[2,3]]}"},"fields":{"title":"The Basepoint Component Is Uniquely Inclusion-Maximal and Closed","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 4, 5], [2, 3]], basepoint=0. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (basepoint, component, all_containing_connected_subsets, inclusion_maximal, closed, certificate).","answer":{"kind":"expression","value":"(0, [0, 1, 4, 5], [[0], [0, 1], [0, 4], [0, 1, 4], [0, 5], [0, 1, 5], [0, 4, 5], [0, 1, 4, 5]], Yes, Yes, unique_inclusion_maximal_connected_subset)","explanation":"Retain the actual component and every containing connected subset, check inclusion of each in the component and check that the complement is open. Another disjoint component can have larger cardinality without changing this basepoint component. The exact result, retaining the requested order, is (0, [0, 1, 4, 5], [[0], [0, 1], [0, 4], [0, 1, 4], [0, 5], [0, 1, 5], [0, 4, 5], [0, 1, 4, 5]], Yes, Yes, unique_inclusion_maximal_connected_subset)."},"skills":["The Basepoint Component Is Uniquely Inclusion-Maximal and Closed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0051","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aed96f958c33dbb23017f73646b5e067eeb196869df9604aae4a847dc109d7ea","family_id":"qf-mod48-component-partition","seed":202611177051,"semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1],[2,3,4,5]]}"},"fields":{"title":"Finite Topological Components Partition the Space and Are Clopen","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1], [2, 3, 4, 5]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 1], [2, 3, 4, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)","explanation":"Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 1], [2, 3, 4, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)."},"skills":["Finite Topological Components Partition the Space and Are Clopen"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0052","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d530f7bd80d7ca4102e81b514d8a8b67c31c4d8590f4bd5ed8b32749c247a8ea","family_id":"qf-mod48-component-partition","seed":202611177052,"semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2],[1,3,4,5]]}"},"fields":{"title":"Finite Topological Components Partition the Space and Are Clopen","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2], [1, 3, 4, 5]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 2], [1, 3, 4, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)","explanation":"Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 2], [1, 3, 4, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)."},"skills":["Finite Topological Components Partition the Space and Are Clopen"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0053","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"181389deef385863ee34364e1988fbbe50ea28a454b46700ef8e602b924d71cd","family_id":"qf-mod48-component-partition","seed":202611177053,"semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2],[3,4,5]]}"},"fields":{"title":"Finite Topological Components Partition the Space and Are Clopen","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2], [3, 4, 5]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 1, 2], [3, 4, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)","explanation":"Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 1, 2], [3, 4, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)."},"skills":["Finite Topological Components Partition the Space and Are Clopen"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0054","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"077c0b6289fc25b3e8661128302bfa85728c3efe8f271ae4e511403fab404447","family_id":"qf-mod48-component-partition","seed":202611177054,"semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,3],[1,2,4,5]]}"},"fields":{"title":"Finite Topological Components Partition the Space and Are Clopen","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 3], [1, 2, 4, 5]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 3], [1, 2, 4, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)","explanation":"Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 3], [1, 2, 4, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)."},"skills":["Finite Topological Components Partition the Space and Are Clopen"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0055","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73a5ec4afa8ad608dd81e19ad37513db6842e10a6e60ac2b131a340a34cd7c8f","family_id":"qf-mod48-component-partition","seed":202611177055,"semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,3],[2,4,5]]}"},"fields":{"title":"Finite Topological Components Partition the Space and Are Clopen","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 3], [2, 4, 5]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 1, 3], [2, 4, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)","explanation":"Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 1, 3], [2, 4, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)."},"skills":["Finite Topological Components Partition the Space and Are Clopen"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0056","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad7eccf175f8357f175c4708c8daf0e01ee05d1bd95f87971ff0d204f7be1b14","family_id":"qf-mod48-component-partition","seed":202611177056,"semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2,3],[1,4,5]]}"},"fields":{"title":"Finite Topological Components Partition the Space and Are Clopen","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 3], [1, 4, 5]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 2, 3], [1, 4, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)","explanation":"Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 2, 3], [1, 4, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)."},"skills":["Finite Topological Components Partition the Space and Are Clopen"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0057","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"792440d6b4b1bfb50dc674a43f740c8559d480956973f8b41810d4dc8850e1bf","family_id":"qf-mod48-component-partition","seed":202611177057,"semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2,3],[4,5]]}"},"fields":{"title":"Finite Topological Components Partition the Space and Are Clopen","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3], [4, 5]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 1, 2, 3], [4, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)","explanation":"Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 1, 2, 3], [4, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)."},"skills":["Finite Topological Components Partition the Space and Are Clopen"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0058","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ddf8570e1a2c67a3cee0ab93bcba69f6b9a9c07d5f5ae8fb7e491ceafb0966d3","family_id":"qf-mod48-component-partition","seed":202611177058,"semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,4],[1,2,3,5]]}"},"fields":{"title":"Finite Topological Components Partition the Space and Are Clopen","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 4], [1, 2, 3, 5]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 4], [1, 2, 3, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)","explanation":"Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 4], [1, 2, 3, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)."},"skills":["Finite Topological Components Partition the Space and Are Clopen"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0059","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d30ef42cac700b550b71f30a7ee8026029f1abe94ba3005f1a300dbb3acbd36","family_id":"qf-mod48-component-partition","seed":202611177059,"semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,4],[2,3,5]]}"},"fields":{"title":"Finite Topological Components Partition the Space and Are Clopen","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 4], [2, 3, 5]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 1, 4], [2, 3, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)","explanation":"Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 1, 4], [2, 3, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)."},"skills":["Finite Topological Components Partition the Space and Are Clopen"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0060","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"631839440e96535b19f9558205de0974555872d1a4c60278763b7b86ac7ca8f0","family_id":"qf-mod48-component-partition","seed":202611177060,"semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2,4],[1,3,5]]}"},"fields":{"title":"Finite Topological Components Partition the Space and Are Clopen","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 4], [1, 3, 5]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 2, 4], [1, 3, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)","explanation":"Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 2, 4], [1, 3, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)."},"skills":["Finite Topological Components Partition the Space and Are Clopen"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0061","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f43c9c955ce1c2bca79a269d50a8791f5f4dabb22c0485112d0d6a5ccf72056a","family_id":"qf-mod48-component-partition","seed":202611177061,"semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2,4],[3,5]]}"},"fields":{"title":"Finite Topological Components Partition the Space and Are Clopen","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 4], [3, 5]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 1, 2, 4], [3, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)","explanation":"Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 1, 2, 4], [3, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)."},"skills":["Finite Topological Components Partition the Space and Are Clopen"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0062","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b6b747428eb022ce5146aa75604a6f5fa17751fa72a574a22116c504bdbde169","family_id":"qf-mod48-component-partition","seed":202611177062,"semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,3,4],[1,2,5]]}"},"fields":{"title":"Finite Topological Components Partition the Space and Are Clopen","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 3, 4], [1, 2, 5]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 3, 4], [1, 2, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)","explanation":"Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 3, 4], [1, 2, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)."},"skills":["Finite Topological Components Partition the Space and Are Clopen"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0063","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eebcb74ff220e18e77623446cfd3436bb40420595abf84b275673dd96f4b7992","family_id":"qf-mod48-component-partition","seed":202611177063,"semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,3,4],[2,5]]}"},"fields":{"title":"Finite Topological Components Partition the Space and Are Clopen","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 3, 4], [2, 5]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 1, 3, 4], [2, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)","explanation":"Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 1, 3, 4], [2, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)."},"skills":["Finite Topological Components Partition the Space and Are Clopen"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0064","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e3bb3ce8511ea22a4f47b76f2e80bd60f88678f81c325c59ead5d0745afcfbf","family_id":"qf-mod48-component-partition","seed":202611177064,"semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2,3,4],[1,5]]}"},"fields":{"title":"Finite Topological Components Partition the Space and Are Clopen","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 3, 4], [1, 5]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 2, 3, 4], [1, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)","explanation":"Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 2, 3, 4], [1, 5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)."},"skills":["Finite Topological Components Partition the Space and Are Clopen"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0065","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"675a10af24d4e05ca64ea47aa6b9129e5fb790d1f2218f34aff005ed76bc0bbb","family_id":"qf-mod48-component-partition","seed":202611177065,"semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2,3,4],[5]]}"},"fields":{"title":"Finite Topological Components Partition the Space and Are Clopen","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4], [5]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 4], [5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)","explanation":"Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 4], [5]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)."},"skills":["Finite Topological Components Partition the Space and Are Clopen"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0066","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a0aef6850ef6e6765e62a1e07ab4bfb514bec71f63512a13f960425d5b0abfa5","family_id":"qf-mod48-component-partition","seed":202611177066,"semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,5],[1,2,3,4]]}"},"fields":{"title":"Finite Topological Components Partition the Space and Are Clopen","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 5], [1, 2, 3, 4]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 5], [1, 2, 3, 4]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)","explanation":"Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 5], [1, 2, 3, 4]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)."},"skills":["Finite Topological Components Partition the Space and Are Clopen"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0067","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc65d48ff1f7eb67270ea1eeda49d455f1fe7b7751896192f481980f5d8a1ca8","family_id":"qf-mod48-component-partition","seed":202611177067,"semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,5],[2,3,4]]}"},"fields":{"title":"Finite Topological Components Partition the Space and Are Clopen","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 5], [2, 3, 4]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 1, 5], [2, 3, 4]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)","explanation":"Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 1, 5], [2, 3, 4]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)."},"skills":["Finite Topological Components Partition the Space and Are Clopen"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0068","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4526dad046fc24912205fe85bab7d0e06eaeee03e648953d76eeaef91a9a698d","family_id":"qf-mod48-component-partition","seed":202611177068,"semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2,5],[1,3,4]]}"},"fields":{"title":"Finite Topological Components Partition the Space and Are Clopen","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 5], [1, 3, 4]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 2, 5], [1, 3, 4]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)","explanation":"Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 2, 5], [1, 3, 4]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)."},"skills":["Finite Topological Components Partition the Space and Are Clopen"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0069","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a35d0e4e9b5a2f595ac891e05ff94160a90cd641ce7a7247125ccfc182bc5ab","family_id":"qf-mod48-component-partition","seed":202611177069,"semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2,5],[3,4]]}"},"fields":{"title":"Finite Topological Components Partition the Space and Are Clopen","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 5], [3, 4]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 1, 2, 5], [3, 4]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)","explanation":"Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 1, 2, 5], [3, 4]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)."},"skills":["Finite Topological Components Partition the Space and Are Clopen"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0070","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01bb72c2019b174d8a402512a43d370d37043f324dfaf9092006bcb0eed0264a","family_id":"qf-mod48-component-partition","seed":202611177070,"semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,3,5],[1,2,4]]}"},"fields":{"title":"Finite Topological Components Partition the Space and Are Clopen","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 3, 5], [1, 2, 4]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 3, 5], [1, 2, 4]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)","explanation":"Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 3, 5], [1, 2, 4]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)."},"skills":["Finite Topological Components Partition the Space and Are Clopen"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0071","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a8d11676b4448f1715cb141ae8cf18c00cdf74fe9797e8653de1e2ce873ff68","family_id":"qf-mod48-component-partition","seed":202611177071,"semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,3,5],[2,4]]}"},"fields":{"title":"Finite Topological Components Partition the Space and Are Clopen","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 3, 5], [2, 4]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 1, 3, 5], [2, 4]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)","explanation":"Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 1, 3, 5], [2, 4]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)."},"skills":["Finite Topological Components Partition the Space and Are Clopen"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0072","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70e28cc6c794a1810b5843da922be88d91a5ccbe72393cbf484691022cf3d2a7","family_id":"qf-mod48-component-partition","seed":202611177072,"semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2,3,5],[1,4]]}"},"fields":{"title":"Finite Topological Components Partition the Space and Are Clopen","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 2, 3, 5], [1, 4]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 2, 3, 5], [1, 4]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)","explanation":"Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 2, 3, 5], [1, 4]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)."},"skills":["Finite Topological Components Partition the Space and Are Clopen"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0073","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c82e3dcc503d8b2af348308b8ae9fe71734160631fcba6bf66cdd6eab49a53b","family_id":"qf-mod48-component-partition","seed":202611177073,"semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2,3,5],[4]]}"},"fields":{"title":"Finite Topological Components Partition the Space and Are Clopen","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 5], [4]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 1, 2, 3, 5], [4]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)","explanation":"Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 1, 2, 3, 5], [4]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)."},"skills":["Finite Topological Components Partition the Space and Are Clopen"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0074","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19202ccab25b27282b712165d3b679a200753554ed6d59db53c49d7be128781b","family_id":"qf-mod48-component-partition","seed":202611177074,"semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,4,5],[1,2,3]]}"},"fields":{"title":"Finite Topological Components Partition the Space and Are Clopen","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 4, 5], [1, 2, 3]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 4, 5], [1, 2, 3]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)","explanation":"Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 4, 5], [1, 2, 3]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)."},"skills":["Finite Topological Components Partition the Space and Are Clopen"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0075","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"084046a0413b47d83b4a1658092d44150385e337583522dec8753b16679c53b0","family_id":"qf-mod48-component-partition","seed":202611177075,"semantic_template":"component-partition|{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,4,5],[2,3]]}"},"fields":{"title":"Finite Topological Components Partition the Space and Are Clopen","stem":"Use complete finite points=[0, 1, 2, 3, 4, 5], topology=[[], [0, 1, 2, 3, 4, 5], [0, 1, 4, 5], [2, 3]]. Connectedness uses the actual induced topology on each subset, not an arbitrary graph on the same carrier. Give (components, pairwise_disjoint, cover, closed_flags, open_flags, finite_partition_certificate, certificate).","answer":{"kind":"expression","value":"([[0, 1, 4, 5], [2, 3]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)","explanation":"Check every component for disjoint coverage, closedness and openness. Openness follows here from finitely many closed components, not from a general-space theorem. In a general space components are closed but need not be open; the rational subspace example remains separate from this finite enumeration. The exact result, retaining the requested order, is ([[0, 1, 4, 5], [2, 3]], Yes, Yes, [Yes, Yes], [Yes, Yes], Yes, finite_components_partition_and_are_clopen_finite_specific)."},"skills":["Finite Topological Components Partition the Space and Are Clopen"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0076","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c41f7128d3087be402e5e084c5dc9ebc548ad6681289dc57cc31359fec579113","family_id":"qf-mod48-component-boundary","seed":202611177076,"semantic_template":"component-boundary|{mode:\"finite_clopen\",model:{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1],[2,3,4,5]]}}"},"fields":{"title":"Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition","stem":"Use mode=finite_clopen and complete model={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1],[2,3,4,5]]}. Give (mode, components, closed_flags, open_flags, all_clopen, certificate).","answer":{"kind":"expression","value":"(finite_clopen, [[0, 1], [2, 3, 4, 5]], [Yes, Yes], [Yes, Yes], Yes, finite_number_of_closed_components_makes_each_open)","explanation":"Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (finite_clopen, [[0, 1], [2, 3, 4, 5]], [Yes, Yes], [Yes, Yes], Yes, finite_number_of_closed_components_makes_each_open)."},"skills":["Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0077","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aaf9628422da5a85f521081132cda14c5040e52eb6804c7be5078662ace727b8","family_id":"qf-mod48-component-boundary","seed":202611177077,"semantic_template":"component-boundary|{mode:\"finite_no_nonopen_witness\",model:{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2],[1,3,4,5]]}}"},"fields":{"title":"Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition","stem":"Use mode=finite_no_nonopen_witness and complete model={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,2],[1,3,4,5]]}. Give (mode, components, all_clopen, general_nonopen_witness_claim, boundary).","answer":{"kind":"expression","value":"(finite_no_nonopen_witness, [[0, 2], [1, 3, 4, 5]], Yes, No, finite_models_cannot_witness_a_general_nonopen_component)","explanation":"Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (finite_no_nonopen_witness, [[0, 2], [1, 3, 4, 5]], Yes, No, finite_models_cannot_witness_a_general_nonopen_component)."},"skills":["Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0078","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb3805d9e9168a71fcd339705e23ba35d6aea3e83e2d0ad57ab5d5641d496c15","family_id":"qf-mod48-component-boundary","seed":202611177078,"semantic_template":"component-boundary|{mode:\"graph_confusion\",model:{graph_edges:[[0,1],[1,2],[2,3],[3,4],[4,5]],points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2],[3,4,5]]}}"},"fields":{"title":"Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition","stem":"Use mode=graph_confusion and complete model={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,2],[3,4,5]],\"graph_edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]]}. Give (mode, topological_components, graph_components, different_partitions, structure_identification_claim, boundary).","answer":{"kind":"expression","value":"(graph_confusion, [[0, 1, 2], [3, 4, 5]], [[0, 1, 2, 3, 4, 5]], Yes, No, same_carrier_different_structures_give_different_components)","explanation":"Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (graph_confusion, [[0, 1, 2], [3, 4, 5]], [[0, 1, 2, 3, 4, 5]], Yes, No, same_carrier_different_structures_give_different_components)."},"skills":["Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0079","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8231fade5de1d673d87ae0e8c11ddc596a61ffcf0e7c1304aedc2cb1d761cc0d","family_id":"qf-mod48-component-boundary","seed":202611177079,"semantic_template":"component-boundary|{mode:\"cardinality_not_definition\",model:{basepoint:0,points:[0,1,2],topology:[[],[0,1,2],[0],[1,2]]}}"},"fields":{"title":"Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition","stem":"Use mode=cardinality_not_definition and complete model={\"points\":[0,1,2],\"topology\":[[],[0,1,2],[0],[1,2]],\"basepoint\":0}. Give (mode, basepoint, own_component, larger_other_component, own_cardinality, other_cardinality, global_cardinality_definition_claim, boundary).","answer":{"kind":"expression","value":"(cardinality_not_definition, 0, [0], [1, 2], 1, 2, No, component_is_inclusion_maximal_not_global_maximum_cardinality)","explanation":"Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (cardinality_not_definition, 0, [0], [1, 2], 1, 2, No, component_is_inclusion_maximal_not_global_maximum_cardinality)."},"skills":["Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0080","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a78253ac5cbe6510cd86e6ca6e3b803bb69a068aa72791abde8cec8a0b551645","family_id":"qf-mod48-component-boundary","seed":202611177080,"semantic_template":"component-boundary|{mode:\"finite_clopen\",model:{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,3],[2,4,5]]}}"},"fields":{"title":"Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition","stem":"Use mode=finite_clopen and complete model={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,3],[2,4,5]]}. Give (mode, components, closed_flags, open_flags, all_clopen, certificate).","answer":{"kind":"expression","value":"(finite_clopen, [[0, 1, 3], [2, 4, 5]], [Yes, Yes], [Yes, Yes], Yes, finite_number_of_closed_components_makes_each_open)","explanation":"Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (finite_clopen, [[0, 1, 3], [2, 4, 5]], [Yes, Yes], [Yes, Yes], Yes, finite_number_of_closed_components_makes_each_open)."},"skills":["Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0081","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88b3897420f0a59bdcbd2f6cefd2df136e6b1d918ed57c0d3dd3c926e0e0b89c","family_id":"qf-mod48-component-boundary","seed":202611177081,"semantic_template":"component-boundary|{mode:\"finite_no_nonopen_witness\",model:{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2,3],[1,4,5]]}}"},"fields":{"title":"Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition","stem":"Use mode=finite_no_nonopen_witness and complete model={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,2,3],[1,4,5]]}. Give (mode, components, all_clopen, general_nonopen_witness_claim, boundary).","answer":{"kind":"expression","value":"(finite_no_nonopen_witness, [[0, 2, 3], [1, 4, 5]], Yes, No, finite_models_cannot_witness_a_general_nonopen_component)","explanation":"Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (finite_no_nonopen_witness, [[0, 2, 3], [1, 4, 5]], Yes, No, finite_models_cannot_witness_a_general_nonopen_component)."},"skills":["Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0082","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8035de975a2fa3084c86c024f96f792d15252981a103c49d9caf4e9d0c29826c","family_id":"qf-mod48-component-boundary","seed":202611177082,"semantic_template":"component-boundary|{mode:\"graph_confusion\",model:{graph_edges:[[0,1],[1,2],[2,3],[3,4],[4,5]],points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2,3],[4,5]]}}"},"fields":{"title":"Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition","stem":"Use mode=graph_confusion and complete model={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,2,3],[4,5]],\"graph_edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]]}. Give (mode, topological_components, graph_components, different_partitions, structure_identification_claim, boundary).","answer":{"kind":"expression","value":"(graph_confusion, [[0, 1, 2, 3], [4, 5]], [[0, 1, 2, 3, 4, 5]], Yes, No, same_carrier_different_structures_give_different_components)","explanation":"Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (graph_confusion, [[0, 1, 2, 3], [4, 5]], [[0, 1, 2, 3, 4, 5]], Yes, No, same_carrier_different_structures_give_different_components)."},"skills":["Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0083","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e4d3969e035824bf3e3e708b39a8a4fde0f068effabdc41f660e6cc75e54442","family_id":"qf-mod48-component-boundary","seed":202611177083,"semantic_template":"component-boundary|{mode:\"cardinality_not_definition\",model:{basepoint:0,points:[0,1,2,3],topology:[[],[0,1,2,3],[0],[1,2,3]]}}"},"fields":{"title":"Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition","stem":"Use mode=cardinality_not_definition and complete model={\"points\":[0,1,2,3],\"topology\":[[],[0,1,2,3],[0],[1,2,3]],\"basepoint\":0}. Give (mode, basepoint, own_component, larger_other_component, own_cardinality, other_cardinality, global_cardinality_definition_claim, boundary).","answer":{"kind":"expression","value":"(cardinality_not_definition, 0, [0], [1, 2, 3], 1, 3, No, component_is_inclusion_maximal_not_global_maximum_cardinality)","explanation":"Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (cardinality_not_definition, 0, [0], [1, 2, 3], 1, 3, No, component_is_inclusion_maximal_not_global_maximum_cardinality)."},"skills":["Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0084","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c74fcccc8cf49ed0cd5e89956f156950e8d0ff0de41e7cdc7f8647012eb203d0","family_id":"qf-mod48-component-boundary","seed":202611177084,"semantic_template":"component-boundary|{mode:\"finite_clopen\",model:{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,4],[2,3,5]]}}"},"fields":{"title":"Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition","stem":"Use mode=finite_clopen and complete model={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,4],[2,3,5]]}. Give (mode, components, closed_flags, open_flags, all_clopen, certificate).","answer":{"kind":"expression","value":"(finite_clopen, [[0, 1, 4], [2, 3, 5]], [Yes, Yes], [Yes, Yes], Yes, finite_number_of_closed_components_makes_each_open)","explanation":"Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (finite_clopen, [[0, 1, 4], [2, 3, 5]], [Yes, Yes], [Yes, Yes], Yes, finite_number_of_closed_components_makes_each_open)."},"skills":["Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0085","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72170dd8afeca557889f908c9ae8857770e010c91213077ec1508ee253b01128","family_id":"qf-mod48-component-boundary","seed":202611177085,"semantic_template":"component-boundary|{mode:\"finite_no_nonopen_witness\",model:{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2,4],[1,3,5]]}}"},"fields":{"title":"Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition","stem":"Use mode=finite_no_nonopen_witness and complete model={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,2,4],[1,3,5]]}. Give (mode, components, all_clopen, general_nonopen_witness_claim, boundary).","answer":{"kind":"expression","value":"(finite_no_nonopen_witness, [[0, 2, 4], [1, 3, 5]], Yes, No, finite_models_cannot_witness_a_general_nonopen_component)","explanation":"Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (finite_no_nonopen_witness, [[0, 2, 4], [1, 3, 5]], Yes, No, finite_models_cannot_witness_a_general_nonopen_component)."},"skills":["Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0086","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84b5d3d69edb95ca7ce0f709357e0c119d22776e65377093a6a8137c5e14bbda","family_id":"qf-mod48-component-boundary","seed":202611177086,"semantic_template":"component-boundary|{mode:\"graph_confusion\",model:{graph_edges:[[0,1],[1,2],[2,3],[3,4],[4,5]],points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2,4],[3,5]]}}"},"fields":{"title":"Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition","stem":"Use mode=graph_confusion and complete model={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,2,4],[3,5]],\"graph_edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]]}. Give (mode, topological_components, graph_components, different_partitions, structure_identification_claim, boundary).","answer":{"kind":"expression","value":"(graph_confusion, [[0, 1, 2, 4], [3, 5]], [[0, 1, 2, 3, 4, 5]], Yes, No, same_carrier_different_structures_give_different_components)","explanation":"Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (graph_confusion, [[0, 1, 2, 4], [3, 5]], [[0, 1, 2, 3, 4, 5]], Yes, No, same_carrier_different_structures_give_different_components)."},"skills":["Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0087","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd72b8ec419c1107789faa92e288f07354ee1540509b5a8211740867a9c1264c","family_id":"qf-mod48-component-boundary","seed":202611177087,"semantic_template":"component-boundary|{mode:\"cardinality_not_definition\",model:{basepoint:0,points:[0,1,2,3,4],topology:[[],[0,1,2,3,4],[0],[1,2,3,4]]}}"},"fields":{"title":"Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition","stem":"Use mode=cardinality_not_definition and complete model={\"points\":[0,1,2,3,4],\"topology\":[[],[0,1,2,3,4],[0],[1,2,3,4]],\"basepoint\":0}. Give (mode, basepoint, own_component, larger_other_component, own_cardinality, other_cardinality, global_cardinality_definition_claim, boundary).","answer":{"kind":"expression","value":"(cardinality_not_definition, 0, [0], [1, 2, 3, 4], 1, 4, No, component_is_inclusion_maximal_not_global_maximum_cardinality)","explanation":"Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (cardinality_not_definition, 0, [0], [1, 2, 3, 4], 1, 4, No, component_is_inclusion_maximal_not_global_maximum_cardinality)."},"skills":["Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0088","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f237b5650784d5febf8fa0d5d72c6f53e5743ef76d094d307301dd9a2336119","family_id":"qf-mod48-component-boundary","seed":202611177088,"semantic_template":"component-boundary|{mode:\"finite_clopen\",model:{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,3,4],[2,5]]}}"},"fields":{"title":"Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition","stem":"Use mode=finite_clopen and complete model={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,3,4],[2,5]]}. Give (mode, components, closed_flags, open_flags, all_clopen, certificate).","answer":{"kind":"expression","value":"(finite_clopen, [[0, 1, 3, 4], [2, 5]], [Yes, Yes], [Yes, Yes], Yes, finite_number_of_closed_components_makes_each_open)","explanation":"Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (finite_clopen, [[0, 1, 3, 4], [2, 5]], [Yes, Yes], [Yes, Yes], Yes, finite_number_of_closed_components_makes_each_open)."},"skills":["Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0089","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b2ccef273afe1c2b420178190640afa8dbe3138144c44979a4c088d9153a19a","family_id":"qf-mod48-component-boundary","seed":202611177089,"semantic_template":"component-boundary|{mode:\"finite_no_nonopen_witness\",model:{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2,3,4],[1,5]]}}"},"fields":{"title":"Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition","stem":"Use mode=finite_no_nonopen_witness and complete model={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,2,3,4],[1,5]]}. Give (mode, components, all_clopen, general_nonopen_witness_claim, boundary).","answer":{"kind":"expression","value":"(finite_no_nonopen_witness, [[0, 2, 3, 4], [1, 5]], Yes, No, finite_models_cannot_witness_a_general_nonopen_component)","explanation":"Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (finite_no_nonopen_witness, [[0, 2, 3, 4], [1, 5]], Yes, No, finite_models_cannot_witness_a_general_nonopen_component)."},"skills":["Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0090","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"137f38cdbd5abb9be065e922ccd37051e6cc505254fe7ab59a799d2ba6fda00b","family_id":"qf-mod48-component-boundary","seed":202611177090,"semantic_template":"component-boundary|{mode:\"graph_confusion\",model:{graph_edges:[[0,1],[1,2],[2,3],[3,4],[4,5]],points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2,3,4],[5]]}}"},"fields":{"title":"Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition","stem":"Use mode=graph_confusion and complete model={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,2,3,4],[5]],\"graph_edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]]}. Give (mode, topological_components, graph_components, different_partitions, structure_identification_claim, boundary).","answer":{"kind":"expression","value":"(graph_confusion, [[0, 1, 2, 3, 4], [5]], [[0, 1, 2, 3, 4, 5]], Yes, No, same_carrier_different_structures_give_different_components)","explanation":"Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (graph_confusion, [[0, 1, 2, 3, 4], [5]], [[0, 1, 2, 3, 4, 5]], Yes, No, same_carrier_different_structures_give_different_components)."},"skills":["Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0091","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a12b58bb1ece41b6aa3092b8254a62ea1454f84cbe42dea81247c9e62a777041","family_id":"qf-mod48-component-boundary","seed":202611177091,"semantic_template":"component-boundary|{mode:\"cardinality_not_definition\",model:{basepoint:0,points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0],[1,2,3,4,5]]}}"},"fields":{"title":"Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition","stem":"Use mode=cardinality_not_definition and complete model={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0],[1,2,3,4,5]],\"basepoint\":0}. Give (mode, basepoint, own_component, larger_other_component, own_cardinality, other_cardinality, global_cardinality_definition_claim, boundary).","answer":{"kind":"expression","value":"(cardinality_not_definition, 0, [0], [1, 2, 3, 4, 5], 1, 5, No, component_is_inclusion_maximal_not_global_maximum_cardinality)","explanation":"Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (cardinality_not_definition, 0, [0], [1, 2, 3, 4, 5], 1, 5, No, component_is_inclusion_maximal_not_global_maximum_cardinality)."},"skills":["Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0092","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a1a79bc2010c078f6f77a0fa6bdab3c26f3cdc0e2e8657f56fc3971a8db58ad","family_id":"qf-mod48-component-boundary","seed":202611177092,"semantic_template":"component-boundary|{mode:\"finite_clopen\",model:{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,5],[2,3,4]]}}"},"fields":{"title":"Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition","stem":"Use mode=finite_clopen and complete model={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,5],[2,3,4]]}. Give (mode, components, closed_flags, open_flags, all_clopen, certificate).","answer":{"kind":"expression","value":"(finite_clopen, [[0, 1, 5], [2, 3, 4]], [Yes, Yes], [Yes, Yes], Yes, finite_number_of_closed_components_makes_each_open)","explanation":"Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (finite_clopen, [[0, 1, 5], [2, 3, 4]], [Yes, Yes], [Yes, Yes], Yes, finite_number_of_closed_components_makes_each_open)."},"skills":["Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0093","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35d62776a1af18227ad6b2ad7649683deadf9a1f938e2874bac443f13f0331f3","family_id":"qf-mod48-component-boundary","seed":202611177093,"semantic_template":"component-boundary|{mode:\"finite_no_nonopen_witness\",model:{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2,5],[1,3,4]]}}"},"fields":{"title":"Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition","stem":"Use mode=finite_no_nonopen_witness and complete model={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,2,5],[1,3,4]]}. Give (mode, components, all_clopen, general_nonopen_witness_claim, boundary).","answer":{"kind":"expression","value":"(finite_no_nonopen_witness, [[0, 2, 5], [1, 3, 4]], Yes, No, finite_models_cannot_witness_a_general_nonopen_component)","explanation":"Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (finite_no_nonopen_witness, [[0, 2, 5], [1, 3, 4]], Yes, No, finite_models_cannot_witness_a_general_nonopen_component)."},"skills":["Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0094","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb321f2bdc654c05db36a0f5b104ef52917dd226358aa139ee79f1278f0625f9","family_id":"qf-mod48-component-boundary","seed":202611177094,"semantic_template":"component-boundary|{mode:\"graph_confusion\",model:{graph_edges:[[0,1],[1,2],[2,3],[3,4],[4,5]],points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2,5],[3,4]]}}"},"fields":{"title":"Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition","stem":"Use mode=graph_confusion and complete model={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,2,5],[3,4]],\"graph_edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]]}. Give (mode, topological_components, graph_components, different_partitions, structure_identification_claim, boundary).","answer":{"kind":"expression","value":"(graph_confusion, [[0, 1, 2, 5], [3, 4]], [[0, 1, 2, 3, 4, 5]], Yes, No, same_carrier_different_structures_give_different_components)","explanation":"Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (graph_confusion, [[0, 1, 2, 5], [3, 4]], [[0, 1, 2, 3, 4, 5]], Yes, No, same_carrier_different_structures_give_different_components)."},"skills":["Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0095","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"09f910886c9041bf8df94cf9ea71cdd503e78c2f811ccb4d82b19f99b05a5950","family_id":"qf-mod48-component-boundary","seed":202611177095,"semantic_template":"component-boundary|{mode:\"cardinality_not_definition\",model:{basepoint:0,points:[0,1,2,3,4,5,6],topology:[[],[0,1,2,3,4,5,6],[0],[1,2,3,4,5,6]]}}"},"fields":{"title":"Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition","stem":"Use mode=cardinality_not_definition and complete model={\"points\":[0,1,2,3,4,5,6],\"topology\":[[],[0,1,2,3,4,5,6],[0],[1,2,3,4,5,6]],\"basepoint\":0}. Give (mode, basepoint, own_component, larger_other_component, own_cardinality, other_cardinality, global_cardinality_definition_claim, boundary).","answer":{"kind":"expression","value":"(cardinality_not_definition, 0, [0], [1, 2, 3, 4, 5, 6], 1, 6, No, component_is_inclusion_maximal_not_global_maximum_cardinality)","explanation":"Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (cardinality_not_definition, 0, [0], [1, 2, 3, 4, 5, 6], 1, 6, No, component_is_inclusion_maximal_not_global_maximum_cardinality)."},"skills":["Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0096","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71cd81f5e3e37b3a6102347cfd2e92217ccba57df144a699757d88c0f2ed2b0c","family_id":"qf-mod48-component-boundary","seed":202611177096,"semantic_template":"component-boundary|{mode:\"finite_clopen\",model:{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,3,5],[2,4]]}}"},"fields":{"title":"Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition","stem":"Use mode=finite_clopen and complete model={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,3,5],[2,4]]}. Give (mode, components, closed_flags, open_flags, all_clopen, certificate).","answer":{"kind":"expression","value":"(finite_clopen, [[0, 1, 3, 5], [2, 4]], [Yes, Yes], [Yes, Yes], Yes, finite_number_of_closed_components_makes_each_open)","explanation":"Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (finite_clopen, [[0, 1, 3, 5], [2, 4]], [Yes, Yes], [Yes, Yes], Yes, finite_number_of_closed_components_makes_each_open)."},"skills":["Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0097","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ee7bf2f3b2b8c2063fcf152473d90e2a4d3659bea59d36c3b579e7fad9af39a","family_id":"qf-mod48-component-boundary","seed":202611177097,"semantic_template":"component-boundary|{mode:\"finite_no_nonopen_witness\",model:{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,2,3,5],[1,4]]}}"},"fields":{"title":"Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition","stem":"Use mode=finite_no_nonopen_witness and complete model={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,2,3,5],[1,4]]}. Give (mode, components, all_clopen, general_nonopen_witness_claim, boundary).","answer":{"kind":"expression","value":"(finite_no_nonopen_witness, [[0, 2, 3, 5], [1, 4]], Yes, No, finite_models_cannot_witness_a_general_nonopen_component)","explanation":"Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (finite_no_nonopen_witness, [[0, 2, 3, 5], [1, 4]], Yes, No, finite_models_cannot_witness_a_general_nonopen_component)."},"skills":["Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0098","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2fa9c9883e4b7d1050d6181164824eff77a355596b5a4dcd44fc709692600df","family_id":"qf-mod48-component-boundary","seed":202611177098,"semantic_template":"component-boundary|{mode:\"graph_confusion\",model:{graph_edges:[[0,1],[1,2],[2,3],[3,4],[4,5]],points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,2,3,5],[4]]}}"},"fields":{"title":"Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition","stem":"Use mode=graph_confusion and complete model={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,2,3,5],[4]],\"graph_edges\":[[0,1],[1,2],[2,3],[3,4],[4,5]]}. Give (mode, topological_components, graph_components, different_partitions, structure_identification_claim, boundary).","answer":{"kind":"expression","value":"(graph_confusion, [[0, 1, 2, 3, 5], [4]], [[0, 1, 2, 3, 4, 5]], Yes, No, same_carrier_different_structures_give_different_components)","explanation":"Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (graph_confusion, [[0, 1, 2, 3, 5], [4]], [[0, 1, 2, 3, 4, 5]], Yes, No, same_carrier_different_structures_give_different_components)."},"skills":["Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0099","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3204a15b5a2cb5ec93f18578000b4a70230257bd8fcd31c0aed2a450aaf7b745","family_id":"qf-mod48-component-boundary","seed":202611177099,"semantic_template":"component-boundary|{mode:\"cardinality_not_definition\",model:{basepoint:0,points:[0,1,2,3,4,5,6,7],topology:[[],[0,1,2,3,4,5,6,7],[0],[1,2,3,4,5,6,7]]}}"},"fields":{"title":"Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition","stem":"Use mode=cardinality_not_definition and complete model={\"points\":[0,1,2,3,4,5,6,7],\"topology\":[[],[0,1,2,3,4,5,6,7],[0],[1,2,3,4,5,6,7]],\"basepoint\":0}. Give (mode, basepoint, own_component, larger_other_component, own_cardinality, other_cardinality, global_cardinality_definition_claim, boundary).","answer":{"kind":"expression","value":"(cardinality_not_definition, 0, [0], [1, 2, 3, 4, 5, 6, 7], 1, 7, No, component_is_inclusion_maximal_not_global_maximum_cardinality)","explanation":"Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (cardinality_not_definition, 0, [0], [1, 2, 3, 4, 5, 6, 7], 1, 7, No, component_is_inclusion_maximal_not_global_maximum_cardinality)."},"skills":["Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r07-topological-connected-component-0100","batch_id":"batch-20261015-advance-20261117-r07-topological-connected-component-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c232c0e30b57aa40b926a3d7e510134c1ecaa7e6a07a777fc73c5a06bfa293d","family_id":"qf-mod48-component-boundary","seed":202611177100,"semantic_template":"component-boundary|{mode:\"finite_clopen\",model:{points:[0,1,2,3,4,5],topology:[[],[0,1,2,3,4,5],[0,1,4,5],[2,3]]}}"},"fields":{"title":"Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition","stem":"Use mode=finite_clopen and complete model={\"points\":[0,1,2,3,4,5],\"topology\":[[],[0,1,2,3,4,5],[0,1,4,5],[2,3]]}. Give (mode, components, closed_flags, open_flags, all_clopen, certificate).","answer":{"kind":"expression","value":"(finite_clopen, [[0, 1, 4, 5], [2, 3]], [Yes, Yes], [Yes, Yes], Yes, finite_number_of_closed_components_makes_each_open)","explanation":"Finite-clopen modes retain the actual topology components and cannot fabricate a nonopen finite example. Graph-confusion mode compares a separately declared graph structure. Cardinality mode retains the basepoint component and a larger distinct one, showing inclusion-maximal is not globally largest. Keep each mode field meaning separate. The exact result, retaining the requested order, is (finite_clopen, [[0, 1, 4, 5], [2, 3]], [Yes, Yes], [Yes, Yes], Yes, finite_number_of_closed_components_makes_each_open)."},"skills":["Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}