{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0001","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"481de07b115c6023b72d03d62a8e27ac9ae8502d851687b448db9c65487223fe","family_id":"qf-mod40-compact-finite-compactness","seed":202611093001,"semantic_template":"qf-mod40-compact-finite-compactness/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"The Complete Finite Relative Topology Supplies Compactness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).","answer":{"kind":"expression","value":"([1, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)","explanation":"Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([1, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)."},"skills":["The Complete Finite Relative Topology Supplies Compactness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0002","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64966689567d83d8cb859ae74c726448624102f1dddfd8f3a8aa6e5cfe904045","family_id":"qf-mod40-compact-finite-compactness","seed":202611093002,"semantic_template":"qf-mod40-compact-finite-compactness/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,1],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"The Complete Finite Relative Topology Supplies Compactness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3, 4], [0, 1, 2, 4]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).","answer":{"kind":"expression","value":"([1, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)","explanation":"Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([1, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)."},"skills":["The Complete Finite Relative Topology Supplies Compactness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0003","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"33a573282972fa24c85c7f607f687aa954c7a75713b88daa686f822d6d33cbff","family_id":"qf-mod40-compact-finite-compactness","seed":202611093003,"semantic_template":"qf-mod40-compact-finite-compactness/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,1],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"The Complete Finite Relative Topology Supplies Compactness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).","answer":{"kind":"expression","value":"([1, 2, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)","explanation":"Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([1, 2, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)."},"skills":["The Complete Finite Relative Topology Supplies Compactness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0004","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7c19686bf9185b738849a5cdc1c32f947103302bbd7b48fd7845415d4b6b87c","family_id":"qf-mod40-compact-finite-compactness","seed":202611093004,"semantic_template":"qf-mod40-compact-finite-compactness/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,1],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"The Complete Finite Relative Topology Supplies Compactness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).","answer":{"kind":"expression","value":"([1, 2, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)","explanation":"Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([1, 2, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)."},"skills":["The Complete Finite Relative Topology Supplies Compactness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0005","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0bc403e9fa8d98751ba25780c7ac4546175a1962edb88022c0ceba86963b3fa7","family_id":"qf-mod40-compact-finite-compactness","seed":202611093005,"semantic_template":"qf-mod40-compact-finite-compactness/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,1],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"The Complete Finite Relative Topology Supplies Compactness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).","answer":{"kind":"expression","value":"([1, 2, 3], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)","explanation":"Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([1, 2, 3], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)."},"skills":["The Complete Finite Relative Topology Supplies Compactness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0006","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e89fa0b4702c8938baa6b205d02512756eacc3239ad6c86c9fb902b8e204d76","family_id":"qf-mod40-compact-finite-compactness","seed":202611093006,"semantic_template":"qf-mod40-compact-finite-compactness/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,1],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"The Complete Finite Relative Topology Supplies Compactness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 1, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).","answer":{"kind":"expression","value":"([1, 2, 3], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)","explanation":"Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([1, 2, 3], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)."},"skills":["The Complete Finite Relative Topology Supplies Compactness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0007","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e19d803165dbeae00111706ff3db0da9b83d2561a46f50b8a1a93ca96a24dabb","family_id":"qf-mod40-compact-finite-compactness","seed":202611093007,"semantic_template":"qf-mod40-compact-finite-compactness/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,2],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"The Complete Finite Relative Topology Supplies Compactness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).","answer":{"kind":"expression","value":"([2, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)","explanation":"Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([2, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)."},"skills":["The Complete Finite Relative Topology Supplies Compactness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0008","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"10a261413a8a504036423614503b2b639ee72d90580bf2559bf374bf78db6290","family_id":"qf-mod40-compact-finite-compactness","seed":202611093008,"semantic_template":"qf-mod40-compact-finite-compactness/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,2],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"The Complete Finite Relative Topology Supplies Compactness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).","answer":{"kind":"expression","value":"([2, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)","explanation":"Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([2, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)."},"skills":["The Complete Finite Relative Topology Supplies Compactness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0009","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7586bff89c74489af3a2d57698c897c57dd7d7cfa31c416626091e263714f466","family_id":"qf-mod40-compact-finite-compactness","seed":202611093009,"semantic_template":"qf-mod40-compact-finite-compactness/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,2],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"The Complete Finite Relative Topology Supplies Compactness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).","answer":{"kind":"expression","value":"([1, 2, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)","explanation":"Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([1, 2, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)."},"skills":["The Complete Finite Relative Topology Supplies Compactness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0010","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2612791165ffd900996da12169876af9049c14f730820903bd2798801ec4852b","family_id":"qf-mod40-compact-finite-compactness","seed":202611093010,"semantic_template":"qf-mod40-compact-finite-compactness/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,2],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"The Complete Finite Relative Topology Supplies Compactness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).","answer":{"kind":"expression","value":"([1, 2, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)","explanation":"Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([1, 2, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)."},"skills":["The Complete Finite Relative Topology Supplies Compactness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0011","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14835a3e11cbed62c80128086490c75389dc44b4cf84c308b7e069c575726024","family_id":"qf-mod40-compact-finite-compactness","seed":202611093011,"semantic_template":"qf-mod40-compact-finite-compactness/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,2],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"The Complete Finite Relative Topology Supplies Compactness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2], [0, 2, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).","answer":{"kind":"expression","value":"([1, 2, 3], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)","explanation":"Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([1, 2, 3], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)."},"skills":["The Complete Finite Relative Topology Supplies Compactness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0012","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9eecd2949733a742a2c8511ecfc2ea61cd0d417af0e608d1fbb27627e6f59b28","family_id":"qf-mod40-compact-finite-compactness","seed":202611093012,"semantic_template":"qf-mod40-compact-finite-compactness/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,2],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"The Complete Finite Relative Topology Supplies Compactness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3, 4], [0, 2, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).","answer":{"kind":"expression","value":"([1, 2, 3], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)","explanation":"Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([1, 2, 3], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)."},"skills":["The Complete Finite Relative Topology Supplies Compactness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0013","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"787e79b2b91415966508a11feea2f01c715613da7b152fd413c2f149cfb9ea41","family_id":"qf-mod40-compact-finite-compactness","seed":202611093013,"semantic_template":"qf-mod40-compact-finite-compactness/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,3],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"The Complete Finite Relative Topology Supplies Compactness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3], [0, 3]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).","answer":{"kind":"expression","value":"([2, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)","explanation":"Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([2, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)."},"skills":["The Complete Finite Relative Topology Supplies Compactness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0014","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d15cc0a6285e5c890ee5958f2939a77cf08943172590c50a27dbd0ca2fec4505","family_id":"qf-mod40-compact-finite-compactness","seed":202611093014,"semantic_template":"qf-mod40-compact-finite-compactness/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,3],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"The Complete Finite Relative Topology Supplies Compactness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4], [0, 3]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).","answer":{"kind":"expression","value":"([2, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)","explanation":"Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([2, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)."},"skills":["The Complete Finite Relative Topology Supplies Compactness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0015","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"005b5e475dee0b29ec372a3ef0d2ca7bb935c0a73c148e42e1452765207e0ed0","family_id":"qf-mod40-compact-finite-compactness","seed":202611093015,"semantic_template":"qf-mod40-compact-finite-compactness/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,3],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"The Complete Finite Relative Topology Supplies Compactness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2, 3], [0, 3]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).","answer":{"kind":"expression","value":"([1, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)","explanation":"Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([1, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)."},"skills":["The Complete Finite Relative Topology Supplies Compactness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0016","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fdd5848a628ea6320465b82ad4fd7dadceee453dc319ec1abc644fd71bbb8eee","family_id":"qf-mod40-compact-finite-compactness","seed":202611093016,"semantic_template":"qf-mod40-compact-finite-compactness/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,3],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"The Complete Finite Relative Topology Supplies Compactness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3], [0, 2, 3, 4], [0, 3]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).","answer":{"kind":"expression","value":"([1, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)","explanation":"Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([1, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)."},"skills":["The Complete Finite Relative Topology Supplies Compactness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0017","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"347e33d53d9994ce7bfe50a58ff2a11be4ff31873044270e8fef8a673c0a2450","family_id":"qf-mod40-compact-finite-compactness","seed":202611093017,"semantic_template":"qf-mod40-compact-finite-compactness/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,3],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"The Complete Finite Relative Topology Supplies Compactness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 3], [0, 3, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).","answer":{"kind":"expression","value":"([1, 2, 3], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)","explanation":"Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([1, 2, 3], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)."},"skills":["The Complete Finite Relative Topology Supplies Compactness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0018","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e77aac12f9127bcb9acc252baff09210d0039b17f2df646b09597875e741e06","family_id":"qf-mod40-compact-finite-compactness","seed":202611093018,"semantic_template":"qf-mod40-compact-finite-compactness/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,3],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"The Complete Finite Relative Topology Supplies Compactness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 3], [0, 3, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).","answer":{"kind":"expression","value":"([1, 2, 3], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)","explanation":"Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([1, 2, 3], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)."},"skills":["The Complete Finite Relative Topology Supplies Compactness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0019","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"98ae825c3ff5293e94e87294deebd9e26c0ad403e6359768e371333340086859","family_id":"qf-mod40-compact-finite-compactness","seed":202611093019,"semantic_template":"qf-mod40-compact-finite-compactness/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,4],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"The Complete Finite Relative Topology Supplies Compactness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 4], [0, 4]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).","answer":{"kind":"expression","value":"([2, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)","explanation":"Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([2, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)."},"skills":["The Complete Finite Relative Topology Supplies Compactness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0020","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4dccc960b74062c4e8717f85addac90fb93f4b0ef088f37d308fa896fc9508bf","family_id":"qf-mod40-compact-finite-compactness","seed":202611093020,"semantic_template":"qf-mod40-compact-finite-compactness/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,4],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"The Complete Finite Relative Topology Supplies Compactness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 1, 4], [0, 4]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).","answer":{"kind":"expression","value":"([2, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)","explanation":"Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([2, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)."},"skills":["The Complete Finite Relative Topology Supplies Compactness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0021","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd9450c09dc8f9f23fa6f22e248731b20e83d8fe7b47c008ef52169999164fa6","family_id":"qf-mod40-compact-finite-compactness","seed":202611093021,"semantic_template":"qf-mod40-compact-finite-compactness/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,4],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"The Complete Finite Relative Topology Supplies Compactness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2, 4], [0, 4]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).","answer":{"kind":"expression","value":"([1, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)","explanation":"Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([1, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)."},"skills":["The Complete Finite Relative Topology Supplies Compactness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0022","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16597bd7b88227135f29c907d13afa7f5046d629ad3036d83b75893ea5f17ce8","family_id":"qf-mod40-compact-finite-compactness","seed":202611093022,"semantic_template":"qf-mod40-compact-finite-compactness/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,4],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"The Complete Finite Relative Topology Supplies Compactness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 2, 4], [0, 4]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).","answer":{"kind":"expression","value":"([1, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)","explanation":"Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([1, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)."},"skills":["The Complete Finite Relative Topology Supplies Compactness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0023","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ffbf51ff300e42a87e1ae1f36147df29d0c90e0ccfb9c9f6380832713e624807","family_id":"qf-mod40-compact-finite-compactness","seed":202611093023,"semantic_template":"qf-mod40-compact-finite-compactness/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,4],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"The Complete Finite Relative Topology Supplies Compactness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 3, 4], [0, 4]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).","answer":{"kind":"expression","value":"([1, 2, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)","explanation":"Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([1, 2, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)."},"skills":["The Complete Finite Relative Topology Supplies Compactness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0024","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a7d7d3cec9bf620f7d53a4e0841cf7c0de70de79ff138913d4d6682aaa8a623","family_id":"qf-mod40-compact-finite-compactness","seed":202611093024,"semantic_template":"qf-mod40-compact-finite-compactness/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,4],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"The Complete Finite Relative Topology Supplies Compactness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 3, 4], [0, 4]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).","answer":{"kind":"expression","value":"([1, 2, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)","explanation":"Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([1, 2, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)."},"skills":["The Complete Finite Relative Topology Supplies Compactness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0025","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bde98443157c0eb22f5f08e352e5e6337420c905d52801271a14d942cf56d2b0","family_id":"qf-mod40-compact-finite-compactness","seed":202611093025,"semantic_template":"qf-mod40-compact-finite-compactness/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[1]],\"subset\":[0,3,4]}"},"fields":{"title":"The Complete Finite Relative Topology Supplies Compactness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [1]], subset K=[0, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, relative_open_count, compact, empty_subset, certificate).","answer":{"kind":"expression","value":"([0, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)","explanation":"Compute the number of distinct relative opens. A cover drawn from this finite open-set family has only finitely many distinct members, giving a finite subcover whenever it covers K. This finite-model theorem includes the empty subset; it is not a sampled-cover proof for an infinite topology. The exact result, retaining the requested order, is ([0, 3, 4], 4, Yes, No, every_open_cover_drawn_from_finite_topology_is_finite)."},"skills":["The Complete Finite Relative Topology Supplies Compactness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0026","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f1e9d82be9ad02f1a237451dadf9fc748eb202e86c406d1407f8c415b0cfc13","family_id":"qf-mod40-compact-open-cover-audit","seed":202611093026,"semantic_template":"qf-mod40-compact-open-cover-audit/{\"cover\":[[],[1,3,4],[1,3],[1]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [1], [1, 3], [1, 3, 4]]. Give (cover, covers_subset, union, boundary).","answer":{"kind":"expression","value":"([[], [1], [1, 3], [1, 3, 4]], Yes, [1, 3, 4], single_cover_audit_not_general_compactness_proof)","explanation":"Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [1], [1, 3], [1, 3, 4]], Yes, [1, 3, 4], single_cover_audit_not_general_compactness_proof)."},"skills":["Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0027","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fbde15253dce34296a97c0b9ba41e2cf7917598b306618bc2c7e34cab31b00b2","family_id":"qf-mod40-compact-open-cover-audit","seed":202611093027,"semantic_template":"qf-mod40-compact-open-cover-audit/{\"cover\":[[],[1,3,4],[1,4],[1]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,1],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3, 4], [0, 1, 2, 4]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [1], [1, 3, 4], [1, 4]]. Give (cover, covers_subset, union, boundary).","answer":{"kind":"expression","value":"([[], [1], [1, 3, 4], [1, 4]], Yes, [1, 3, 4], single_cover_audit_not_general_compactness_proof)","explanation":"Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [1], [1, 3, 4], [1, 4]], Yes, [1, 3, 4], single_cover_audit_not_general_compactness_proof)."},"skills":["Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0028","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ec4723d9926c0fb4f361b1125e9f4c1b14e2536c3f0ae828b210430f9f3adbc","family_id":"qf-mod40-compact-open-cover-audit","seed":202611093028,"semantic_template":"qf-mod40-compact-open-cover-audit/{\"cover\":[[],[1,2,4],[1,2],[1]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,1],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [1], [1, 2], [1, 2, 4]]. Give (cover, covers_subset, union, boundary).","answer":{"kind":"expression","value":"([[], [1], [1, 2], [1, 2, 4]], Yes, [1, 2, 4], single_cover_audit_not_general_compactness_proof)","explanation":"Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [1], [1, 2], [1, 2, 4]], Yes, [1, 2, 4], single_cover_audit_not_general_compactness_proof)."},"skills":["Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0029","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eff414c8dcf70500ae92cb3accd229017b87504b0b0f1f483a2c7d47fbd86a3c","family_id":"qf-mod40-compact-open-cover-audit","seed":202611093029,"semantic_template":"qf-mod40-compact-open-cover-audit/{\"cover\":[[],[1,2,4],[1,4],[1]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,1],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [1], [1, 2, 4], [1, 4]]. Give (cover, covers_subset, union, boundary).","answer":{"kind":"expression","value":"([[], [1], [1, 2, 4], [1, 4]], Yes, [1, 2, 4], single_cover_audit_not_general_compactness_proof)","explanation":"Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [1], [1, 2, 4], [1, 4]], Yes, [1, 2, 4], single_cover_audit_not_general_compactness_proof)."},"skills":["Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0030","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06b636e1026b77ce1f75c98af8609ddcd48f71710804ac12c4634d71351a6df3","family_id":"qf-mod40-compact-open-cover-audit","seed":202611093030,"semantic_template":"qf-mod40-compact-open-cover-audit/{\"cover\":[[],[1,2,3],[1,2],[1]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,1],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [1], [1, 2], [1, 2, 3]]. Give (cover, covers_subset, union, boundary).","answer":{"kind":"expression","value":"([[], [1], [1, 2], [1, 2, 3]], Yes, [1, 2, 3], single_cover_audit_not_general_compactness_proof)","explanation":"Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [1], [1, 2], [1, 2, 3]], Yes, [1, 2, 3], single_cover_audit_not_general_compactness_proof)."},"skills":["Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0031","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dbfe7e09f9b8854c493ac742141832cfda690a08e8f8e604352e57842a8a39a5","family_id":"qf-mod40-compact-open-cover-audit","seed":202611093031,"semantic_template":"qf-mod40-compact-open-cover-audit/{\"cover\":[[],[1,2,3],[1,3],[1]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,1],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 1, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [1], [1, 2, 3], [1, 3]]. Give (cover, covers_subset, union, boundary).","answer":{"kind":"expression","value":"([[], [1], [1, 2, 3], [1, 3]], Yes, [1, 2, 3], single_cover_audit_not_general_compactness_proof)","explanation":"Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [1], [1, 2, 3], [1, 3]], Yes, [1, 2, 3], single_cover_audit_not_general_compactness_proof)."},"skills":["Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0032","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"967e708ab7f29fce3cfbb370f80585474d983b7ac51236e05a922de90d20c60a","family_id":"qf-mod40-compact-open-cover-audit","seed":202611093032,"semantic_template":"qf-mod40-compact-open-cover-audit/{\"cover\":[[],[2,3,4],[2,3],[2]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,2],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [2], [2, 3], [2, 3, 4]]. Give (cover, covers_subset, union, boundary).","answer":{"kind":"expression","value":"([[], [2], [2, 3], [2, 3, 4]], Yes, [2, 3, 4], single_cover_audit_not_general_compactness_proof)","explanation":"Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [2], [2, 3], [2, 3, 4]], Yes, [2, 3, 4], single_cover_audit_not_general_compactness_proof)."},"skills":["Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0033","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a5d76fb7fd1a086145454c6d22fc40882ea0285b795d23da497b09e8c688603","family_id":"qf-mod40-compact-open-cover-audit","seed":202611093033,"semantic_template":"qf-mod40-compact-open-cover-audit/{\"cover\":[[],[2,3,4],[2,4],[2]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,2],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [2], [2, 3, 4], [2, 4]]. Give (cover, covers_subset, union, boundary).","answer":{"kind":"expression","value":"([[], [2], [2, 3, 4], [2, 4]], Yes, [2, 3, 4], single_cover_audit_not_general_compactness_proof)","explanation":"Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [2], [2, 3, 4], [2, 4]], Yes, [2, 3, 4], single_cover_audit_not_general_compactness_proof)."},"skills":["Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0034","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"adc95eea3533bca9e62d017a1b41d4d9e2577278a10f287a6b5a7bb7e8c0693b","family_id":"qf-mod40-compact-open-cover-audit","seed":202611093034,"semantic_template":"qf-mod40-compact-open-cover-audit/{\"cover\":[[],[1,2,4],[1,2],[2]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,2],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [1, 2], [1, 2, 4], [2]]. Give (cover, covers_subset, union, boundary).","answer":{"kind":"expression","value":"([[], [1, 2], [1, 2, 4], [2]], Yes, [1, 2, 4], single_cover_audit_not_general_compactness_proof)","explanation":"Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [1, 2], [1, 2, 4], [2]], Yes, [1, 2, 4], single_cover_audit_not_general_compactness_proof)."},"skills":["Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0035","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9082f6b26decfc98dcbc7bc4fa96803d2fb1a66df854a2474cad4580ea7febe","family_id":"qf-mod40-compact-open-cover-audit","seed":202611093035,"semantic_template":"qf-mod40-compact-open-cover-audit/{\"cover\":[[],[1,2,4],[2,4],[2]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,2],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [1, 2, 4], [2], [2, 4]]. Give (cover, covers_subset, union, boundary).","answer":{"kind":"expression","value":"([[], [1, 2, 4], [2], [2, 4]], Yes, [1, 2, 4], single_cover_audit_not_general_compactness_proof)","explanation":"Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [1, 2, 4], [2], [2, 4]], Yes, [1, 2, 4], single_cover_audit_not_general_compactness_proof)."},"skills":["Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0036","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e299e656bafb9bd94c23537cd29ba19511df301e870b5aebb876184151671bea","family_id":"qf-mod40-compact-open-cover-audit","seed":202611093036,"semantic_template":"qf-mod40-compact-open-cover-audit/{\"cover\":[[],[1,2,3],[1,2],[2]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,2],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2], [0, 2, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [1, 2], [1, 2, 3], [2]]. Give (cover, covers_subset, union, boundary).","answer":{"kind":"expression","value":"([[], [1, 2], [1, 2, 3], [2]], Yes, [1, 2, 3], single_cover_audit_not_general_compactness_proof)","explanation":"Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [1, 2], [1, 2, 3], [2]], Yes, [1, 2, 3], single_cover_audit_not_general_compactness_proof)."},"skills":["Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0037","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"727794032da46734adad4610ac1a51faf70156cfb7ed83524fbb7319beee9f94","family_id":"qf-mod40-compact-open-cover-audit","seed":202611093037,"semantic_template":"qf-mod40-compact-open-cover-audit/{\"cover\":[[],[1,2,3],[2,3],[2]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,2],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3, 4], [0, 2, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [1, 2, 3], [2], [2, 3]]. Give (cover, covers_subset, union, boundary).","answer":{"kind":"expression","value":"([[], [1, 2, 3], [2], [2, 3]], Yes, [1, 2, 3], single_cover_audit_not_general_compactness_proof)","explanation":"Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [1, 2, 3], [2], [2, 3]], Yes, [1, 2, 3], single_cover_audit_not_general_compactness_proof)."},"skills":["Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0038","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e1042e06cac34d8a124a55afda3ce75b4fc7e205647d88b6038f3527df039392","family_id":"qf-mod40-compact-open-cover-audit","seed":202611093038,"semantic_template":"qf-mod40-compact-open-cover-audit/{\"cover\":[[],[2,3,4],[2,3],[3]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,3],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3], [0, 3]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [2, 3], [2, 3, 4], [3]]. Give (cover, covers_subset, union, boundary).","answer":{"kind":"expression","value":"([[], [2, 3], [2, 3, 4], [3]], Yes, [2, 3, 4], single_cover_audit_not_general_compactness_proof)","explanation":"Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [2, 3], [2, 3, 4], [3]], Yes, [2, 3, 4], single_cover_audit_not_general_compactness_proof)."},"skills":["Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0039","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c0f1d4ed4a492707b24732c580443a84bfd861bbaa4150ea38a38b1cf4adab2","family_id":"qf-mod40-compact-open-cover-audit","seed":202611093039,"semantic_template":"qf-mod40-compact-open-cover-audit/{\"cover\":[[],[2,3,4],[3,4],[3]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,3],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4], [0, 3]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [2, 3, 4], [3], [3, 4]]. Give (cover, covers_subset, union, boundary).","answer":{"kind":"expression","value":"([[], [2, 3, 4], [3], [3, 4]], Yes, [2, 3, 4], single_cover_audit_not_general_compactness_proof)","explanation":"Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [2, 3, 4], [3], [3, 4]], Yes, [2, 3, 4], single_cover_audit_not_general_compactness_proof)."},"skills":["Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0040","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"472d62d381e7e5926fecc7493662f7287c4e9b8f65e3b1fde693889153d27a4d","family_id":"qf-mod40-compact-open-cover-audit","seed":202611093040,"semantic_template":"qf-mod40-compact-open-cover-audit/{\"cover\":[[],[1,3,4],[1,3],[3]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,3],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2, 3], [0, 3]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [1, 3], [1, 3, 4], [3]]. Give (cover, covers_subset, union, boundary).","answer":{"kind":"expression","value":"([[], [1, 3], [1, 3, 4], [3]], Yes, [1, 3, 4], single_cover_audit_not_general_compactness_proof)","explanation":"Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [1, 3], [1, 3, 4], [3]], Yes, [1, 3, 4], single_cover_audit_not_general_compactness_proof)."},"skills":["Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0041","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4ddae6d1558b80ebdc5c337a576e8b8579e566989894b8b6ac0626d496c8f2f","family_id":"qf-mod40-compact-open-cover-audit","seed":202611093041,"semantic_template":"qf-mod40-compact-open-cover-audit/{\"cover\":[[],[1,3,4],[3,4],[3]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,3],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3], [0, 2, 3, 4], [0, 3]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [1, 3, 4], [3], [3, 4]]. Give (cover, covers_subset, union, boundary).","answer":{"kind":"expression","value":"([[], [1, 3, 4], [3], [3, 4]], Yes, [1, 3, 4], single_cover_audit_not_general_compactness_proof)","explanation":"Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [1, 3, 4], [3], [3, 4]], Yes, [1, 3, 4], single_cover_audit_not_general_compactness_proof)."},"skills":["Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0042","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"307fc386e9b2e397a7cc3996c8c586f59c855d81622c7bc100a86c26eed34ea8","family_id":"qf-mod40-compact-open-cover-audit","seed":202611093042,"semantic_template":"qf-mod40-compact-open-cover-audit/{\"cover\":[[],[1,2,3],[1,3],[3]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,3],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 3], [0, 3, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [1, 2, 3], [1, 3], [3]]. Give (cover, covers_subset, union, boundary).","answer":{"kind":"expression","value":"([[], [1, 2, 3], [1, 3], [3]], Yes, [1, 2, 3], single_cover_audit_not_general_compactness_proof)","explanation":"Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [1, 2, 3], [1, 3], [3]], Yes, [1, 2, 3], single_cover_audit_not_general_compactness_proof)."},"skills":["Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0043","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24816a2d249ae13e90a9e92b8611c18fdd1f0b08520e80185149f78175244824","family_id":"qf-mod40-compact-open-cover-audit","seed":202611093043,"semantic_template":"qf-mod40-compact-open-cover-audit/{\"cover\":[[],[1,2,3],[2,3],[3]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,3],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 3], [0, 3, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [1, 2, 3], [2, 3], [3]]. Give (cover, covers_subset, union, boundary).","answer":{"kind":"expression","value":"([[], [1, 2, 3], [2, 3], [3]], Yes, [1, 2, 3], single_cover_audit_not_general_compactness_proof)","explanation":"Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [1, 2, 3], [2, 3], [3]], Yes, [1, 2, 3], single_cover_audit_not_general_compactness_proof)."},"skills":["Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0044","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c9e536bdb7213e1b4e2cbbae40cd1a5de0bece1137c3dfaf5e2fd102e4b5caf","family_id":"qf-mod40-compact-open-cover-audit","seed":202611093044,"semantic_template":"qf-mod40-compact-open-cover-audit/{\"cover\":[[],[2,3,4],[2,4],[4]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,4],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 4], [0, 4]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [2, 3, 4], [2, 4], [4]]. Give (cover, covers_subset, union, boundary).","answer":{"kind":"expression","value":"([[], [2, 3, 4], [2, 4], [4]], Yes, [2, 3, 4], single_cover_audit_not_general_compactness_proof)","explanation":"Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [2, 3, 4], [2, 4], [4]], Yes, [2, 3, 4], single_cover_audit_not_general_compactness_proof)."},"skills":["Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0045","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb95df9153ff4fd9cecdaad74ac91aa03c04cd7261efb8cf686823e8513edfbd","family_id":"qf-mod40-compact-open-cover-audit","seed":202611093045,"semantic_template":"qf-mod40-compact-open-cover-audit/{\"cover\":[[],[2,3,4],[3,4],[4]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,4],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 1, 4], [0, 4]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [2, 3, 4], [3, 4], [4]]. Give (cover, covers_subset, union, boundary).","answer":{"kind":"expression","value":"([[], [2, 3, 4], [3, 4], [4]], Yes, [2, 3, 4], single_cover_audit_not_general_compactness_proof)","explanation":"Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [2, 3, 4], [3, 4], [4]], Yes, [2, 3, 4], single_cover_audit_not_general_compactness_proof)."},"skills":["Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0046","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1dc73202385f03cecd740eb84ef0fe62a33d80023d102bc5b47c059c17b4da2a","family_id":"qf-mod40-compact-open-cover-audit","seed":202611093046,"semantic_template":"qf-mod40-compact-open-cover-audit/{\"cover\":[[],[1,3,4],[1,4],[4]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,4],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2, 4], [0, 4]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [1, 3, 4], [1, 4], [4]]. Give (cover, covers_subset, union, boundary).","answer":{"kind":"expression","value":"([[], [1, 3, 4], [1, 4], [4]], Yes, [1, 3, 4], single_cover_audit_not_general_compactness_proof)","explanation":"Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [1, 3, 4], [1, 4], [4]], Yes, [1, 3, 4], single_cover_audit_not_general_compactness_proof)."},"skills":["Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0047","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"517ea36317c73f323c22b038c901d2631c056bd3c0e02ad79b7ea4c638dea8bd","family_id":"qf-mod40-compact-open-cover-audit","seed":202611093047,"semantic_template":"qf-mod40-compact-open-cover-audit/{\"cover\":[[],[1,3,4],[3,4],[4]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,4],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 2, 4], [0, 4]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [1, 3, 4], [3, 4], [4]]. Give (cover, covers_subset, union, boundary).","answer":{"kind":"expression","value":"([[], [1, 3, 4], [3, 4], [4]], Yes, [1, 3, 4], single_cover_audit_not_general_compactness_proof)","explanation":"Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [1, 3, 4], [3, 4], [4]], Yes, [1, 3, 4], single_cover_audit_not_general_compactness_proof)."},"skills":["Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0048","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a5a9939c058adb5fb373904fec263204c518205d8a8c674fcb40004afb934c5","family_id":"qf-mod40-compact-open-cover-audit","seed":202611093048,"semantic_template":"qf-mod40-compact-open-cover-audit/{\"cover\":[[],[1,2,4],[1,4],[4]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,4],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 3, 4], [0, 4]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [1, 2, 4], [1, 4], [4]]. Give (cover, covers_subset, union, boundary).","answer":{"kind":"expression","value":"([[], [1, 2, 4], [1, 4], [4]], Yes, [1, 2, 4], single_cover_audit_not_general_compactness_proof)","explanation":"Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [1, 2, 4], [1, 4], [4]], Yes, [1, 2, 4], single_cover_audit_not_general_compactness_proof)."},"skills":["Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0049","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6dce0109f1afe8494d53a0260727c6a192c45cb765cfb5930f116a5fbfe61d4f","family_id":"qf-mod40-compact-open-cover-audit","seed":202611093049,"semantic_template":"qf-mod40-compact-open-cover-audit/{\"cover\":[[],[1,2,4],[2,4],[4]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,4],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 3, 4], [0, 4]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [1, 2, 4], [2, 4], [4]]. Give (cover, covers_subset, union, boundary).","answer":{"kind":"expression","value":"([[], [1, 2, 4], [2, 4], [4]], Yes, [1, 2, 4], single_cover_audit_not_general_compactness_proof)","explanation":"Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [1, 2, 4], [2, 4], [4]], Yes, [1, 2, 4], single_cover_audit_not_general_compactness_proof)."},"skills":["Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0050","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f59174cfadf9301298962f463821fe713d1727c5eae918945426ab34c02298e","family_id":"qf-mod40-compact-open-cover-audit","seed":202611093050,"semantic_template":"qf-mod40-compact-open-cover-audit/{\"cover\":[[],[0,3,4],[0,3],[0]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[1]],\"subset\":[0,3,4]}"},"fields":{"title":"Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [1]], subset K=[0, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Supplied indexed cover=[[], [0], [0, 3], [0, 3, 4]]. Give (cover, covers_subset, union, boundary).","answer":{"kind":"expression","value":"([[], [0], [0, 3], [0, 3, 4]], Yes, [0, 3, 4], single_cover_audit_not_general_compactness_proof)","explanation":"Check that every supplied member is a relative open, compute its union and test whether it covers K. A failing list is not a cover of K; one passing list alone does not certify general compactness. The exact result, retaining the requested order, is ([[], [0], [0, 3], [0, 3, 4]], Yes, [0, 3, 4], single_cover_audit_not_general_compactness_proof)."},"skills":["Audit the Actual Relative-Open Cover without Promoting One Success to a General Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0051","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8082fa9c1d6dc9f349058a7f3791b0ed418c2e126379abc8018a575b7bc438b0","family_id":"qf-mod40-compact-subcover-certificate","seed":202611093051,"semantic_template":"qf-mod40-compact-subcover-certificate/{\"cover\":[[],[1,3,4],[1,3],[1]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[0]],\"selected\":[3],\"subset\":[1,3,4]}"},"fields":{"title":"Original Cover Membership and Actual Coverage Are Separate Checks","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [1], [1, 3], [1, 3, 4]], selected indices=[3]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).","answer":{"kind":"expression","value":"([3], [[1, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)","explanation":"Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([3], [[1, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)."},"skills":["Original Cover Membership and Actual Coverage Are Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0052","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5dc1aa31b6f75a92ef0a28d12b1bc092eb64ee545bb574a4b0cfb45a66c003a1","family_id":"qf-mod40-compact-subcover-certificate","seed":202611093052,"semantic_template":"qf-mod40-compact-subcover-certificate/{\"cover\":[[],[1,3,4],[1,4],[1]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,1],[0]],\"selected\":[2],\"subset\":[1,3,4]}"},"fields":{"title":"Original Cover Membership and Actual Coverage Are Separate Checks","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3, 4], [0, 1, 2, 4]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [1], [1, 3, 4], [1, 4]], selected indices=[2]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).","answer":{"kind":"expression","value":"([2], [[1, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)","explanation":"Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([2], [[1, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)."},"skills":["Original Cover Membership and Actual Coverage Are Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0053","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd11ace2433e9d5acb26c02610f39fc635eadc1e057b11c498cc933b9460cac4","family_id":"qf-mod40-compact-subcover-certificate","seed":202611093053,"semantic_template":"qf-mod40-compact-subcover-certificate/{\"cover\":[[],[1,2,4],[1,2],[1]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,1],[0]],\"selected\":[3],\"subset\":[1,2,4]}"},"fields":{"title":"Original Cover Membership and Actual Coverage Are Separate Checks","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [1], [1, 2], [1, 2, 4]], selected indices=[3]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).","answer":{"kind":"expression","value":"([3], [[1, 2, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)","explanation":"Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([3], [[1, 2, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)."},"skills":["Original Cover Membership and Actual Coverage Are Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0054","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba37afdd49fca1faa48fa16a020b02eb61cb078c83003d32f1510520c921fb4d","family_id":"qf-mod40-compact-subcover-certificate","seed":202611093054,"semantic_template":"qf-mod40-compact-subcover-certificate/{\"cover\":[[],[1,2,4],[1,4],[1]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,1],[0]],\"selected\":[2],\"subset\":[1,2,4]}"},"fields":{"title":"Original Cover Membership and Actual Coverage Are Separate Checks","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [1], [1, 2, 4], [1, 4]], selected indices=[2]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).","answer":{"kind":"expression","value":"([2], [[1, 2, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)","explanation":"Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([2], [[1, 2, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)."},"skills":["Original Cover Membership and Actual Coverage Are Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0055","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c3f98cfcee35e9d33788c82fce1288dd6378124e8cec8faee8f36279bd26b77e","family_id":"qf-mod40-compact-subcover-certificate","seed":202611093055,"semantic_template":"qf-mod40-compact-subcover-certificate/{\"cover\":[[],[1,2,3],[1,2],[1]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,1],[0]],\"selected\":[3],\"subset\":[1,2,3]}"},"fields":{"title":"Original Cover Membership and Actual Coverage Are Separate Checks","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [1], [1, 2], [1, 2, 3]], selected indices=[3]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).","answer":{"kind":"expression","value":"([3], [[1, 2, 3]], Yes, Yes, finite_subcover_selected_only_from_original_cover)","explanation":"Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([3], [[1, 2, 3]], Yes, Yes, finite_subcover_selected_only_from_original_cover)."},"skills":["Original Cover Membership and Actual Coverage Are Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0056","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"afef946abd50c8f0f339528db95483704b7f0a4a3b77fbdfbb87de2e0e30d469","family_id":"qf-mod40-compact-subcover-certificate","seed":202611093056,"semantic_template":"qf-mod40-compact-subcover-certificate/{\"cover\":[[],[1,2,3],[1,3],[1]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,1],[0]],\"selected\":[2],\"subset\":[1,2,3]}"},"fields":{"title":"Original Cover Membership and Actual Coverage Are Separate Checks","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 1, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [1], [1, 2, 3], [1, 3]], selected indices=[2]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).","answer":{"kind":"expression","value":"([2], [[1, 2, 3]], Yes, Yes, finite_subcover_selected_only_from_original_cover)","explanation":"Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([2], [[1, 2, 3]], Yes, Yes, finite_subcover_selected_only_from_original_cover)."},"skills":["Original Cover Membership and Actual Coverage Are Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0057","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"10080267abd206be5eb8f6aa0de4d209fb0bbb42780084426e3612fa113a47fa","family_id":"qf-mod40-compact-subcover-certificate","seed":202611093057,"semantic_template":"qf-mod40-compact-subcover-certificate/{\"cover\":[[],[2,3,4],[2,3],[2]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,2],[0]],\"selected\":[3],\"subset\":[2,3,4]}"},"fields":{"title":"Original Cover Membership and Actual Coverage Are Separate Checks","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [2], [2, 3], [2, 3, 4]], selected indices=[3]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).","answer":{"kind":"expression","value":"([3], [[2, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)","explanation":"Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([3], [[2, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)."},"skills":["Original Cover Membership and Actual Coverage Are Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0058","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7ca20de1d6e64818b4545da08ca9ba753cbb263426bf40fc417a7b679402780","family_id":"qf-mod40-compact-subcover-certificate","seed":202611093058,"semantic_template":"qf-mod40-compact-subcover-certificate/{\"cover\":[[],[2,3,4],[2,4],[2]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,2],[0]],\"selected\":[2],\"subset\":[2,3,4]}"},"fields":{"title":"Original Cover Membership and Actual Coverage Are Separate Checks","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [2], [2, 3, 4], [2, 4]], selected indices=[2]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).","answer":{"kind":"expression","value":"([2], [[2, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)","explanation":"Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([2], [[2, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)."},"skills":["Original Cover Membership and Actual Coverage Are Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0059","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1579a2a5ef7a07ebbb0bc54db4c44fa38e2b7befab8344127533a67414bada5","family_id":"qf-mod40-compact-subcover-certificate","seed":202611093059,"semantic_template":"qf-mod40-compact-subcover-certificate/{\"cover\":[[],[1,2,4],[1,2],[2]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,2],[0]],\"selected\":[2],\"subset\":[1,2,4]}"},"fields":{"title":"Original Cover Membership and Actual Coverage Are Separate Checks","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [1, 2], [1, 2, 4], [2]], selected indices=[2]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).","answer":{"kind":"expression","value":"([2], [[1, 2, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)","explanation":"Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([2], [[1, 2, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)."},"skills":["Original Cover Membership and Actual Coverage Are Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0060","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a65fe6969ea3300eb984ba56e0235be6b72ac651b548461ab5262a61e3151bb","family_id":"qf-mod40-compact-subcover-certificate","seed":202611093060,"semantic_template":"qf-mod40-compact-subcover-certificate/{\"cover\":[[],[1,2,4],[2,4],[2]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,2],[0]],\"selected\":[1],\"subset\":[1,2,4]}"},"fields":{"title":"Original Cover Membership and Actual Coverage Are Separate Checks","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [1, 2, 4], [2], [2, 4]], selected indices=[1]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).","answer":{"kind":"expression","value":"([1], [[1, 2, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)","explanation":"Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([1], [[1, 2, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)."},"skills":["Original Cover Membership and Actual Coverage Are Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0061","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1baeb596305a98368e04f08926758e331035bf921139e7285ab1cf9acbc05d5e","family_id":"qf-mod40-compact-subcover-certificate","seed":202611093061,"semantic_template":"qf-mod40-compact-subcover-certificate/{\"cover\":[[],[1,2,3],[1,2],[2]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,2],[0]],\"selected\":[2],\"subset\":[1,2,3]}"},"fields":{"title":"Original Cover Membership and Actual Coverage Are Separate Checks","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2], [0, 2, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [1, 2], [1, 2, 3], [2]], selected indices=[2]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).","answer":{"kind":"expression","value":"([2], [[1, 2, 3]], Yes, Yes, finite_subcover_selected_only_from_original_cover)","explanation":"Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([2], [[1, 2, 3]], Yes, Yes, finite_subcover_selected_only_from_original_cover)."},"skills":["Original Cover Membership and Actual Coverage Are Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0062","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0787985bc1c13c0805a48d5a06b6e54be2dffb88fe1abcc0239b3398e953ca5","family_id":"qf-mod40-compact-subcover-certificate","seed":202611093062,"semantic_template":"qf-mod40-compact-subcover-certificate/{\"cover\":[[],[1,2,3],[2,3],[2]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,2],[0]],\"selected\":[1],\"subset\":[1,2,3]}"},"fields":{"title":"Original Cover Membership and Actual Coverage Are Separate Checks","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3, 4], [0, 2, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [1, 2, 3], [2], [2, 3]], selected indices=[1]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).","answer":{"kind":"expression","value":"([1], [[1, 2, 3]], Yes, Yes, finite_subcover_selected_only_from_original_cover)","explanation":"Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([1], [[1, 2, 3]], Yes, Yes, finite_subcover_selected_only_from_original_cover)."},"skills":["Original Cover Membership and Actual Coverage Are Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0063","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5bae783fb091e9f7359334502c7305928d2fa62cd4436003564c35a5c8cc4322","family_id":"qf-mod40-compact-subcover-certificate","seed":202611093063,"semantic_template":"qf-mod40-compact-subcover-certificate/{\"cover\":[[],[2,3,4],[2,3],[3]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,3],[0]],\"selected\":[2],\"subset\":[2,3,4]}"},"fields":{"title":"Original Cover Membership and Actual Coverage Are Separate Checks","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3], [0, 3]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [2, 3], [2, 3, 4], [3]], selected indices=[2]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).","answer":{"kind":"expression","value":"([2], [[2, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)","explanation":"Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([2], [[2, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)."},"skills":["Original Cover Membership and Actual Coverage Are Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0064","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e698e28dfda41d2865e6a62eb4b7636372f08a4c6d8e8a170dd2ec05b1fb3c3","family_id":"qf-mod40-compact-subcover-certificate","seed":202611093064,"semantic_template":"qf-mod40-compact-subcover-certificate/{\"cover\":[[],[2,3,4],[3,4],[3]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,3],[0]],\"selected\":[1],\"subset\":[2,3,4]}"},"fields":{"title":"Original Cover Membership and Actual Coverage Are Separate Checks","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4], [0, 3]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [2, 3, 4], [3], [3, 4]], selected indices=[1]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).","answer":{"kind":"expression","value":"([1], [[2, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)","explanation":"Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([1], [[2, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)."},"skills":["Original Cover Membership and Actual Coverage Are Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0065","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"260061c1d45c3b5da572c5b1bf49f31840abda3902e83e3f450d75fdad084f9e","family_id":"qf-mod40-compact-subcover-certificate","seed":202611093065,"semantic_template":"qf-mod40-compact-subcover-certificate/{\"cover\":[[],[1,3,4],[1,3],[3]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,3],[0]],\"selected\":[2],\"subset\":[1,3,4]}"},"fields":{"title":"Original Cover Membership and Actual Coverage Are Separate Checks","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2, 3], [0, 3]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [1, 3], [1, 3, 4], [3]], selected indices=[2]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).","answer":{"kind":"expression","value":"([2], [[1, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)","explanation":"Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([2], [[1, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)."},"skills":["Original Cover Membership and Actual Coverage Are Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0066","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c861cb6683cd1ab57cc88cb3b750b588fd4f1d72ab7eeb5c810cc1a93f9a734a","family_id":"qf-mod40-compact-subcover-certificate","seed":202611093066,"semantic_template":"qf-mod40-compact-subcover-certificate/{\"cover\":[[],[1,3,4],[3,4],[3]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,3],[0]],\"selected\":[1],\"subset\":[1,3,4]}"},"fields":{"title":"Original Cover Membership and Actual Coverage Are Separate Checks","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3], [0, 2, 3, 4], [0, 3]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [1, 3, 4], [3], [3, 4]], selected indices=[1]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).","answer":{"kind":"expression","value":"([1], [[1, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)","explanation":"Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([1], [[1, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)."},"skills":["Original Cover Membership and Actual Coverage Are Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0067","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5c2097578bb2c1fd207312e5902ae28787db54b73d3d99274ef1922fcd9319d","family_id":"qf-mod40-compact-subcover-certificate","seed":202611093067,"semantic_template":"qf-mod40-compact-subcover-certificate/{\"cover\":[[],[1,2,3],[1,3],[3]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,3],[0]],\"selected\":[1],\"subset\":[1,2,3]}"},"fields":{"title":"Original Cover Membership and Actual Coverage Are Separate Checks","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 3], [0, 3, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [1, 2, 3], [1, 3], [3]], selected indices=[1]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).","answer":{"kind":"expression","value":"([1], [[1, 2, 3]], Yes, Yes, finite_subcover_selected_only_from_original_cover)","explanation":"Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([1], [[1, 2, 3]], Yes, Yes, finite_subcover_selected_only_from_original_cover)."},"skills":["Original Cover Membership and Actual Coverage Are Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0068","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3802d1e7c752ab09adebdaeef1798c3329e837abe575110ef98530fff890bfb","family_id":"qf-mod40-compact-subcover-certificate","seed":202611093068,"semantic_template":"qf-mod40-compact-subcover-certificate/{\"cover\":[[],[1,2,3],[2,3],[3]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,3],[0]],\"selected\":[1],\"subset\":[1,2,3]}"},"fields":{"title":"Original Cover Membership and Actual Coverage Are Separate Checks","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 3], [0, 3, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [1, 2, 3], [2, 3], [3]], selected indices=[1]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).","answer":{"kind":"expression","value":"([1], [[1, 2, 3]], Yes, Yes, finite_subcover_selected_only_from_original_cover)","explanation":"Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([1], [[1, 2, 3]], Yes, Yes, finite_subcover_selected_only_from_original_cover)."},"skills":["Original Cover Membership and Actual Coverage Are Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0069","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11c9d48ee5a5b83196e8bf9f66efab915037d54c3089d413f24e8fcfafd4baf4","family_id":"qf-mod40-compact-subcover-certificate","seed":202611093069,"semantic_template":"qf-mod40-compact-subcover-certificate/{\"cover\":[[],[2,3,4],[2,4],[4]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,4],[0]],\"selected\":[1],\"subset\":[2,3,4]}"},"fields":{"title":"Original Cover Membership and Actual Coverage Are Separate Checks","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 4], [0, 4]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [2, 3, 4], [2, 4], [4]], selected indices=[1]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).","answer":{"kind":"expression","value":"([1], [[2, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)","explanation":"Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([1], [[2, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)."},"skills":["Original Cover Membership and Actual Coverage Are Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0070","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"98245480481d3b05e01415ab1f9e86561b8b2a465b1e40a45c58d1f0a7d3ab66","family_id":"qf-mod40-compact-subcover-certificate","seed":202611093070,"semantic_template":"qf-mod40-compact-subcover-certificate/{\"cover\":[[],[2,3,4],[3,4],[4]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,4],[0]],\"selected\":[1],\"subset\":[2,3,4]}"},"fields":{"title":"Original Cover Membership and Actual Coverage Are Separate Checks","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 1, 4], [0, 4]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [2, 3, 4], [3, 4], [4]], selected indices=[1]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).","answer":{"kind":"expression","value":"([1], [[2, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)","explanation":"Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([1], [[2, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)."},"skills":["Original Cover Membership and Actual Coverage Are Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0071","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b86c8e013433d4e1e6a8993a66d67cc7c557a330325770f0f226b7647cc1744d","family_id":"qf-mod40-compact-subcover-certificate","seed":202611093071,"semantic_template":"qf-mod40-compact-subcover-certificate/{\"cover\":[[],[1,3,4],[1,4],[4]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,4],[0]],\"selected\":[1],\"subset\":[1,3,4]}"},"fields":{"title":"Original Cover Membership and Actual Coverage Are Separate Checks","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2, 4], [0, 4]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [1, 3, 4], [1, 4], [4]], selected indices=[1]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).","answer":{"kind":"expression","value":"([1], [[1, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)","explanation":"Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([1], [[1, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)."},"skills":["Original Cover Membership and Actual Coverage Are Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0072","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea90ad1ab94fe40b8a4d25cac07d9d2151561cbcbcc6e693e03a06dea8f67d06","family_id":"qf-mod40-compact-subcover-certificate","seed":202611093072,"semantic_template":"qf-mod40-compact-subcover-certificate/{\"cover\":[[],[1,3,4],[3,4],[4]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,4],[0]],\"selected\":[1],\"subset\":[1,3,4]}"},"fields":{"title":"Original Cover Membership and Actual Coverage Are Separate Checks","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 2, 4], [0, 4]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [1, 3, 4], [3, 4], [4]], selected indices=[1]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).","answer":{"kind":"expression","value":"([1], [[1, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)","explanation":"Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([1], [[1, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)."},"skills":["Original Cover Membership and Actual Coverage Are Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0073","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a385300dc0d92c2df6f42fd29fd1c65ab86f4f866188f506d2502ff86066313d","family_id":"qf-mod40-compact-subcover-certificate","seed":202611093073,"semantic_template":"qf-mod40-compact-subcover-certificate/{\"cover\":[[],[1,2,4],[1,4],[4]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,4],[0]],\"selected\":[1],\"subset\":[1,2,4]}"},"fields":{"title":"Original Cover Membership and Actual Coverage Are Separate Checks","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 3, 4], [0, 4]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [1, 2, 4], [1, 4], [4]], selected indices=[1]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).","answer":{"kind":"expression","value":"([1], [[1, 2, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)","explanation":"Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([1], [[1, 2, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)."},"skills":["Original Cover Membership and Actual Coverage Are Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0074","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"405f29d76e52092a944e4754bc6d86a7dab0722b9026d567904d2cf7a8a06f56","family_id":"qf-mod40-compact-subcover-certificate","seed":202611093074,"semantic_template":"qf-mod40-compact-subcover-certificate/{\"cover\":[[],[1,2,4],[2,4],[4]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,4],[0]],\"selected\":[1],\"subset\":[1,2,4]}"},"fields":{"title":"Original Cover Membership and Actual Coverage Are Separate Checks","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 3, 4], [0, 4]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [1, 2, 4], [2, 4], [4]], selected indices=[1]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).","answer":{"kind":"expression","value":"([1], [[1, 2, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)","explanation":"Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([1], [[1, 2, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)."},"skills":["Original Cover Membership and Actual Coverage Are Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0075","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4012c271806e9ba1afe0be46c8e91501257a830ccc9f6aefe38c4258f8ef48b7","family_id":"qf-mod40-compact-subcover-certificate","seed":202611093075,"semantic_template":"qf-mod40-compact-subcover-certificate/{\"cover\":[[],[0,3,4],[0,3],[0]],\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[1]],\"selected\":[3],\"subset\":[0,3,4]}"},"fields":{"title":"Original Cover Membership and Actual Coverage Are Separate Checks","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [1]], subset K=[0, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Original indexed cover=[[], [0], [0, 3], [0, 3, 4]], selected indices=[3]. Give (selected_indices, selected_sets, covers_subset, selected_from_original, certificate).","answer":{"kind":"expression","value":"([3], [[0, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)","explanation":"Select only distinct valid original indices and retain the increasingly ordered index list and its selected sets. Their union must still cover K. Being selected from the original list alone does not imply coverage, and no minimum or minimal subcover is asserted. The exact result, retaining the requested order, is ([3], [[0, 3, 4]], Yes, Yes, finite_subcover_selected_only_from_original_cover)."},"skills":["Original Cover Membership and Actual Coverage Are Separate Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0076","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9cc24ce7b74a2c1fcf729f6ce219f6b4b022873790551dd66b3601c0f1c45b7","family_id":"qf-mod40-compact-nonproperty-boundary","seed":202611093076,"semantic_template":"qf-mod40-compact-nonproperty-boundary/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"Compactness Does Not Supply General Ambient Closedness or Boundedness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).","answer":{"kind":"expression","value":"([1, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)","explanation":"Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([1, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)."},"skills":["Compactness Does Not Supply General Ambient Closedness or Boundedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0077","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e792085c872d42a5bb817d75cf59de63d5836d6b5e08966f80c78c7d01d1a077","family_id":"qf-mod40-compact-nonproperty-boundary","seed":202611093077,"semantic_template":"qf-mod40-compact-nonproperty-boundary/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,1],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"Compactness Does Not Supply General Ambient Closedness or Boundedness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3, 4], [0, 1, 2, 4]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).","answer":{"kind":"expression","value":"([1, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)","explanation":"Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([1, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)."},"skills":["Compactness Does Not Supply General Ambient Closedness or Boundedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0078","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73310dd1179e71f8b46444c8aacd2300e5912698fd32287d229cabcfade3330b","family_id":"qf-mod40-compact-nonproperty-boundary","seed":202611093078,"semantic_template":"qf-mod40-compact-nonproperty-boundary/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,1],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"Compactness Does Not Supply General Ambient Closedness or Boundedness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).","answer":{"kind":"expression","value":"([1, 2, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)","explanation":"Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([1, 2, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)."},"skills":["Compactness Does Not Supply General Ambient Closedness or Boundedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0079","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"319ed868f24d486a351ba4d9467008e92e6c0450de7dc395ecd86f389bb6e238","family_id":"qf-mod40-compact-nonproperty-boundary","seed":202611093079,"semantic_template":"qf-mod40-compact-nonproperty-boundary/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,1],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"Compactness Does Not Supply General Ambient Closedness or Boundedness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).","answer":{"kind":"expression","value":"([1, 2, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)","explanation":"Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([1, 2, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)."},"skills":["Compactness Does Not Supply General Ambient Closedness or Boundedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0080","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f4673246548f91657ab885a4abd0f05c4ce877df9a5317b56872236359e5c72","family_id":"qf-mod40-compact-nonproperty-boundary","seed":202611093080,"semantic_template":"qf-mod40-compact-nonproperty-boundary/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,1],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"Compactness Does Not Supply General Ambient Closedness or Boundedness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).","answer":{"kind":"expression","value":"([1, 2, 3], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)","explanation":"Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([1, 2, 3], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)."},"skills":["Compactness Does Not Supply General Ambient Closedness or Boundedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0081","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a83fd9a73a3d31daaf3292a867fd804badad27d88f6f74c73b0a9e84bee4003e","family_id":"qf-mod40-compact-nonproperty-boundary","seed":202611093081,"semantic_template":"qf-mod40-compact-nonproperty-boundary/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,1],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"Compactness Does Not Supply General Ambient Closedness or Boundedness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 1, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).","answer":{"kind":"expression","value":"([1, 2, 3], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)","explanation":"Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([1, 2, 3], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)."},"skills":["Compactness Does Not Supply General Ambient Closedness or Boundedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0082","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f350717269b7349ab137efd7692ac4641f0f2e5128342bd297bda0611dd9bdc","family_id":"qf-mod40-compact-nonproperty-boundary","seed":202611093082,"semantic_template":"qf-mod40-compact-nonproperty-boundary/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,2],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"Compactness Does Not Supply General Ambient Closedness or Boundedness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).","answer":{"kind":"expression","value":"([2, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)","explanation":"Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([2, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)."},"skills":["Compactness Does Not Supply General Ambient Closedness or Boundedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0083","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5983a9b376c2ff5024d1c01e5c1a93bc91506c9f67b09a1f42cc3337efd133ba","family_id":"qf-mod40-compact-nonproperty-boundary","seed":202611093083,"semantic_template":"qf-mod40-compact-nonproperty-boundary/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,2],[0,2],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"Compactness Does Not Supply General Ambient Closedness or Boundedness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).","answer":{"kind":"expression","value":"([2, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)","explanation":"Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([2, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)."},"skills":["Compactness Does Not Supply General Ambient Closedness or Boundedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0084","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c8d927b506ae2ff7e28db909d06ce7eeeca0d223775668f62ffc39a06420be4d","family_id":"qf-mod40-compact-nonproperty-boundary","seed":202611093084,"semantic_template":"qf-mod40-compact-nonproperty-boundary/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,2],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"Compactness Does Not Supply General Ambient Closedness or Boundedness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).","answer":{"kind":"expression","value":"([1, 2, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)","explanation":"Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([1, 2, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)."},"skills":["Compactness Does Not Supply General Ambient Closedness or Boundedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0085","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4625a9cb8e1286813e37a767c950d56826e92be5aceec66946e31bc648a6ec4","family_id":"qf-mod40-compact-nonproperty-boundary","seed":202611093085,"semantic_template":"qf-mod40-compact-nonproperty-boundary/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,2],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"Compactness Does Not Supply General Ambient Closedness or Boundedness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).","answer":{"kind":"expression","value":"([1, 2, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)","explanation":"Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([1, 2, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)."},"skills":["Compactness Does Not Supply General Ambient Closedness or Boundedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0086","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"09ed1f4817391e95adc1146887f8d4226f7f26aad32ea732ba7e73f68d7a2092","family_id":"qf-mod40-compact-nonproperty-boundary","seed":202611093086,"semantic_template":"qf-mod40-compact-nonproperty-boundary/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,2],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"Compactness Does Not Supply General Ambient Closedness or Boundedness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2], [0, 2, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).","answer":{"kind":"expression","value":"([1, 2, 3], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)","explanation":"Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([1, 2, 3], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)."},"skills":["Compactness Does Not Supply General Ambient Closedness or Boundedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0087","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bae81390fcec7611e2666f32c6f807dbdf742f04c6e7199fe3ca670cfdd31d2b","family_id":"qf-mod40-compact-nonproperty-boundary","seed":202611093087,"semantic_template":"qf-mod40-compact-nonproperty-boundary/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,2],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"Compactness Does Not Supply General Ambient Closedness or Boundedness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2], [0, 2, 3, 4], [0, 2, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).","answer":{"kind":"expression","value":"([1, 2, 3], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)","explanation":"Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([1, 2, 3], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)."},"skills":["Compactness Does Not Supply General Ambient Closedness or Boundedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0088","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"852d1f3d85191d15d1998111596f28d44176aebc0e07027a1af692d9bebfeb5a","family_id":"qf-mod40-compact-nonproperty-boundary","seed":202611093088,"semantic_template":"qf-mod40-compact-nonproperty-boundary/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,3],[0,3],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"Compactness Does Not Supply General Ambient Closedness or Boundedness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 1, 3], [0, 3]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).","answer":{"kind":"expression","value":"([2, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)","explanation":"Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([2, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)."},"skills":["Compactness Does Not Supply General Ambient Closedness or Boundedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0089","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7cb597fe657c3d1db597b7b2fedc218d5a5c5f7da21d1c77211a4a9c5a782f3b","family_id":"qf-mod40-compact-nonproperty-boundary","seed":202611093089,"semantic_template":"qf-mod40-compact-nonproperty-boundary/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,3],[0,3],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"Compactness Does Not Supply General Ambient Closedness or Boundedness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3], [0, 1, 3, 4], [0, 3]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).","answer":{"kind":"expression","value":"([2, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)","explanation":"Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([2, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)."},"skills":["Compactness Does Not Supply General Ambient Closedness or Boundedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0090","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9cee23c2663a64a1b2e9695ace58649a07f581d092d58b21d75714de84b6df33","family_id":"qf-mod40-compact-nonproperty-boundary","seed":202611093090,"semantic_template":"qf-mod40-compact-nonproperty-boundary/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,2,3],[0,3],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"Compactness Does Not Supply General Ambient Closedness or Boundedness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2, 3], [0, 3]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).","answer":{"kind":"expression","value":"([1, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)","explanation":"Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([1, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)."},"skills":["Compactness Does Not Supply General Ambient Closedness or Boundedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0091","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"56b05c1db5c51a7dee68da2f320bcbd39f01bbd93bfcaf634ed35c975fdf98c2","family_id":"qf-mod40-compact-nonproperty-boundary","seed":202611093091,"semantic_template":"qf-mod40-compact-nonproperty-boundary/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,3],[0,3],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"Compactness Does Not Supply General Ambient Closedness or Boundedness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3], [0, 2, 3, 4], [0, 3]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).","answer":{"kind":"expression","value":"([1, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)","explanation":"Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([1, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)."},"skills":["Compactness Does Not Supply General Ambient Closedness or Boundedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0092","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b4e7797cd0820f355ba426c6b48c1d584c8b9b96d91d8fc12153096678bf366","family_id":"qf-mod40-compact-nonproperty-boundary","seed":202611093092,"semantic_template":"qf-mod40-compact-nonproperty-boundary/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,3],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"Compactness Does Not Supply General Ambient Closedness or Boundedness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 3], [0, 3, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).","answer":{"kind":"expression","value":"([1, 2, 3], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)","explanation":"Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([1, 2, 3], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)."},"skills":["Compactness Does Not Supply General Ambient Closedness or Boundedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0093","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"09f53f89895ba223e45d4ca5115fa3ae82c1d977f9ec7d2e961ed72c90838286","family_id":"qf-mod40-compact-nonproperty-boundary","seed":202611093093,"semantic_template":"qf-mod40-compact-nonproperty-boundary/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,3],[0]],\"subset\":[1,2,3]}"},"fields":{"title":"Compactness Does Not Supply General Ambient Closedness or Boundedness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 3], [0, 3, 4]], subset K=[1, 2, 3] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).","answer":{"kind":"expression","value":"([1, 2, 3], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)","explanation":"Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([1, 2, 3], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)."},"skills":["Compactness Does Not Supply General Ambient Closedness or Boundedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0094","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b107e34bd2a73f8f7fd6c9131fdeb79fd85d04d5aab5143872913468626695b3","family_id":"qf-mod40-compact-nonproperty-boundary","seed":202611093094,"semantic_template":"qf-mod40-compact-nonproperty-boundary/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,1,4],[0,4],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"Compactness Does Not Supply General Ambient Closedness or Boundedness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 1, 4], [0, 4]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).","answer":{"kind":"expression","value":"([2, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)","explanation":"Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([2, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)."},"skills":["Compactness Does Not Supply General Ambient Closedness or Boundedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0095","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2bbb33304f7dc784ebee11e36287c842d369b2380a8c72ede2d8c1240619b262","family_id":"qf-mod40-compact-nonproperty-boundary","seed":202611093095,"semantic_template":"qf-mod40-compact-nonproperty-boundary/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,1,4],[0,4],[0]],\"subset\":[2,3,4]}"},"fields":{"title":"Compactness Does Not Supply General Ambient Closedness or Boundedness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 1, 4], [0, 4]], subset K=[2, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).","answer":{"kind":"expression","value":"([2, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)","explanation":"Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([2, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)."},"skills":["Compactness Does Not Supply General Ambient Closedness or Boundedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0096","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ac24030bd65ca2bd8ec5627ce41f8c92eb31837909f41d8e085442fe8b08d7d","family_id":"qf-mod40-compact-nonproperty-boundary","seed":202611093096,"semantic_template":"qf-mod40-compact-nonproperty-boundary/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,4],[0,2,4],[0,4],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"Compactness Does Not Supply General Ambient Closedness or Boundedness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 2, 4], [0, 2, 4], [0, 4]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).","answer":{"kind":"expression","value":"([1, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)","explanation":"Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([1, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)."},"skills":["Compactness Does Not Supply General Ambient Closedness or Boundedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0097","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24d96fc5dda307e6468b7e79952376f6da4278a7bc9a17d0c05ca533972e330f","family_id":"qf-mod40-compact-nonproperty-boundary","seed":202611093097,"semantic_template":"qf-mod40-compact-nonproperty-boundary/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,2,4],[0,4],[0]],\"subset\":[1,3,4]}"},"fields":{"title":"Compactness Does Not Supply General Ambient Closedness or Boundedness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 2, 4], [0, 4]], subset K=[1, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).","answer":{"kind":"expression","value":"([1, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)","explanation":"Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([1, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)."},"skills":["Compactness Does Not Supply General Ambient Closedness or Boundedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0098","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f34cb1f823eefa1d98ca400c3fe104371a6a02c73142a1c713381b200a07bc63","family_id":"qf-mod40-compact-nonproperty-boundary","seed":202611093098,"semantic_template":"qf-mod40-compact-nonproperty-boundary/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,3,4],[0,3,4],[0,4],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"Compactness Does Not Supply General Ambient Closedness or Boundedness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 1, 3, 4], [0, 3, 4], [0, 4]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).","answer":{"kind":"expression","value":"([1, 2, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)","explanation":"Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([1, 2, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)."},"skills":["Compactness Does Not Supply General Ambient Closedness or Boundedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0099","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fcadc92f78e96d4b0364730dec2437a5daf81a56d6bd68cbc1879755cbbbf2d0","family_id":"qf-mod40-compact-nonproperty-boundary","seed":202611093099,"semantic_template":"qf-mod40-compact-nonproperty-boundary/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,2,3,4],[0,3,4],[0,4],[0]],\"subset\":[1,2,4]}"},"fields":{"title":"Compactness Does Not Supply General Ambient Closedness or Boundedness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0], [0, 1, 2, 3, 4], [0, 2, 3, 4], [0, 3, 4], [0, 4]], subset K=[1, 2, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).","answer":{"kind":"expression","value":"([1, 2, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)","explanation":"Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([1, 2, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)."},"skills":["Compactness Does Not Supply General Ambient Closedness or Boundedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r03-compact-subset-0100","batch_id":"batch-20261014-advance-20261109-r03-compact-subset-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd6e1f1c7df2fcf59f2aa94f71d8c49ba88be4fa2294c2e2e588be29f65cc0a1","family_id":"qf-mod40-compact-nonproperty-boundary","seed":202611093100,"semantic_template":"qf-mod40-compact-nonproperty-boundary/{\"n\":5,\"parent_topology\":[[],[0,1,2,3,4],[0,1,2,3],[0,1,2],[0,1],[1]],\"subset\":[0,3,4]}"},"fields":{"title":"Compactness Does Not Supply General Ambient Closedness or Boundedness","stem":"Use ambient carrier Fin(5)={0,…,4}, complete canonical parent_topology=[[], [0, 1], [0, 1, 2], [0, 1, 2, 3], [0, 1, 2, 3, 4], [1]], subset K=[0, 3, 4] and its full induced topology {O intersect K: O is parent-open}, identifying duplicate intersections. Relative opens are subsets of K; ambient witnesses O need not lie in K. Empty K is compact by convention. Give (subset, compact, general_ambient_closedness_claim, general_boundedness_claim, boundary).","answer":{"kind":"expression","value":"([0, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)","explanation":"Retain the finite compactness flag. The two fixed No values withhold general closedness and boundedness inferences without extra hypotheses; they do not classify this particular subset as nonclosed or unbounded. A general topology has no assumed metric. The exact result, retaining the requested order, is ([0, 3, 4], Yes, No, No, compact_does_not_generally_imply_ambient_closed_or_bounded_without_extra_structure)."},"skills":["Compactness Does Not Supply General Ambient Closedness or Boundedness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}