{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0001","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"978b3f4ef3c9870c6e9df90f71b3940ffe34577cc4ee8b0ff16dd01add75cb00","family_id":"qf-mod41-compactimg-image-set","seed":202611103001,"semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[0,0,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"The Exact Finite Image Is Separate from a Continuity Audit","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 0, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], [0, 0, 0], [0], Yes, finite_compact_source_subset_exact_image)","explanation":"Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [0, 0, 0], [0], Yes, finite_compact_source_subset_exact_image)."},"skills":["The Exact Finite Image Is Separate from a Continuity Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0002","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52cdb313ef9bbb6cc47a469406a97af6cac3c855955560420b2fcb09fe0c71c2","family_id":"qf-mod41-compactimg-image-set","seed":202611103002,"semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[1,0,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"The Exact Finite Image Is Separate from a Continuity Audit","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 0, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], [1, 0, 0], [0, 1], Yes, finite_compact_source_subset_exact_image)","explanation":"Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [1, 0, 0], [0, 1], Yes, finite_compact_source_subset_exact_image)."},"skills":["The Exact Finite Image Is Separate from a Continuity Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0003","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50c20d16010d41062016dd52d7f14706a9f75b8d39c5990e87bc4307ba7f7a91","family_id":"qf-mod41-compactimg-image-set","seed":202611103003,"semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[2,0,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"The Exact Finite Image Is Separate from a Continuity Audit","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 0, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], [2, 0, 0], [0, 2], Yes, finite_compact_source_subset_exact_image)","explanation":"Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [2, 0, 0], [0, 2], Yes, finite_compact_source_subset_exact_image)."},"skills":["The Exact Finite Image Is Separate from a Continuity Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0004","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed6c250a589664beef382a6aa4a9f0451e94bc25c79e6698505b0b75b17fd11c","family_id":"qf-mod41-compactimg-image-set","seed":202611103004,"semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[0,1,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"The Exact Finite Image Is Separate from a Continuity Audit","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 1, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], [0, 1, 0], [0, 1], Yes, finite_compact_source_subset_exact_image)","explanation":"Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [0, 1, 0], [0, 1], Yes, finite_compact_source_subset_exact_image)."},"skills":["The Exact Finite Image Is Separate from a Continuity Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0005","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8830df5ee108dab37e82500205cf137b417c40204d47573fd508782846d1dcef","family_id":"qf-mod41-compactimg-image-set","seed":202611103005,"semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[1,1,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"The Exact Finite Image Is Separate from a Continuity Audit","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 1, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], [1, 1, 0], [0, 1], Yes, finite_compact_source_subset_exact_image)","explanation":"Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [1, 1, 0], [0, 1], Yes, finite_compact_source_subset_exact_image)."},"skills":["The Exact Finite Image Is Separate from a Continuity Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0006","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55b53ce5533edf829c1a3c5b89d7e1b904c19a9af7fab015d0d8ecdfa1c81f41","family_id":"qf-mod41-compactimg-image-set","seed":202611103006,"semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[2,1,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"The Exact Finite Image Is Separate from a Continuity Audit","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 1, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], [2, 1, 0], [0, 1, 2], Yes, finite_compact_source_subset_exact_image)","explanation":"Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [2, 1, 0], [0, 1, 2], Yes, finite_compact_source_subset_exact_image)."},"skills":["The Exact Finite Image Is Separate from a Continuity Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0007","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fda1674bc14d8a05b87a55258b2d8ade76a47ebbd4f57d6a5d2d2f3e4608eb42","family_id":"qf-mod41-compactimg-image-set","seed":202611103007,"semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[0,2,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"The Exact Finite Image Is Separate from a Continuity Audit","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 2, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], [0, 2, 0], [0, 2], Yes, finite_compact_source_subset_exact_image)","explanation":"Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [0, 2, 0], [0, 2], Yes, finite_compact_source_subset_exact_image)."},"skills":["The Exact Finite Image Is Separate from a Continuity Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0008","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f25330a0fa0a99408902a12a40de348b0566c4fe71f0eca8ef8a49b41d63198","family_id":"qf-mod41-compactimg-image-set","seed":202611103008,"semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[1,2,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"The Exact Finite Image Is Separate from a Continuity Audit","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 2, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], [1, 2, 0], [0, 1, 2], Yes, finite_compact_source_subset_exact_image)","explanation":"Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [1, 2, 0], [0, 1, 2], Yes, finite_compact_source_subset_exact_image)."},"skills":["The Exact Finite Image Is Separate from a Continuity Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0009","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"33fcc84ae36bcadc96726923cb9be52937e1c4ecbc977ea4074f43ac6b15c134","family_id":"qf-mod41-compactimg-image-set","seed":202611103009,"semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[2,2,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"The Exact Finite Image Is Separate from a Continuity Audit","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 2, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], [2, 2, 0], [0, 2], Yes, finite_compact_source_subset_exact_image)","explanation":"Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [2, 2, 0], [0, 2], Yes, finite_compact_source_subset_exact_image)."},"skills":["The Exact Finite Image Is Separate from a Continuity Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0010","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"903f9714a08c3715a1ea9a647ee5ee79249a49e06b3e6ad80f39a35a65da5d35","family_id":"qf-mod41-compactimg-image-set","seed":202611103010,"semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[0,0,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"The Exact Finite Image Is Separate from a Continuity Audit","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 0, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], [0, 0, 1], [0, 1], Yes, finite_compact_source_subset_exact_image)","explanation":"Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [0, 0, 1], [0, 1], Yes, finite_compact_source_subset_exact_image)."},"skills":["The Exact Finite Image Is Separate from a Continuity Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0011","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68e25a6e2f77096c79e1a68c0ce9856f959808ad2da811bb2b774ad605bc8485","family_id":"qf-mod41-compactimg-image-set","seed":202611103011,"semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[1,0,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"The Exact Finite Image Is Separate from a Continuity Audit","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 0, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], [1, 0, 1], [0, 1], Yes, finite_compact_source_subset_exact_image)","explanation":"Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [1, 0, 1], [0, 1], Yes, finite_compact_source_subset_exact_image)."},"skills":["The Exact Finite Image Is Separate from a Continuity Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0012","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"880743f454b12029b327517045830cddf94dc80cf210436cbb1b9ed7ec838d32","family_id":"qf-mod41-compactimg-image-set","seed":202611103012,"semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[2,0,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"The Exact Finite Image Is Separate from a Continuity Audit","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 0, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], [2, 0, 1], [0, 1, 2], Yes, finite_compact_source_subset_exact_image)","explanation":"Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [2, 0, 1], [0, 1, 2], Yes, finite_compact_source_subset_exact_image)."},"skills":["The Exact Finite Image Is Separate from a Continuity Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0013","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ceaaf23ab1a0953b8b39fbf11a785f4ecc24e2be99eeb5ea52ade9166a1a9eb2","family_id":"qf-mod41-compactimg-image-set","seed":202611103013,"semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[0,1,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"The Exact Finite Image Is Separate from a Continuity Audit","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 1, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], [0, 1, 1], [0, 1], Yes, finite_compact_source_subset_exact_image)","explanation":"Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [0, 1, 1], [0, 1], Yes, finite_compact_source_subset_exact_image)."},"skills":["The Exact Finite Image Is Separate from a Continuity Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0014","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae6d3494880ef8b82e8ccc0cfc0ddc75ae53cd77160535d46c832ddaa3885692","family_id":"qf-mod41-compactimg-image-set","seed":202611103014,"semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[1,1,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"The Exact Finite Image Is Separate from a Continuity Audit","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 1, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], [1, 1, 1], [1], Yes, finite_compact_source_subset_exact_image)","explanation":"Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [1, 1, 1], [1], Yes, finite_compact_source_subset_exact_image)."},"skills":["The Exact Finite Image Is Separate from a Continuity Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0015","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81c8e01e7814c2030a3164944e1219846fb99186d9eb4c5a267207498a1a39e6","family_id":"qf-mod41-compactimg-image-set","seed":202611103015,"semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[2,1,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"The Exact Finite Image Is Separate from a Continuity Audit","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 1, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], [2, 1, 1], [1, 2], Yes, finite_compact_source_subset_exact_image)","explanation":"Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [2, 1, 1], [1, 2], Yes, finite_compact_source_subset_exact_image)."},"skills":["The Exact Finite Image Is Separate from a Continuity Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0016","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49a1aaadae8c672af1b5a3aaf2140231aed88dcfb8cbf5b9eb771b1a987fa730","family_id":"qf-mod41-compactimg-image-set","seed":202611103016,"semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[0,2,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"The Exact Finite Image Is Separate from a Continuity Audit","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 2, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], [0, 2, 1], [0, 1, 2], Yes, finite_compact_source_subset_exact_image)","explanation":"Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [0, 2, 1], [0, 1, 2], Yes, finite_compact_source_subset_exact_image)."},"skills":["The Exact Finite Image Is Separate from a Continuity Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0017","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6dcbeb9c48f19475759cdc480401824dfb4842a3a276c1564c841cbc60593a2","family_id":"qf-mod41-compactimg-image-set","seed":202611103017,"semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[1,2,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"The Exact Finite Image Is Separate from a Continuity Audit","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 2, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], [1, 2, 1], [1, 2], Yes, finite_compact_source_subset_exact_image)","explanation":"Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [1, 2, 1], [1, 2], Yes, finite_compact_source_subset_exact_image)."},"skills":["The Exact Finite Image Is Separate from a Continuity Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0018","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01e96a44bfb4282cbcba42c3029efbc5a6b2ba3eb95acf526eae17a47f1f4edc","family_id":"qf-mod41-compactimg-image-set","seed":202611103018,"semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[2,2,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"The Exact Finite Image Is Separate from a Continuity Audit","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 2, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], [2, 2, 1], [1, 2], Yes, finite_compact_source_subset_exact_image)","explanation":"Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [2, 2, 1], [1, 2], Yes, finite_compact_source_subset_exact_image)."},"skills":["The Exact Finite Image Is Separate from a Continuity Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0019","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13cb780341e19a4d2bf64942ee2a1c55d3c6f3cc8bd88853a214bd2b6ab61b32","family_id":"qf-mod41-compactimg-image-set","seed":202611103019,"semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[0,0,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"The Exact Finite Image Is Separate from a Continuity Audit","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 0, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], [0, 0, 2], [0, 2], Yes, finite_compact_source_subset_exact_image)","explanation":"Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [0, 0, 2], [0, 2], Yes, finite_compact_source_subset_exact_image)."},"skills":["The Exact Finite Image Is Separate from a Continuity Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0020","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd370cfe8ea159b1bd84cea286e452ba87e89d8759e0ae046353f407a6a9e487","family_id":"qf-mod41-compactimg-image-set","seed":202611103020,"semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[1,0,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"The Exact Finite Image Is Separate from a Continuity Audit","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 0, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], [1, 0, 2], [0, 1, 2], Yes, finite_compact_source_subset_exact_image)","explanation":"Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [1, 0, 2], [0, 1, 2], Yes, finite_compact_source_subset_exact_image)."},"skills":["The Exact Finite Image Is Separate from a Continuity Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0021","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"132937c34bb82c07d5f211095bcde8d3896ea2ee767c2aaa0fbd2ba9e18c5d00","family_id":"qf-mod41-compactimg-image-set","seed":202611103021,"semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[2,0,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"The Exact Finite Image Is Separate from a Continuity Audit","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 0, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], [2, 0, 2], [0, 2], Yes, finite_compact_source_subset_exact_image)","explanation":"Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [2, 0, 2], [0, 2], Yes, finite_compact_source_subset_exact_image)."},"skills":["The Exact Finite Image Is Separate from a Continuity Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0022","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31119fe3c4a384ba8f8e9179de88c3042aaac54164757018c162c0c7cf94bbb0","family_id":"qf-mod41-compactimg-image-set","seed":202611103022,"semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[0,1,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"The Exact Finite Image Is Separate from a Continuity Audit","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 1, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], [0, 1, 2], [0, 1, 2], Yes, finite_compact_source_subset_exact_image)","explanation":"Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [0, 1, 2], [0, 1, 2], Yes, finite_compact_source_subset_exact_image)."},"skills":["The Exact Finite Image Is Separate from a Continuity Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0023","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"94d20ce08d3542ef8cae16afa7a32109026b9a797034835aafae40850356fd2c","family_id":"qf-mod41-compactimg-image-set","seed":202611103023,"semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[1,1,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"The Exact Finite Image Is Separate from a Continuity Audit","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 1, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], [1, 1, 2], [1, 2], Yes, finite_compact_source_subset_exact_image)","explanation":"Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [1, 1, 2], [1, 2], Yes, finite_compact_source_subset_exact_image)."},"skills":["The Exact Finite Image Is Separate from a Continuity Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0024","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee23ad50e78055e7c2757c40638c674716c4fab0ea16916958c18599e9327c01","family_id":"qf-mod41-compactimg-image-set","seed":202611103024,"semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[2,1,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"The Exact Finite Image Is Separate from a Continuity Audit","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 1, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], [2, 1, 2], [1, 2], Yes, finite_compact_source_subset_exact_image)","explanation":"Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [2, 1, 2], [1, 2], Yes, finite_compact_source_subset_exact_image)."},"skills":["The Exact Finite Image Is Separate from a Continuity Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0025","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"524dae086b8bce0dcd10a67b08be8569b79ba7959d5cc3a9fc0f082263bf4ed6","family_id":"qf-mod41-compactimg-image-set","seed":202611103025,"semantic_template":"qf-mod41-compactimg-image-set/{\"map\":[0,2,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"The Exact Finite Image Is Separate from a Continuity Audit","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 2, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (subset, map, image, finite_compactness_certificate, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], [0, 2, 2], [0, 2], Yes, finite_compact_source_subset_exact_image)","explanation":"Enumerate f(K) and remove repeated values. Finiteness gives compactness of K and its finite image even before continuity is separately audited. This finite observation is not the general continuous-image theorem or its converse. The exact result, retaining the requested order, is ([0, 1, 2], [0, 2, 2], [0, 2], Yes, finite_compact_source_subset_exact_image)."},"skills":["The Exact Finite Image Is Separate from a Continuity Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0026","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6017dacf5be43fc2ca7a25799d038c189662d87c5333254e8663f3020d912d26","family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103026,"semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[0,0,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Every Target Open Must Have an Open Source Preimage","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 0, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).","answer":{"kind":"expression","value":"([[[], []], [[0], [0, 1, 2]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], []], [[2], []]], Yes, every_target_open_has_source_open_preimage)","explanation":"List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], [0, 1, 2]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], []], [[2], []]], Yes, every_target_open_has_source_open_preimage)."},"skills":["Every Target Open Must Have an Open Source Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0027","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e34b03b345f278ef31971ddb836efb45a232757496b2c3b28a07e548e8f325a","family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103027,"semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[1,0,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Every Target Open Must Have an Open Source Preimage","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 0, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).","answer":{"kind":"expression","value":"([[[], []], [[0], [1, 2]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [1, 2]], [[1], [0]], [[1, 2], [0]], [[2], []]], Yes, every_target_open_has_source_open_preimage)","explanation":"List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], [1, 2]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [1, 2]], [[1], [0]], [[1, 2], [0]], [[2], []]], Yes, every_target_open_has_source_open_preimage)."},"skills":["Every Target Open Must Have an Open Source Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0028","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96dba3fa0576c645d5b124b7f4ae6259897976ca8047e54d73ccb85c0ce6700a","family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103028,"semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[2,0,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Every Target Open Must Have an Open Source Preimage","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 0, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).","answer":{"kind":"expression","value":"([[[], []], [[0], [1, 2]], [[0, 1], [1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], [0]], [[2], [0]]], Yes, every_target_open_has_source_open_preimage)","explanation":"List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], [1, 2]], [[0, 1], [1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], [0]], [[2], [0]]], Yes, every_target_open_has_source_open_preimage)."},"skills":["Every Target Open Must Have an Open Source Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0029","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2621d617742815411978c6ad17b38c1f90c5b75379562f43b8232f7340bd86c6","family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103029,"semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[0,1,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Every Target Open Must Have an Open Source Preimage","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 1, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).","answer":{"kind":"expression","value":"([[[], []], [[0], [0, 2]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 2]], [[1], [1]], [[1, 2], [1]], [[2], []]], Yes, every_target_open_has_source_open_preimage)","explanation":"List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], [0, 2]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 2]], [[1], [1]], [[1, 2], [1]], [[2], []]], Yes, every_target_open_has_source_open_preimage)."},"skills":["Every Target Open Must Have an Open Source Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0030","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7c612a3ac82896cde5ee9d335184e63c8b31f60d18eb7642706472d57eacd8f","family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103030,"semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[1,1,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Every Target Open Must Have an Open Source Preimage","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 1, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).","answer":{"kind":"expression","value":"([[[], []], [[0], [2]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [2]], [[1], [0, 1]], [[1, 2], [0, 1]], [[2], []]], Yes, every_target_open_has_source_open_preimage)","explanation":"List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], [2]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [2]], [[1], [0, 1]], [[1, 2], [0, 1]], [[2], []]], Yes, every_target_open_has_source_open_preimage)."},"skills":["Every Target Open Must Have an Open Source Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0031","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"192f6d3711e5e105f420a3fe8b05d659f07fa795a3b7aaa05e278a67c4e73329","family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103031,"semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[2,1,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Every Target Open Must Have an Open Source Preimage","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 1, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).","answer":{"kind":"expression","value":"([[[], []], [[0], [2]], [[0, 1], [1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 2]], [[1], [1]], [[1, 2], [0, 1]], [[2], [0]]], Yes, every_target_open_has_source_open_preimage)","explanation":"List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], [2]], [[0, 1], [1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 2]], [[1], [1]], [[1, 2], [0, 1]], [[2], [0]]], Yes, every_target_open_has_source_open_preimage)."},"skills":["Every Target Open Must Have an Open Source Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0032","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91c482fcf1cf21eae97a9103d048a802717eb38eaab18f01b549acbc712dc961","family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103032,"semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[0,2,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Every Target Open Must Have an Open Source Preimage","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 2, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).","answer":{"kind":"expression","value":"([[[], []], [[0], [0, 2]], [[0, 1], [0, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], [1]], [[2], [1]]], Yes, every_target_open_has_source_open_preimage)","explanation":"List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], [0, 2]], [[0, 1], [0, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], [1]], [[2], [1]]], Yes, every_target_open_has_source_open_preimage)."},"skills":["Every Target Open Must Have an Open Source Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0033","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b015b0153c87202634be83a3fe44eeb2c74992f194f85beb3d6dc3c606071006","family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103033,"semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[1,2,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Every Target Open Must Have an Open Source Preimage","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 2, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).","answer":{"kind":"expression","value":"([[[], []], [[0], [2]], [[0, 1], [0, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [1, 2]], [[1], [0]], [[1, 2], [0, 1]], [[2], [1]]], Yes, every_target_open_has_source_open_preimage)","explanation":"List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], [2]], [[0, 1], [0, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [1, 2]], [[1], [0]], [[1, 2], [0, 1]], [[2], [1]]], Yes, every_target_open_has_source_open_preimage)."},"skills":["Every Target Open Must Have an Open Source Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0034","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4cd7bd3e8a4f28736b8bde0a1b2c92f72a6410a46d1e0f6c86a3cd8c879cf6a1","family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103034,"semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[2,2,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Every Target Open Must Have an Open Source Preimage","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 2, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).","answer":{"kind":"expression","value":"([[[], []], [[0], [2]], [[0, 1], [2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], [0, 1]], [[2], [0, 1]]], Yes, every_target_open_has_source_open_preimage)","explanation":"List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], [2]], [[0, 1], [2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], [0, 1]], [[2], [0, 1]]], Yes, every_target_open_has_source_open_preimage)."},"skills":["Every Target Open Must Have an Open Source Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0035","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2797caefbec624653451214f7d8dfb345b936f3ece6b63b7f18d713b230ba0c3","family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103035,"semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[0,0,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Every Target Open Must Have an Open Source Preimage","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 0, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).","answer":{"kind":"expression","value":"([[[], []], [[0], [0, 1]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1]], [[1], [2]], [[1, 2], [2]], [[2], []]], Yes, every_target_open_has_source_open_preimage)","explanation":"List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], [0, 1]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1]], [[1], [2]], [[1, 2], [2]], [[2], []]], Yes, every_target_open_has_source_open_preimage)."},"skills":["Every Target Open Must Have an Open Source Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0036","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bfcb7b74d56aac413b387c3ee8227cd95a2a6340f53e0c2862012870df80b131","family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103036,"semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[1,0,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Every Target Open Must Have an Open Source Preimage","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 0, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).","answer":{"kind":"expression","value":"([[[], []], [[0], [1]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [1]], [[1], [0, 2]], [[1, 2], [0, 2]], [[2], []]], Yes, every_target_open_has_source_open_preimage)","explanation":"List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], [1]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [1]], [[1], [0, 2]], [[1, 2], [0, 2]], [[2], []]], Yes, every_target_open_has_source_open_preimage)."},"skills":["Every Target Open Must Have an Open Source Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0037","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0be1942bf98ee9efe1adf48576ef0acc6519932bd1dfe967b268a885101df015","family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103037,"semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[2,0,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Every Target Open Must Have an Open Source Preimage","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 0, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).","answer":{"kind":"expression","value":"([[[], []], [[0], [1]], [[0, 1], [1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1]], [[1], [2]], [[1, 2], [0, 2]], [[2], [0]]], Yes, every_target_open_has_source_open_preimage)","explanation":"List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], [1]], [[0, 1], [1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1]], [[1], [2]], [[1, 2], [0, 2]], [[2], [0]]], Yes, every_target_open_has_source_open_preimage)."},"skills":["Every Target Open Must Have an Open Source Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0038","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be7c40e292d330c12ee6b17bada8c5cc03a8065ad1090ef33bcc24aa66dec52e","family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103038,"semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[0,1,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Every Target Open Must Have an Open Source Preimage","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 1, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).","answer":{"kind":"expression","value":"([[[], []], [[0], [0]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0]], [[1], [1, 2]], [[1, 2], [1, 2]], [[2], []]], Yes, every_target_open_has_source_open_preimage)","explanation":"List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], [0]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0]], [[1], [1, 2]], [[1, 2], [1, 2]], [[2], []]], Yes, every_target_open_has_source_open_preimage)."},"skills":["Every Target Open Must Have an Open Source Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0039","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"815926e7861a6ba415089cc3ff242d7bf0b7c33939eaa99a749ab67d175b876f","family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103039,"semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[1,1,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Every Target Open Must Have an Open Source Preimage","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 1, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).","answer":{"kind":"expression","value":"([[[], []], [[0], []], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], []], [[1], [0, 1, 2]], [[1, 2], [0, 1, 2]], [[2], []]], Yes, every_target_open_has_source_open_preimage)","explanation":"List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], []], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], []], [[1], [0, 1, 2]], [[1, 2], [0, 1, 2]], [[2], []]], Yes, every_target_open_has_source_open_preimage)."},"skills":["Every Target Open Must Have an Open Source Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0040","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"881e786794d571c7f4a8941b628cd3269fcd52259abfc3f40d3ef1aafe0a0924","family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103040,"semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[2,1,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Every Target Open Must Have an Open Source Preimage","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 1, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).","answer":{"kind":"expression","value":"([[[], []], [[0], []], [[0, 1], [1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0]], [[1], [1, 2]], [[1, 2], [0, 1, 2]], [[2], [0]]], Yes, every_target_open_has_source_open_preimage)","explanation":"List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], []], [[0, 1], [1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0]], [[1], [1, 2]], [[1, 2], [0, 1, 2]], [[2], [0]]], Yes, every_target_open_has_source_open_preimage)."},"skills":["Every Target Open Must Have an Open Source Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0041","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"595eea5d301966f0fbf01cb7f2d05880483ba27df4ff42556f5925cccd371967","family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103041,"semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[0,2,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Every Target Open Must Have an Open Source Preimage","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 2, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).","answer":{"kind":"expression","value":"([[[], []], [[0], [0]], [[0, 1], [0, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1]], [[1], [2]], [[1, 2], [1, 2]], [[2], [1]]], Yes, every_target_open_has_source_open_preimage)","explanation":"List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], [0]], [[0, 1], [0, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1]], [[1], [2]], [[1, 2], [1, 2]], [[2], [1]]], Yes, every_target_open_has_source_open_preimage)."},"skills":["Every Target Open Must Have an Open Source Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0042","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"010a96465926c6e3534bcaef0f8de5f86cc8a26fcc70122acea9e9745eb817ae","family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103042,"semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[1,2,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Every Target Open Must Have an Open Source Preimage","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 2, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).","answer":{"kind":"expression","value":"([[[], []], [[0], []], [[0, 1], [0, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [1]], [[1], [0, 2]], [[1, 2], [0, 1, 2]], [[2], [1]]], Yes, every_target_open_has_source_open_preimage)","explanation":"List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], []], [[0, 1], [0, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [1]], [[1], [0, 2]], [[1, 2], [0, 1, 2]], [[2], [1]]], Yes, every_target_open_has_source_open_preimage)."},"skills":["Every Target Open Must Have an Open Source Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0043","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"371b7d8b2e9e01a93e032ef1ebf0164597e7e9a9745b0a483cca706f5c769da7","family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103043,"semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[2,2,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Every Target Open Must Have an Open Source Preimage","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 2, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).","answer":{"kind":"expression","value":"([[[], []], [[0], []], [[0, 1], [2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1]], [[1], [2]], [[1, 2], [0, 1, 2]], [[2], [0, 1]]], Yes, every_target_open_has_source_open_preimage)","explanation":"List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], []], [[0, 1], [2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1]], [[1], [2]], [[1, 2], [0, 1, 2]], [[2], [0, 1]]], Yes, every_target_open_has_source_open_preimage)."},"skills":["Every Target Open Must Have an Open Source Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0044","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78fc321ee4c9c1ac0756ad87236c4f0f4c84900addc02027831759c9445113a1","family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103044,"semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[0,0,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Every Target Open Must Have an Open Source Preimage","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 0, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).","answer":{"kind":"expression","value":"([[[], []], [[0], [0, 1]], [[0, 1], [0, 1]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], [2]], [[2], [2]]], Yes, every_target_open_has_source_open_preimage)","explanation":"List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], [0, 1]], [[0, 1], [0, 1]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], [2]], [[2], [2]]], Yes, every_target_open_has_source_open_preimage)."},"skills":["Every Target Open Must Have an Open Source Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0045","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb1f8c11b9d4788cc0a73a54c0743ab5111ca65f93e9f9f6cd6ec14c888f55fb","family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103045,"semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[1,0,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Every Target Open Must Have an Open Source Preimage","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 0, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).","answer":{"kind":"expression","value":"([[[], []], [[0], [1]], [[0, 1], [0, 1]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [1, 2]], [[1], [0]], [[1, 2], [0, 2]], [[2], [2]]], Yes, every_target_open_has_source_open_preimage)","explanation":"List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], [1]], [[0, 1], [0, 1]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [1, 2]], [[1], [0]], [[1, 2], [0, 2]], [[2], [2]]], Yes, every_target_open_has_source_open_preimage)."},"skills":["Every Target Open Must Have an Open Source Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0046","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0f9348e27f95aaf1a6b48897d4221e3e1b08920bfb7beba71ec336cf03999b4","family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103046,"semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[2,0,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Every Target Open Must Have an Open Source Preimage","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 0, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).","answer":{"kind":"expression","value":"([[[], []], [[0], [1]], [[0, 1], [1]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], [0, 2]], [[2], [0, 2]]], Yes, every_target_open_has_source_open_preimage)","explanation":"List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], [1]], [[0, 1], [1]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], [0, 2]], [[2], [0, 2]]], Yes, every_target_open_has_source_open_preimage)."},"skills":["Every Target Open Must Have an Open Source Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0047","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6fbe85e08d418f5ec9a7c9c4034cf67c54b23c19ee88b0e7e5a1224153b83126","family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103047,"semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[0,1,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Every Target Open Must Have an Open Source Preimage","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 1, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).","answer":{"kind":"expression","value":"([[[], []], [[0], [0]], [[0, 1], [0, 1]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 2]], [[1], [1]], [[1, 2], [1, 2]], [[2], [2]]], Yes, every_target_open_has_source_open_preimage)","explanation":"List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], [0]], [[0, 1], [0, 1]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 2]], [[1], [1]], [[1, 2], [1, 2]], [[2], [2]]], Yes, every_target_open_has_source_open_preimage)."},"skills":["Every Target Open Must Have an Open Source Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0048","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7c3cf095336705243be67a76382824a27f36545ca5bf6c710690df62a78efb34","family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103048,"semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[1,1,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Every Target Open Must Have an Open Source Preimage","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 1, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).","answer":{"kind":"expression","value":"([[[], []], [[0], []], [[0, 1], [0, 1]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [2]], [[1], [0, 1]], [[1, 2], [0, 1, 2]], [[2], [2]]], Yes, every_target_open_has_source_open_preimage)","explanation":"List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], []], [[0, 1], [0, 1]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [2]], [[1], [0, 1]], [[1, 2], [0, 1, 2]], [[2], [2]]], Yes, every_target_open_has_source_open_preimage)."},"skills":["Every Target Open Must Have an Open Source Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0049","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ffc796e37f53613eb95fc55fb089354c978d6e67110d2d598902b0aa4d8aa3a","family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103049,"semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[2,1,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Every Target Open Must Have an Open Source Preimage","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 1, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).","answer":{"kind":"expression","value":"([[[], []], [[0], []], [[0, 1], [1]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 2]], [[1], [1]], [[1, 2], [0, 1, 2]], [[2], [0, 2]]], Yes, every_target_open_has_source_open_preimage)","explanation":"List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], []], [[0, 1], [1]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 2]], [[1], [1]], [[1, 2], [0, 1, 2]], [[2], [0, 2]]], Yes, every_target_open_has_source_open_preimage)."},"skills":["Every Target Open Must Have an Open Source Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0050","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d681f1e804e5d961c1c1d88bb10504252d378a4060c38c654458f89ad29eee8a","family_id":"qf-mod41-compactimg-continuity-audit","seed":202611103050,"semantic_template":"qf-mod41-compactimg-continuity-audit/{\"map\":[0,2,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Every Target Open Must Have an Open Source Preimage","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 2, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).","answer":{"kind":"expression","value":"([[[], []], [[0], [0]], [[0, 1], [0]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], [1, 2]], [[2], [1, 2]]], Yes, every_target_open_has_source_open_preimage)","explanation":"List [target open,full source preimage] for every target open in canonical order. The final continuity flag is their conjunction and may be No; computing a finite compact image alone does not establish continuity. The exact result, retaining the requested order, is ([[[], []], [[0], [0]], [[0, 1], [0]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [0, 1, 2]], [[1], []], [[1, 2], [1, 2]], [[2], [1, 2]]], Yes, every_target_open_has_source_open_preimage)."},"skills":["Every Target Open Must Have an Open Source Preimage"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0051","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1d95345662414e94e2c90d0e684e09a96cfe294fe60c1d79ab4a684f715818d","family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103051,"semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[0]],\"map\":[0,0,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Pull an Image Cover Back to K and Preserve Its Subcover Indices","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 0, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[0]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).","answer":{"kind":"expression","value":"([0], [[0]], [[0, 1, 2]], [0], [[0]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation":"Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([0], [[0]], [[0, 1, 2]], [0], [[0]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)."},"skills":["Pull an Image Cover Back to K and Preserve Its Subcover Indices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0052","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f99e543673881f54f43b9fecff63141a09ede6b9e0bbfc7786f31b0d900ac29e","family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103052,"semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[0],[1]],\"map\":[1,0,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Pull an Image Cover Back to K and Preserve Its Subcover Indices","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 0, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[0], [1]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).","answer":{"kind":"expression","value":"([0, 1], [[0], [1]], [[1, 2], [0]], [0, 1], [[0], [1]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation":"Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([0, 1], [[0], [1]], [[1, 2], [0]], [0, 1], [[0], [1]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)."},"skills":["Pull an Image Cover Back to K and Preserve Its Subcover Indices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0053","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42a1beb622e30719f73269cfbb7fba783a165e779aef820b795ffc5c8c3923a7","family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103053,"semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[0],[2]],\"map\":[2,0,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Pull an Image Cover Back to K and Preserve Its Subcover Indices","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 0, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[0], [2]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).","answer":{"kind":"expression","value":"([0, 2], [[0], [2]], [[1, 2], [0]], [0, 1], [[0], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation":"Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([0, 2], [[0], [2]], [[1, 2], [0]], [0, 1], [[0], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)."},"skills":["Pull an Image Cover Back to K and Preserve Its Subcover Indices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0054","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a513946e2ad58c31f6b51d7fcfc4e3a4d7f2a2f6cf4a1c1e5948c6abbcf6d4d9","family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103054,"semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[0],[1]],\"map\":[0,1,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Pull an Image Cover Back to K and Preserve Its Subcover Indices","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 1, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[0], [1]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).","answer":{"kind":"expression","value":"([0, 1], [[0], [1]], [[0, 2], [1]], [0, 1], [[0], [1]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation":"Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([0, 1], [[0], [1]], [[0, 2], [1]], [0, 1], [[0], [1]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)."},"skills":["Pull an Image Cover Back to K and Preserve Its Subcover Indices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0055","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5abad99f57ab223ceb8cf5c478b5983f4e5f30721c28586af49c12129766688e","family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103055,"semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[0],[1]],\"map\":[1,1,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Pull an Image Cover Back to K and Preserve Its Subcover Indices","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 1, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[0], [1]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).","answer":{"kind":"expression","value":"([0, 1], [[0], [1]], [[2], [0, 1]], [0, 1], [[0], [1]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation":"Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([0, 1], [[0], [1]], [[2], [0, 1]], [0, 1], [[0], [1]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)."},"skills":["Pull an Image Cover Back to K and Preserve Its Subcover Indices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0056","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bbc089cbbd7f8344da254983f4f2cd22ef91b8af5499f1a9543e19c484e290d5","family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103056,"semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[0],[1],[2]],\"map\":[2,1,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Pull an Image Cover Back to K and Preserve Its Subcover Indices","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 1, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[0], [1], [2]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], [[0], [1], [2]], [[2], [1], [0]], [0, 1, 2], [[0], [1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation":"Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([0, 1, 2], [[0], [1], [2]], [[2], [1], [0]], [0, 1, 2], [[0], [1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)."},"skills":["Pull an Image Cover Back to K and Preserve Its Subcover Indices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0057","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63e6fb83e6b93650ed326f7f0aaaa00d42b72980411283b86e248abf046c47e3","family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103057,"semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[0],[2]],\"map\":[0,2,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Pull an Image Cover Back to K and Preserve Its Subcover Indices","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 2, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[0], [2]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).","answer":{"kind":"expression","value":"([0, 2], [[0], [2]], [[0, 2], [1]], [0, 1], [[0], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation":"Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([0, 2], [[0], [2]], [[0, 2], [1]], [0, 1], [[0], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)."},"skills":["Pull an Image Cover Back to K and Preserve Its Subcover Indices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0058","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5cf4d17f092c2191bd68fc7a6decbb26b5e61b451adff43495c40f0137523d3","family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103058,"semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[0],[1],[2]],\"map\":[1,2,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Pull an Image Cover Back to K and Preserve Its Subcover Indices","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 2, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[0], [1], [2]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], [[0], [1], [2]], [[2], [0], [1]], [0, 1, 2], [[0], [1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation":"Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([0, 1, 2], [[0], [1], [2]], [[2], [0], [1]], [0, 1, 2], [[0], [1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)."},"skills":["Pull an Image Cover Back to K and Preserve Its Subcover Indices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0059","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9dc6bad936ddf9dee90de97e6f6072f10c6533bd9021099e464e95cd03a3e8b6","family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103059,"semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[0],[2]],\"map\":[2,2,0],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Pull an Image Cover Back to K and Preserve Its Subcover Indices","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 2, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[0], [2]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).","answer":{"kind":"expression","value":"([0, 2], [[0], [2]], [[2], [0, 1]], [0, 1], [[0], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation":"Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([0, 2], [[0], [2]], [[2], [0, 1]], [0, 1], [[0], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)."},"skills":["Pull an Image Cover Back to K and Preserve Its Subcover Indices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0060","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae7fd05e0536c0ebf468e809a2d354c9bdb2cfa95859ee87a4a8ec36e5135649","family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103060,"semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[0],[1]],\"map\":[0,0,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Pull an Image Cover Back to K and Preserve Its Subcover Indices","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 0, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[0], [1]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).","answer":{"kind":"expression","value":"([0, 1], [[0], [1]], [[0, 1], [2]], [0, 1], [[0], [1]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation":"Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([0, 1], [[0], [1]], [[0, 1], [2]], [0, 1], [[0], [1]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)."},"skills":["Pull an Image Cover Back to K and Preserve Its Subcover Indices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0061","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48da31856a40be6b99d48a227e1ecc25670d46ba5d5e740694fcebdd03f679b1","family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103061,"semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[0],[1]],\"map\":[1,0,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Pull an Image Cover Back to K and Preserve Its Subcover Indices","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 0, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[0], [1]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).","answer":{"kind":"expression","value":"([0, 1], [[0], [1]], [[1], [0, 2]], [0, 1], [[0], [1]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation":"Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([0, 1], [[0], [1]], [[1], [0, 2]], [0, 1], [[0], [1]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)."},"skills":["Pull an Image Cover Back to K and Preserve Its Subcover Indices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0062","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"317b155949c572117f063ff0254aaf455b26127d6a7151df62c770dfe0809bf6","family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103062,"semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[0],[1],[2]],\"map\":[2,0,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Pull an Image Cover Back to K and Preserve Its Subcover Indices","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 0, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[0], [1], [2]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], [[0], [1], [2]], [[1], [2], [0]], [0, 1, 2], [[0], [1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation":"Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([0, 1, 2], [[0], [1], [2]], [[1], [2], [0]], [0, 1, 2], [[0], [1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)."},"skills":["Pull an Image Cover Back to K and Preserve Its Subcover Indices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0063","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3523a2513029eb31736af558ccb4f2b4577144f221b009cf4ec49b9782fb5b9b","family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103063,"semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[0],[1]],\"map\":[0,1,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Pull an Image Cover Back to K and Preserve Its Subcover Indices","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 1, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[0], [1]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).","answer":{"kind":"expression","value":"([0, 1], [[0], [1]], [[0], [1, 2]], [0, 1], [[0], [1]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation":"Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([0, 1], [[0], [1]], [[0], [1, 2]], [0, 1], [[0], [1]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)."},"skills":["Pull an Image Cover Back to K and Preserve Its Subcover Indices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0064","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3556a4cb3bd7499cb0fc40713a21bdc49df837d24525a6b0520132ef017bf507","family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103064,"semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[1]],\"map\":[1,1,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Pull an Image Cover Back to K and Preserve Its Subcover Indices","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 1, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[1]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).","answer":{"kind":"expression","value":"([1], [[1]], [[0, 1, 2]], [0], [[1]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation":"Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([1], [[1]], [[0, 1, 2]], [0], [[1]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)."},"skills":["Pull an Image Cover Back to K and Preserve Its Subcover Indices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0065","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db72897ed48df7b321ec81e3441b5334eacae2919aa549a0bdc89125e9fb1eaa","family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103065,"semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[1],[2]],\"map\":[2,1,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Pull an Image Cover Back to K and Preserve Its Subcover Indices","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 1, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[1], [2]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).","answer":{"kind":"expression","value":"([1, 2], [[1], [2]], [[1, 2], [0]], [0, 1], [[1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation":"Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([1, 2], [[1], [2]], [[1, 2], [0]], [0, 1], [[1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)."},"skills":["Pull an Image Cover Back to K and Preserve Its Subcover Indices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0066","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d07107a7be90a0f32aeaeec433e2a0b253f33f4d9ae5fdb1204b25397bc4cac5","family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103066,"semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[0],[1],[2]],\"map\":[0,2,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Pull an Image Cover Back to K and Preserve Its Subcover Indices","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 2, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[0], [1], [2]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], [[0], [1], [2]], [[0], [2], [1]], [0, 1, 2], [[0], [1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation":"Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([0, 1, 2], [[0], [1], [2]], [[0], [2], [1]], [0, 1, 2], [[0], [1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)."},"skills":["Pull an Image Cover Back to K and Preserve Its Subcover Indices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0067","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2bd81145931aab153b64fb06b13dbcdffb84e435e31c3328f892c03070c06e8","family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103067,"semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[1],[2]],\"map\":[1,2,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Pull an Image Cover Back to K and Preserve Its Subcover Indices","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 2, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[1], [2]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).","answer":{"kind":"expression","value":"([1, 2], [[1], [2]], [[0, 2], [1]], [0, 1], [[1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation":"Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([1, 2], [[1], [2]], [[0, 2], [1]], [0, 1], [[1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)."},"skills":["Pull an Image Cover Back to K and Preserve Its Subcover Indices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0068","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d93514022d278dbb2052bf4cf45b8fb67da81e6655038eaf87c36a871a5643e","family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103068,"semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[1],[2]],\"map\":[2,2,1],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Pull an Image Cover Back to K and Preserve Its Subcover Indices","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 2, 1] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[1], [2]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).","answer":{"kind":"expression","value":"([1, 2], [[1], [2]], [[2], [0, 1]], [0, 1], [[1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation":"Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([1, 2], [[1], [2]], [[2], [0, 1]], [0, 1], [[1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)."},"skills":["Pull an Image Cover Back to K and Preserve Its Subcover Indices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0069","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7adbe0cf971302bbdf791a30071e8938edf49180cd24b3727691c54a449ecbb5","family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103069,"semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[0],[2]],\"map\":[0,0,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Pull an Image Cover Back to K and Preserve Its Subcover Indices","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 0, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[0], [2]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).","answer":{"kind":"expression","value":"([0, 2], [[0], [2]], [[0, 1], [2]], [0, 1], [[0], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation":"Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([0, 2], [[0], [2]], [[0, 1], [2]], [0, 1], [[0], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)."},"skills":["Pull an Image Cover Back to K and Preserve Its Subcover Indices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0070","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"812892f55549706675e1cb1b7fd06c1fac1a5a40530938b87ef9cdec09156d03","family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103070,"semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[0],[1],[2]],\"map\":[1,0,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Pull an Image Cover Back to K and Preserve Its Subcover Indices","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 0, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[0], [1], [2]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], [[0], [1], [2]], [[1], [0], [2]], [0, 1, 2], [[0], [1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation":"Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([0, 1, 2], [[0], [1], [2]], [[1], [0], [2]], [0, 1, 2], [[0], [1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)."},"skills":["Pull an Image Cover Back to K and Preserve Its Subcover Indices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0071","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc5c6316635c575e5ec545e6b45a4adc302b5eaccdc94eae82326615460d0a96","family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103071,"semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[0],[2]],\"map\":[2,0,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Pull an Image Cover Back to K and Preserve Its Subcover Indices","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 0, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[0], [2]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).","answer":{"kind":"expression","value":"([0, 2], [[0], [2]], [[1], [0, 2]], [0, 1], [[0], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation":"Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([0, 2], [[0], [2]], [[1], [0, 2]], [0, 1], [[0], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)."},"skills":["Pull an Image Cover Back to K and Preserve Its Subcover Indices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0072","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4bc7eba5a583d3a7e5d93a95db95960512317afda046d8c9ab5499e726df888c","family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103072,"semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[0],[1],[2]],\"map\":[0,1,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Pull an Image Cover Back to K and Preserve Its Subcover Indices","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 1, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[0], [1], [2]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], [[0], [1], [2]], [[0], [1], [2]], [0, 1, 2], [[0], [1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation":"Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([0, 1, 2], [[0], [1], [2]], [[0], [1], [2]], [0, 1, 2], [[0], [1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)."},"skills":["Pull an Image Cover Back to K and Preserve Its Subcover Indices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0073","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52afb5bbaa04c89a2295d62082daf10e0f5988f8b6735323354e74862fedf707","family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103073,"semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[1],[2]],\"map\":[1,1,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Pull an Image Cover Back to K and Preserve Its Subcover Indices","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 1, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[1], [2]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).","answer":{"kind":"expression","value":"([1, 2], [[1], [2]], [[0, 1], [2]], [0, 1], [[1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation":"Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([1, 2], [[1], [2]], [[0, 1], [2]], [0, 1], [[1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)."},"skills":["Pull an Image Cover Back to K and Preserve Its Subcover Indices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0074","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1b40398c9e4595ebd06046151afc1c9f8c9d761ade7b1ad6c0ea0d8cb817657","family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103074,"semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[1],[2]],\"map\":[2,1,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Pull an Image Cover Back to K and Preserve Its Subcover Indices","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[2, 1, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[1], [2]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).","answer":{"kind":"expression","value":"([1, 2], [[1], [2]], [[1], [0, 2]], [0, 1], [[1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation":"Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([1, 2], [[1], [2]], [[1], [0, 2]], [0, 1], [[1], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)."},"skills":["Pull an Image Cover Back to K and Preserve Its Subcover Indices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0075","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db23bc992b0eeb9f9cf12ff79776ffeb72ef2424358e2a9a5fce8d2634549312","family_id":"qf-mod41-compactimg-cover-pullback","seed":202611103075,"semantic_template":"qf-mod41-compactimg-cover-pullback/{\"cover\":[[0],[2]],\"map\":[0,2,2],\"source_n\":3,\"source_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]],\"subset\":[0,1,2],\"target_n\":3,\"target_topology\":[[],[0,1,2],[0,1],[0,2],[0],[1,2],[1],[2]]}"},"fields":{"title":"Pull an Image Cover Back to K and Preserve Its Subcover Indices","stem":"Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[0, 2, 2] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Original indexed target-open cover=[[0], [2]]. Give (image, cover, relative_pullbacks, selected_indices, selected_target_opens, finite_subcover_certificate, certificate).","answer":{"kind":"expression","value":"([0, 2], [[0], [2]], [[0], [1, 2]], [0, 1], [[0], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)","explanation":"Require continuity and actual coverage of f(K). Intersect each preimage with K, select a finite subcover of K and use those same indices in the original target cover. The source selects minimum cardinality, then first increasing bitmask among ties, not a unique subcover. No injectivity or surjectivity is required. The exact result, retaining the requested order, is ([0, 2], [[0], [2]], [[0], [1, 2]], [0, 1], [[0], [2]], Yes, finite_subcover_pulled_back_to_K_then_pushed_forward_to_image)."},"skills":["Pull an Image Cover Back to K and Preserve Its Subcover Indices"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0076","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b9d81aafaf516a13c0ad47899066875de02bb2b96348282fc9bc6c95593522d","family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103076,"semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-2,\"right\":2,\"value\":0}"},"fields":{"title":"A Compact Constant Image Does Not Make an Open-Interval Domain Compact","stem":"Use the ordinary real open interval (-2,2) with left<right and constant map value=0 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(-2, 2, 0, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation":"The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-2, 2, 0, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)."},"skills":["A Compact Constant Image Does Not Make an Open-Interval Domain Compact"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0077","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b0be56e199a50183bb781b497852c32b12a6b7a68d870da6fbbd7263afebe09","family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103077,"semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-3,\"right\":3,\"value\":1}"},"fields":{"title":"A Compact Constant Image Does Not Make an Open-Interval Domain Compact","stem":"Use the ordinary real open interval (-3,3) with left<right and constant map value=1 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(-3, 3, 1, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation":"The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-3, 3, 1, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)."},"skills":["A Compact Constant Image Does Not Make an Open-Interval Domain Compact"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0078","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27238e8d3cadcd5df3237312f434e5359a567b4d1a0fe6a41f413b1caf5fe97a","family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103078,"semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-4,\"right\":4,\"value\":2}"},"fields":{"title":"A Compact Constant Image Does Not Make an Open-Interval Domain Compact","stem":"Use the ordinary real open interval (-4,4) with left<right and constant map value=2 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(-4, 4, 2, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation":"The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-4, 4, 2, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)."},"skills":["A Compact Constant Image Does Not Make an Open-Interval Domain Compact"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0079","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00f2bb0b527247501575cc57e00822ed304dac9499af1213fc608e66beae0080","family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103079,"semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-5,\"right\":5,\"value\":3}"},"fields":{"title":"A Compact Constant Image Does Not Make an Open-Interval Domain Compact","stem":"Use the ordinary real open interval (-5,5) with left<right and constant map value=3 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(-5, 5, 3, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation":"The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-5, 5, 3, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)."},"skills":["A Compact Constant Image Does Not Make an Open-Interval Domain Compact"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0080","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ef786de8beca6f73749471ab4e25e2bd0d330be67fa363a4fc9ed2d5924448a","family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103080,"semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-6,\"right\":6,\"value\":4}"},"fields":{"title":"A Compact Constant Image Does Not Make an Open-Interval Domain Compact","stem":"Use the ordinary real open interval (-6,6) with left<right and constant map value=4 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(-6, 6, 4, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation":"The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-6, 6, 4, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)."},"skills":["A Compact Constant Image Does Not Make an Open-Interval Domain Compact"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0081","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fcd864bef7b1e95fabd403e4e1d1d98eca51b79091bc574c066a8d749cbd7df4","family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103081,"semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-7,\"right\":7,\"value\":5}"},"fields":{"title":"A Compact Constant Image Does Not Make an Open-Interval Domain Compact","stem":"Use the ordinary real open interval (-7,7) with left<right and constant map value=5 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(-7, 7, 5, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation":"The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-7, 7, 5, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)."},"skills":["A Compact Constant Image Does Not Make an Open-Interval Domain Compact"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0082","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43f5a40dcf56a33af9a36b9bff42f74cb5e2333091bb67fbf9240d43125d2e66","family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103082,"semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-8,\"right\":8,\"value\":6}"},"fields":{"title":"A Compact Constant Image Does Not Make an Open-Interval Domain Compact","stem":"Use the ordinary real open interval (-8,8) with left<right and constant map value=6 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(-8, 8, 6, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation":"The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-8, 8, 6, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)."},"skills":["A Compact Constant Image Does Not Make an Open-Interval Domain Compact"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0083","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28e979c4c003c22809258a058a68acf306047894f8646b65bd73afeaa806d9a0","family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103083,"semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-9,\"right\":9,\"value\":7}"},"fields":{"title":"A Compact Constant Image Does Not Make an Open-Interval Domain Compact","stem":"Use the ordinary real open interval (-9,9) with left<right and constant map value=7 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(-9, 9, 7, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation":"The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-9, 9, 7, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)."},"skills":["A Compact Constant Image Does Not Make an Open-Interval Domain Compact"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0084","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75fff3c7c1ce7210c45c7949133711f557c98f50810615fa6e9a70ac75bf47d2","family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103084,"semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-10,\"right\":10,\"value\":8}"},"fields":{"title":"A Compact Constant Image Does Not Make an Open-Interval Domain Compact","stem":"Use the ordinary real open interval (-10,10) with left<right and constant map value=8 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(-10, 10, 8, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation":"The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-10, 10, 8, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)."},"skills":["A Compact Constant Image Does Not Make an Open-Interval Domain Compact"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0085","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0cd9cee86934fbd65c34471195079048cadb6ccbf7cd3ed258ba62e40ce61017","family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103085,"semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-11,\"right\":11,\"value\":9}"},"fields":{"title":"A Compact Constant Image Does Not Make an Open-Interval Domain Compact","stem":"Use the ordinary real open interval (-11,11) with left<right and constant map value=9 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(-11, 11, 9, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation":"The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-11, 11, 9, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)."},"skills":["A Compact Constant Image Does Not Make an Open-Interval Domain Compact"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0086","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b021a8d27fb00964b4854d2fd20df1cc1310ad01ccf357ce7dfce452aa1e34f8","family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103086,"semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-12,\"right\":12,\"value\":10}"},"fields":{"title":"A Compact Constant Image Does Not Make an Open-Interval Domain Compact","stem":"Use the ordinary real open interval (-12,12) with left<right and constant map value=10 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(-12, 12, 10, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation":"The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-12, 12, 10, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)."},"skills":["A Compact Constant Image Does Not Make an Open-Interval Domain Compact"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0087","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8847d5d72731b04791a1acd0ca4d011a586e506cb70b74404e8d23158baf8faf","family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103087,"semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-13,\"right\":13,\"value\":11}"},"fields":{"title":"A Compact Constant Image Does Not Make an Open-Interval Domain Compact","stem":"Use the ordinary real open interval (-13,13) with left<right and constant map value=11 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(-13, 13, 11, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation":"The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-13, 13, 11, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)."},"skills":["A Compact Constant Image Does Not Make an Open-Interval Domain Compact"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0088","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07021656bb290a9cd2f70a6b58c9e1eb586d63b079f56a3a1dca1f6f22308e54","family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103088,"semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-14,\"right\":14,\"value\":12}"},"fields":{"title":"A Compact Constant Image Does Not Make an Open-Interval Domain Compact","stem":"Use the ordinary real open interval (-14,14) with left<right and constant map value=12 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(-14, 14, 12, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation":"The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-14, 14, 12, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)."},"skills":["A Compact Constant Image Does Not Make an Open-Interval Domain Compact"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0089","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64bb6a7e1f646405ac562d1c31b416faa8ff979d0fbc7f0ccedfa87b54a94398","family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103089,"semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-15,\"right\":15,\"value\":13}"},"fields":{"title":"A Compact Constant Image Does Not Make an Open-Interval Domain Compact","stem":"Use the ordinary real open interval (-15,15) with left<right and constant map value=13 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(-15, 15, 13, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation":"The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-15, 15, 13, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)."},"skills":["A Compact Constant Image Does Not Make an Open-Interval Domain Compact"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0090","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4dc50470d4570745f0334865765832fd7a74b0ed8d155fb2c44f1d5b4153ba6","family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103090,"semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-16,\"right\":16,\"value\":14}"},"fields":{"title":"A Compact Constant Image Does Not Make an Open-Interval Domain Compact","stem":"Use the ordinary real open interval (-16,16) with left<right and constant map value=14 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(-16, 16, 14, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation":"The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-16, 16, 14, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)."},"skills":["A Compact Constant Image Does Not Make an Open-Interval Domain Compact"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0091","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1036084d739d6150a6ef67b63c599db08aad6bab6300b7d456ce9f214f046c64","family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103091,"semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-17,\"right\":17,\"value\":15}"},"fields":{"title":"A Compact Constant Image Does Not Make an Open-Interval Domain Compact","stem":"Use the ordinary real open interval (-17,17) with left<right and constant map value=15 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(-17, 17, 15, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation":"The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-17, 17, 15, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)."},"skills":["A Compact Constant Image Does Not Make an Open-Interval Domain Compact"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0092","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20537cc2d4573076b4a58a74dd3f9dcfdeeb09334288f8ba59aedb42e2048184","family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103092,"semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-18,\"right\":18,\"value\":16}"},"fields":{"title":"A Compact Constant Image Does Not Make an Open-Interval Domain Compact","stem":"Use the ordinary real open interval (-18,18) with left<right and constant map value=16 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(-18, 18, 16, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation":"The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-18, 18, 16, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)."},"skills":["A Compact Constant Image Does Not Make an Open-Interval Domain Compact"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0093","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d580906f50b0495aeb5f7983789b8714b11cd1d7c984b6d55b697046a4ceaa98","family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103093,"semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-19,\"right\":19,\"value\":17}"},"fields":{"title":"A Compact Constant Image Does Not Make an Open-Interval Domain Compact","stem":"Use the ordinary real open interval (-19,19) with left<right and constant map value=17 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(-19, 19, 17, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation":"The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-19, 19, 17, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)."},"skills":["A Compact Constant Image Does Not Make an Open-Interval Domain Compact"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0094","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34458cb08557290ce2a6b9e0a317786479a688fbe1838a656ee178e3697dc5f1","family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103094,"semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-20,\"right\":20,\"value\":18}"},"fields":{"title":"A Compact Constant Image Does Not Make an Open-Interval Domain Compact","stem":"Use the ordinary real open interval (-20,20) with left<right and constant map value=18 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(-20, 20, 18, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation":"The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-20, 20, 18, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)."},"skills":["A Compact Constant Image Does Not Make an Open-Interval Domain Compact"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0095","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac0c9c5b0d785910141df53e535dc4e6c2e6d6bee8a5f9c38045c9e963ec7cc6","family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103095,"semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-21,\"right\":21,\"value\":19}"},"fields":{"title":"A Compact Constant Image Does Not Make an Open-Interval Domain Compact","stem":"Use the ordinary real open interval (-21,21) with left<right and constant map value=19 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(-21, 21, 19, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation":"The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-21, 21, 19, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)."},"skills":["A Compact Constant Image Does Not Make an Open-Interval Domain Compact"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0096","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb21b1d145b6b820567643323d3eaf9b53bf1fff5eb4496daedd5a642f9d6ac1","family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103096,"semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-22,\"right\":22,\"value\":20}"},"fields":{"title":"A Compact Constant Image Does Not Make an Open-Interval Domain Compact","stem":"Use the ordinary real open interval (-22,22) with left<right and constant map value=20 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(-22, 22, 20, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation":"The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-22, 22, 20, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)."},"skills":["A Compact Constant Image Does Not Make an Open-Interval Domain Compact"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0097","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc54a2fb2f1910987d69e934a9094f19393246ae41a94651f59f632144aa9743","family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103097,"semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-23,\"right\":23,\"value\":21}"},"fields":{"title":"A Compact Constant Image Does Not Make an Open-Interval Domain Compact","stem":"Use the ordinary real open interval (-23,23) with left<right and constant map value=21 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(-23, 23, 21, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation":"The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-23, 23, 21, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)."},"skills":["A Compact Constant Image Does Not Make an Open-Interval Domain Compact"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0098","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67c4f0b174a6d1ff9b6b392da629d88862b5548097fd7a4359bf052e92caf004","family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103098,"semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-24,\"right\":24,\"value\":22}"},"fields":{"title":"A Compact Constant Image Does Not Make an Open-Interval Domain Compact","stem":"Use the ordinary real open interval (-24,24) with left<right and constant map value=22 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(-24, 24, 22, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation":"The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-24, 24, 22, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)."},"skills":["A Compact Constant Image Does Not Make an Open-Interval Domain Compact"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0099","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78d1d02020b9ed933a73b2459d98d267b3ae7aa30a22ef02e8a7dbae7f69677b","family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103099,"semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-25,\"right\":25,\"value\":23}"},"fields":{"title":"A Compact Constant Image Does Not Make an Open-Interval Domain Compact","stem":"Use the ordinary real open interval (-25,25) with left<right and constant map value=23 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(-25, 25, 23, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation":"The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-25, 25, 23, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)."},"skills":["A Compact Constant Image Does Not Make an Open-Interval Domain Compact"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r03-continuous-image-of-compact-0100","batch_id":"batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7270df6db96b8194500c2a59845bca4d0432e5fc16a98d095aa778654ddc34ae","family_id":"qf-mod41-compactimg-reverse-implication-boundary","seed":202611103100,"semantic_template":"qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-26,\"right\":26,\"value\":24}"},"fields":{"title":"A Compact Constant Image Does Not Make an Open-Interval Domain Compact","stem":"Use the ordinary real open interval (-26,26) with left<right and constant map value=24 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(-26, 26, 24, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)","explanation":"The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-26, 26, 24, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)."},"skills":["A Compact Constant Image Does Not Make an Open-Interval Domain Compact"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}