{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0001","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ad3ed66825f12acafa84de7d328c158c97eb0684d2944caea0a449765362a2d","family_id":"qf-density11-tends-principal-image","seed":202610117001,"semantic_template":"qf-density11-tends-principal-image/[]"},"fields":{"title":"Finite Principal-Filter Image Generator","stem":"The complete total map f:Fin(0)→Fin(0) has images=[] indexed from 0; Fin(0)=∅. Every table entry must lie in the target type. S=[]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.","answer":{"kind":"expression","value":"[]","explanation":"Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is []."},"skills":["Finite Principal-Filter Image Generator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0002","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16e5764fbaa8dd3b437c54e5eff580bab1bfbe3b08818385d78a0d2c7d7dabd3","family_id":"qf-density11-tends-principal-image","seed":202610117002,"semantic_template":"qf-density11-tends-principal-image/[0]"},"fields":{"title":"Finite Principal-Filter Image Generator","stem":"The complete total map f:Fin(3)→Fin(1) has images=[0, 0, 0] indexed from 0; Fin(3)={0,…,2}. Every table entry must lie in the target type. S=[0, 1]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.","answer":{"kind":"expression","value":"[0]","explanation":"Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0]."},"skills":["Finite Principal-Filter Image Generator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0003","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2bf04ca274ac80325b0b6d195f12b40f3075cbc5b43ff90a3cf7a7a5439bca79","family_id":"qf-density11-tends-principal-image","seed":202610117003,"semantic_template":"qf-density11-tends-principal-image/[1]"},"fields":{"title":"Finite Principal-Filter Image Generator","stem":"The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[1]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.","answer":{"kind":"expression","value":"[1]","explanation":"Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [1]."},"skills":["Finite Principal-Filter Image Generator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0004","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fcdb2115cb2cb94890d097d02309cd3754e92adb1138f32e59334b24f8ac6b48","family_id":"qf-density11-tends-principal-image","seed":202610117004,"semantic_template":"qf-density11-tends-principal-image/[0,1]"},"fields":{"title":"Finite Principal-Filter Image Generator","stem":"The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.","answer":{"kind":"expression","value":"[0, 1]","explanation":"Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 1]."},"skills":["Finite Principal-Filter Image Generator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0005","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2cf72c3bb6e9bbdc0b7b1f8c1b3adfc13ca05e8e399fc8161dc6f014d4cd9679","family_id":"qf-density11-tends-principal-image","seed":202610117005,"semantic_template":"qf-density11-tends-principal-image/[2]"},"fields":{"title":"Finite Principal-Filter Image Generator","stem":"The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.","answer":{"kind":"expression","value":"[2]","explanation":"Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [2]."},"skills":["Finite Principal-Filter Image Generator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0006","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee3205753e791b4b755fe1608d87e7aa9b0cfeb1446a5ee5eac6e8578428a0c8","family_id":"qf-density11-tends-principal-image","seed":202610117006,"semantic_template":"qf-density11-tends-principal-image/[0,2]"},"fields":{"title":"Finite Principal-Filter Image Generator","stem":"The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.","answer":{"kind":"expression","value":"[0, 2]","explanation":"Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 2]."},"skills":["Finite Principal-Filter Image Generator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0007","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2df108147eeb0239fa74d7da8da702ed6d3782d5b3158474e490b762d0b00aae","family_id":"qf-density11-tends-principal-image","seed":202610117007,"semantic_template":"qf-density11-tends-principal-image/[1,2]"},"fields":{"title":"Finite Principal-Filter Image Generator","stem":"The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.","answer":{"kind":"expression","value":"[1, 2]","explanation":"Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [1, 2]."},"skills":["Finite Principal-Filter Image Generator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0008","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6ab3174cb4a7b3df38b3428b6ccf3fa110b22a1e64a9ebd44085b8c9199f8c1","family_id":"qf-density11-tends-principal-image","seed":202610117008,"semantic_template":"qf-density11-tends-principal-image/[0,1,2]"},"fields":{"title":"Finite Principal-Filter Image Generator","stem":"The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.","answer":{"kind":"expression","value":"[0, 1, 2]","explanation":"Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 1, 2]."},"skills":["Finite Principal-Filter Image Generator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0009","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4095c5f367eaa6c0e6f77a0b2160413ea7b9e2bf38f070ed23b408b40b095866","family_id":"qf-density11-tends-principal-image","seed":202610117009,"semantic_template":"qf-density11-tends-principal-image/[3]"},"fields":{"title":"Finite Principal-Filter Image Generator","stem":"The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.","answer":{"kind":"expression","value":"[3]","explanation":"Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [3]."},"skills":["Finite Principal-Filter Image Generator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0010","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ee472ce9c415d87e7af46f89be486d112f8fcf57bc26c93c82d2107ff51ebc9","family_id":"qf-density11-tends-principal-image","seed":202610117010,"semantic_template":"qf-density11-tends-principal-image/[0,3]"},"fields":{"title":"Finite Principal-Filter Image Generator","stem":"The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.","answer":{"kind":"expression","value":"[0, 3]","explanation":"Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 3]."},"skills":["Finite Principal-Filter Image Generator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0011","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b0eb1976083bbfcedb6e19f30483d04b0caf8ef41de84209fc3bd8ba84c2e22","family_id":"qf-density11-tends-principal-image","seed":202610117011,"semantic_template":"qf-density11-tends-principal-image/[1,3]"},"fields":{"title":"Finite Principal-Filter Image Generator","stem":"The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[1, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.","answer":{"kind":"expression","value":"[1, 3]","explanation":"Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [1, 3]."},"skills":["Finite Principal-Filter Image Generator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0012","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54f44cd78e1eca147ba1c0194777e071e6de9708ac9236e8f8e82a891b9c4463","family_id":"qf-density11-tends-principal-image","seed":202610117012,"semantic_template":"qf-density11-tends-principal-image/[0,1,3]"},"fields":{"title":"Finite Principal-Filter Image Generator","stem":"The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.","answer":{"kind":"expression","value":"[0, 1, 3]","explanation":"Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 1, 3]."},"skills":["Finite Principal-Filter Image Generator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0013","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd3fedb35042941ec21cee088aa85d03ff9f7e0f886e3a868d24fe104a2a70c3","family_id":"qf-density11-tends-principal-image","seed":202610117013,"semantic_template":"qf-density11-tends-principal-image/[2,3]"},"fields":{"title":"Finite Principal-Filter Image Generator","stem":"The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[2, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.","answer":{"kind":"expression","value":"[2, 3]","explanation":"Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [2, 3]."},"skills":["Finite Principal-Filter Image Generator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0014","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8480182793367823fb331a5079662669ba40d0e0bba9eacfb8f09094956bfad9","family_id":"qf-density11-tends-principal-image","seed":202610117014,"semantic_template":"qf-density11-tends-principal-image/[0,2,3]"},"fields":{"title":"Finite Principal-Filter Image Generator","stem":"The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 2, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.","answer":{"kind":"expression","value":"[0, 2, 3]","explanation":"Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 2, 3]."},"skills":["Finite Principal-Filter Image Generator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0015","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c427385d065485e64b026697eb89bd4c18b3787dc71b5c57e5792948eae0a31","family_id":"qf-density11-tends-principal-image","seed":202610117015,"semantic_template":"qf-density11-tends-principal-image/[1,2,3]"},"fields":{"title":"Finite Principal-Filter Image Generator","stem":"The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[1, 2, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.","answer":{"kind":"expression","value":"[1, 2, 3]","explanation":"Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [1, 2, 3]."},"skills":["Finite Principal-Filter Image Generator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0016","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c21a9372e1e31d388116232b306501330abcc672734f1c9cf182943458155639","family_id":"qf-density11-tends-principal-image","seed":202610117016,"semantic_template":"qf-density11-tends-principal-image/[0,1,2,3]"},"fields":{"title":"Finite Principal-Filter Image Generator","stem":"The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.","answer":{"kind":"expression","value":"[0, 1, 2, 3]","explanation":"Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 1, 2, 3]."},"skills":["Finite Principal-Filter Image Generator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0017","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1cf27a5369b0cd5df3fef6dd7d08e21ae8c4a409ce66337441e5e12b5216b551","family_id":"qf-density11-tends-principal-image","seed":202610117017,"semantic_template":"qf-density11-tends-principal-image/[4]"},"fields":{"title":"Finite Principal-Filter Image Generator","stem":"The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.","answer":{"kind":"expression","value":"[4]","explanation":"Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [4]."},"skills":["Finite Principal-Filter Image Generator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0018","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7b92e65117b4ccdfc09a1a322b3a8e82244e011c8a9bb20deac2e4a70875330","family_id":"qf-density11-tends-principal-image","seed":202610117018,"semantic_template":"qf-density11-tends-principal-image/[0,4]"},"fields":{"title":"Finite Principal-Filter Image Generator","stem":"The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.","answer":{"kind":"expression","value":"[0, 4]","explanation":"Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 4]."},"skills":["Finite Principal-Filter Image Generator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0019","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78e97cd57bf4b8591aa07252804c68d98fd2d657eb574e7fcaf1c5758f87f4c6","family_id":"qf-density11-tends-principal-image","seed":202610117019,"semantic_template":"qf-density11-tends-principal-image/[1,4]"},"fields":{"title":"Finite Principal-Filter Image Generator","stem":"The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[1, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.","answer":{"kind":"expression","value":"[1, 4]","explanation":"Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [1, 4]."},"skills":["Finite Principal-Filter Image Generator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0020","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5c8a2dc53a6f7889d5d10925c1cae51c39a48e16f9a34e4d27c8e214cca07e6","family_id":"qf-density11-tends-principal-image","seed":202610117020,"semantic_template":"qf-density11-tends-principal-image/[0,1,4]"},"fields":{"title":"Finite Principal-Filter Image Generator","stem":"The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.","answer":{"kind":"expression","value":"[0, 1, 4]","explanation":"Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 1, 4]."},"skills":["Finite Principal-Filter Image Generator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0021","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc6b066f34525db33e43eb4bb6d8410ddb17b9810028da9357b37f928a68b5c5","family_id":"qf-density11-tends-principal-image","seed":202610117021,"semantic_template":"qf-density11-tends-principal-image/[2,4]"},"fields":{"title":"Finite Principal-Filter Image Generator","stem":"The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[2, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.","answer":{"kind":"expression","value":"[2, 4]","explanation":"Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [2, 4]."},"skills":["Finite Principal-Filter Image Generator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0022","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a181049245d7d8ee1b4b4fcb81167c15a10e39ffd86e8960114add7690a63b39","family_id":"qf-density11-tends-principal-image","seed":202610117022,"semantic_template":"qf-density11-tends-principal-image/[0,2,4]"},"fields":{"title":"Finite Principal-Filter Image Generator","stem":"The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 2, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.","answer":{"kind":"expression","value":"[0, 2, 4]","explanation":"Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 2, 4]."},"skills":["Finite Principal-Filter Image Generator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0023","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bca537b1056cd1b5331375d42fefa60838b01cefb258ff1479dde12d6ba71921","family_id":"qf-density11-tends-principal-image","seed":202610117023,"semantic_template":"qf-density11-tends-principal-image/[1,2,4]"},"fields":{"title":"Finite Principal-Filter Image Generator","stem":"The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[1, 2, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.","answer":{"kind":"expression","value":"[1, 2, 4]","explanation":"Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [1, 2, 4]."},"skills":["Finite Principal-Filter Image Generator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0024","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"997136a486e80cada46b1255969348dda31492a8ed3c9e7f0aef17ea066243d5","family_id":"qf-density11-tends-principal-image","seed":202610117024,"semantic_template":"qf-density11-tends-principal-image/[0,1,2,4]"},"fields":{"title":"Finite Principal-Filter Image Generator","stem":"The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.","answer":{"kind":"expression","value":"[0, 1, 2, 4]","explanation":"Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 1, 2, 4]."},"skills":["Finite Principal-Filter Image Generator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0025","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5498fadea555c6fb548181625d2d4d085d354f40d42628ae769cd4b309548a6e","family_id":"qf-density11-tends-principal-image","seed":202610117025,"semantic_template":"qf-density11-tends-principal-image/[3,4]"},"fields":{"title":"Finite Principal-Filter Image Generator","stem":"The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[3, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.","answer":{"kind":"expression","value":"[3, 4]","explanation":"Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [3, 4]."},"skills":["Finite Principal-Filter Image Generator"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0026","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f44783d5538c5869524ee5f57e9d98b3612e3933d684b746810e73b0a2dad9c8","family_id":"qf-density11-tends-principal-obstructions","seed":202610117026,"semantic_template":"qf-density11-tends-principal-obstructions/[]"},"fields":{"title":"Finite Principal-Filter Direction Obstructions","stem":"The complete total map f:Fin(3)→Fin(1) has images=[0, 0, 0] indexed from 0; Fin(3)={0,…,2}. Every table entry must lie in the target type. S=[0, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.","answer":{"kind":"expression","value":"[]","explanation":"An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is []."},"skills":["Finite Principal-Filter Direction Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0027","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67dc9c3af11e789ce36f1c1110b958122c6cd3cd1fe6cb910585df260d2f7131","family_id":"qf-density11-tends-principal-obstructions","seed":202610117027,"semantic_template":"qf-density11-tends-principal-obstructions/[0]"},"fields":{"title":"Finite Principal-Filter Direction Obstructions","stem":"The complete total map f:Fin(8)→Fin(2) has images=[1, 0, 0, 0, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.","answer":{"kind":"expression","value":"[0]","explanation":"An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0]."},"skills":["Finite Principal-Filter Direction Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0028","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"413ac1aa94c816ff58608df0c2788378d88302646bcc8f24c957ed42b90794f1","family_id":"qf-density11-tends-principal-obstructions","seed":202610117028,"semantic_template":"qf-density11-tends-principal-obstructions/[1]"},"fields":{"title":"Finite Principal-Filter Direction Obstructions","stem":"The complete total map f:Fin(8)→Fin(2) has images=[0, 1, 0, 0, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.","answer":{"kind":"expression","value":"[1]","explanation":"An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [1]."},"skills":["Finite Principal-Filter Direction Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0029","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71502521ad3adb8dee5daec59fbd180e13ba27a012351e657c49db382aa7b36c","family_id":"qf-density11-tends-principal-obstructions","seed":202610117029,"semantic_template":"qf-density11-tends-principal-obstructions/[0,1]"},"fields":{"title":"Finite Principal-Filter Direction Obstructions","stem":"The complete total map f:Fin(8)→Fin(2) has images=[1, 1, 0, 0, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.","answer":{"kind":"expression","value":"[0, 1]","explanation":"An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 1]."},"skills":["Finite Principal-Filter Direction Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0030","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca536c2ae929e15cc1db04d6786c56fd507d617efc29542a6f09b39ddef878b0","family_id":"qf-density11-tends-principal-obstructions","seed":202610117030,"semantic_template":"qf-density11-tends-principal-obstructions/[2]"},"fields":{"title":"Finite Principal-Filter Direction Obstructions","stem":"The complete total map f:Fin(8)→Fin(2) has images=[0, 0, 1, 0, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.","answer":{"kind":"expression","value":"[2]","explanation":"An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [2]."},"skills":["Finite Principal-Filter Direction Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0031","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8661b4005c8029ffeaeff3e8081d00d324002ce8d7c64b6dfccc17057060b356","family_id":"qf-density11-tends-principal-obstructions","seed":202610117031,"semantic_template":"qf-density11-tends-principal-obstructions/[0,2]"},"fields":{"title":"Finite Principal-Filter Direction Obstructions","stem":"The complete total map f:Fin(8)→Fin(2) has images=[1, 0, 1, 0, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.","answer":{"kind":"expression","value":"[0, 2]","explanation":"An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 2]."},"skills":["Finite Principal-Filter Direction Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0032","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"acd00a859765cf14f62a351a403e4be2552d5efc089d2741d1f7cb3046a11070","family_id":"qf-density11-tends-principal-obstructions","seed":202610117032,"semantic_template":"qf-density11-tends-principal-obstructions/[1,2]"},"fields":{"title":"Finite Principal-Filter Direction Obstructions","stem":"The complete total map f:Fin(8)→Fin(2) has images=[0, 1, 1, 0, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.","answer":{"kind":"expression","value":"[1, 2]","explanation":"An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [1, 2]."},"skills":["Finite Principal-Filter Direction Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0033","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b335d6f0f96639f22883c1cb1b88d1a08d9097157260a3e03239f8fd94ff4600","family_id":"qf-density11-tends-principal-obstructions","seed":202610117033,"semantic_template":"qf-density11-tends-principal-obstructions/[0,1,2]"},"fields":{"title":"Finite Principal-Filter Direction Obstructions","stem":"The complete total map f:Fin(8)→Fin(2) has images=[1, 1, 1, 0, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.","answer":{"kind":"expression","value":"[0, 1, 2]","explanation":"An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 1, 2]."},"skills":["Finite Principal-Filter Direction Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0034","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88c94ef1467531d65d4b078d973c43b5784181b17d4706ed92297c1a61b0baae","family_id":"qf-density11-tends-principal-obstructions","seed":202610117034,"semantic_template":"qf-density11-tends-principal-obstructions/[3]"},"fields":{"title":"Finite Principal-Filter Direction Obstructions","stem":"The complete total map f:Fin(8)→Fin(2) has images=[0, 0, 0, 1, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.","answer":{"kind":"expression","value":"[3]","explanation":"An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [3]."},"skills":["Finite Principal-Filter Direction Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0035","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf90e87612f326b86fd8327807e8dacc8cd9c7e2f8db087788296e6e3a360ce5","family_id":"qf-density11-tends-principal-obstructions","seed":202610117035,"semantic_template":"qf-density11-tends-principal-obstructions/[0,3]"},"fields":{"title":"Finite Principal-Filter Direction Obstructions","stem":"The complete total map f:Fin(8)→Fin(2) has images=[1, 0, 0, 1, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.","answer":{"kind":"expression","value":"[0, 3]","explanation":"An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 3]."},"skills":["Finite Principal-Filter Direction Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0036","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23a9c57891665289e306b30b14ce64681a89be2475fceeb1e1dee801ee9a115e","family_id":"qf-density11-tends-principal-obstructions","seed":202610117036,"semantic_template":"qf-density11-tends-principal-obstructions/[1,3]"},"fields":{"title":"Finite Principal-Filter Direction Obstructions","stem":"The complete total map f:Fin(8)→Fin(2) has images=[0, 1, 0, 1, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.","answer":{"kind":"expression","value":"[1, 3]","explanation":"An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [1, 3]."},"skills":["Finite Principal-Filter Direction Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0037","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d498a926387509436558b924894d5d977c9c4ee79ac9bdaba0b554303cf483e5","family_id":"qf-density11-tends-principal-obstructions","seed":202610117037,"semantic_template":"qf-density11-tends-principal-obstructions/[0,1,3]"},"fields":{"title":"Finite Principal-Filter Direction Obstructions","stem":"The complete total map f:Fin(8)→Fin(2) has images=[1, 1, 0, 1, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.","answer":{"kind":"expression","value":"[0, 1, 3]","explanation":"An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 1, 3]."},"skills":["Finite Principal-Filter Direction Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0038","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"943a411f0d7ad7a3e8f67636183ec80884037a48d2cdb4566b06d8ed9d77bcc3","family_id":"qf-density11-tends-principal-obstructions","seed":202610117038,"semantic_template":"qf-density11-tends-principal-obstructions/[2,3]"},"fields":{"title":"Finite Principal-Filter Direction Obstructions","stem":"The complete total map f:Fin(8)→Fin(2) has images=[0, 0, 1, 1, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.","answer":{"kind":"expression","value":"[2, 3]","explanation":"An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [2, 3]."},"skills":["Finite Principal-Filter Direction Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0039","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e12e6fa1b2a56cffd3f5b722831c01d7998258f4ec5995bb6e3f892856b60ef9","family_id":"qf-density11-tends-principal-obstructions","seed":202610117039,"semantic_template":"qf-density11-tends-principal-obstructions/[0,2,3]"},"fields":{"title":"Finite Principal-Filter Direction Obstructions","stem":"The complete total map f:Fin(8)→Fin(2) has images=[1, 0, 1, 1, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.","answer":{"kind":"expression","value":"[0, 2, 3]","explanation":"An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 2, 3]."},"skills":["Finite Principal-Filter Direction Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0040","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9315939cf3fd40683f1fcd4989ca7b3a5cc9dbc0dc1df543b3cec044a2768a3e","family_id":"qf-density11-tends-principal-obstructions","seed":202610117040,"semantic_template":"qf-density11-tends-principal-obstructions/[1,2,3]"},"fields":{"title":"Finite Principal-Filter Direction Obstructions","stem":"The complete total map f:Fin(8)→Fin(2) has images=[0, 1, 1, 1, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.","answer":{"kind":"expression","value":"[1, 2, 3]","explanation":"An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [1, 2, 3]."},"skills":["Finite Principal-Filter Direction Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0041","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9012cdf30e217565267017f12a65491f1def7e19295754cd7881ca2fe0941f8a","family_id":"qf-density11-tends-principal-obstructions","seed":202610117041,"semantic_template":"qf-density11-tends-principal-obstructions/[0,1,2,3]"},"fields":{"title":"Finite Principal-Filter Direction Obstructions","stem":"The complete total map f:Fin(8)→Fin(2) has images=[1, 1, 1, 1, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.","answer":{"kind":"expression","value":"[0, 1, 2, 3]","explanation":"An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 1, 2, 3]."},"skills":["Finite Principal-Filter Direction Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0042","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7924bd56825c57d2b243721bb419a3570c77daf22e9375c3f5009f4330b234c4","family_id":"qf-density11-tends-principal-obstructions","seed":202610117042,"semantic_template":"qf-density11-tends-principal-obstructions/[4]"},"fields":{"title":"Finite Principal-Filter Direction Obstructions","stem":"The complete total map f:Fin(8)→Fin(2) has images=[0, 0, 0, 0, 1, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.","answer":{"kind":"expression","value":"[4]","explanation":"An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [4]."},"skills":["Finite Principal-Filter Direction Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0043","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2d3cab9c93b2b66e054a7ca86521a12a00f38f844c04dabbaef7ef960a204f8","family_id":"qf-density11-tends-principal-obstructions","seed":202610117043,"semantic_template":"qf-density11-tends-principal-obstructions/[0,4]"},"fields":{"title":"Finite Principal-Filter Direction Obstructions","stem":"The complete total map f:Fin(8)→Fin(2) has images=[1, 0, 0, 0, 1, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.","answer":{"kind":"expression","value":"[0, 4]","explanation":"An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 4]."},"skills":["Finite Principal-Filter Direction Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0044","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5aa79c1073c8f0b31108fa7f63b87db12138318685889813891e3ac4a924cec","family_id":"qf-density11-tends-principal-obstructions","seed":202610117044,"semantic_template":"qf-density11-tends-principal-obstructions/[1,4]"},"fields":{"title":"Finite Principal-Filter Direction Obstructions","stem":"The complete total map f:Fin(8)→Fin(2) has images=[0, 1, 0, 0, 1, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.","answer":{"kind":"expression","value":"[1, 4]","explanation":"An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [1, 4]."},"skills":["Finite Principal-Filter Direction Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0045","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff70668fb86cb6510f95a729c71e1bfdac1a0652586c234e65668ef538200d1c","family_id":"qf-density11-tends-principal-obstructions","seed":202610117045,"semantic_template":"qf-density11-tends-principal-obstructions/[0,1,4]"},"fields":{"title":"Finite Principal-Filter Direction Obstructions","stem":"The complete total map f:Fin(8)→Fin(2) has images=[1, 1, 0, 0, 1, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.","answer":{"kind":"expression","value":"[0, 1, 4]","explanation":"An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 1, 4]."},"skills":["Finite Principal-Filter Direction Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0046","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5263df1fb632a89ddf25dd8c64e1fdc5a9c9f0f94432a273327d59e77e2e168","family_id":"qf-density11-tends-principal-obstructions","seed":202610117046,"semantic_template":"qf-density11-tends-principal-obstructions/[2,4]"},"fields":{"title":"Finite Principal-Filter Direction Obstructions","stem":"The complete total map f:Fin(8)→Fin(2) has images=[0, 0, 1, 0, 1, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.","answer":{"kind":"expression","value":"[2, 4]","explanation":"An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [2, 4]."},"skills":["Finite Principal-Filter Direction Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0047","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7171de7332cf46aa7f689b83a1cb9e0fb2308736e2c4eac79601b9fa9ebdc4ac","family_id":"qf-density11-tends-principal-obstructions","seed":202610117047,"semantic_template":"qf-density11-tends-principal-obstructions/[0,2,4]"},"fields":{"title":"Finite Principal-Filter Direction Obstructions","stem":"The complete total map f:Fin(8)→Fin(2) has images=[1, 0, 1, 0, 1, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.","answer":{"kind":"expression","value":"[0, 2, 4]","explanation":"An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 2, 4]."},"skills":["Finite Principal-Filter Direction Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0048","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90d8f0c5a09ea424021a1626fce01526bd9df7dd62845192e9030835e5ec3b6f","family_id":"qf-density11-tends-principal-obstructions","seed":202610117048,"semantic_template":"qf-density11-tends-principal-obstructions/[1,2,4]"},"fields":{"title":"Finite Principal-Filter Direction Obstructions","stem":"The complete total map f:Fin(8)→Fin(2) has images=[0, 1, 1, 0, 1, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.","answer":{"kind":"expression","value":"[1, 2, 4]","explanation":"An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [1, 2, 4]."},"skills":["Finite Principal-Filter Direction Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0049","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"614ed047ac7c049cc18dff33bb39816af36908da83ced79da4f82c8b751fa97b","family_id":"qf-density11-tends-principal-obstructions","seed":202610117049,"semantic_template":"qf-density11-tends-principal-obstructions/[0,1,2,4]"},"fields":{"title":"Finite Principal-Filter Direction Obstructions","stem":"The complete total map f:Fin(8)→Fin(2) has images=[1, 1, 1, 0, 1, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.","answer":{"kind":"expression","value":"[0, 1, 2, 4]","explanation":"An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 1, 2, 4]."},"skills":["Finite Principal-Filter Direction Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0050","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"874a660f9d4712d940c22e36816b0bcb2a90ddfd82759421be5841f5b4f4427e","family_id":"qf-density11-tends-principal-obstructions","seed":202610117050,"semantic_template":"qf-density11-tends-principal-obstructions/[3,4]"},"fields":{"title":"Finite Principal-Filter Direction Obstructions","stem":"The complete total map f:Fin(8)→Fin(2) has images=[0, 0, 0, 1, 1, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.","answer":{"kind":"expression","value":"[3, 4]","explanation":"An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [3, 4]."},"skills":["Finite Principal-Filter Direction Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0051","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fab6bdb7ec2acb3faf6c27e626793cc51115a5403f0bf601d86c23b663552283","family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117051,"semantic_template":"qf-density11-tends-principal-candidate-mask/[\"outside_type\",\"outside_type\",\"outside_type\"]"},"fields":{"title":"Typed Principal-Filter Obstruction Candidates","stem":"The complete total map f:Fin(0)→Fin(0) has images=[] indexed from 0; Fin(0)=∅. Every table entry must lie in the target type. S=[]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[], ordered candidates=[-1, 0, 1]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.","answer":{"kind":"expression","value":"[No, No, No]","explanation":"Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No]."},"skills":["Typed Principal-Filter Obstruction Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0052","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27d5c6dc7d5822235ba748d28b8e8bf72e618a2dd2fee49536e90969eb37b684","family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117052,"semantic_template":"qf-density11-tends-principal-candidate-mask/[\"outside_generator\",\"outside_generator\",\"outside_generator\"]"},"fields":{"title":"Typed Principal-Filter Obstruction Candidates","stem":"The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[3, 4, 5]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.","answer":{"kind":"expression","value":"[No, No, No]","explanation":"Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No]."},"skills":["Typed Principal-Filter Obstruction Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0053","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4fb73b667dd6ec926f072551fa64d230659c604dd5e3d31b0a795432a8aa45ff","family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117053,"semantic_template":"qf-density11-tends-principal-candidate-mask/[\"maps_inside\",\"maps_inside\",\"maps_inside\"]"},"fields":{"title":"Typed Principal-Filter Obstruction Candidates","stem":"The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[0, 0, 0]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.","answer":{"kind":"expression","value":"[No, No, No]","explanation":"Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No]."},"skills":["Typed Principal-Filter Obstruction Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0054","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8fcca63f96e6533b91aa7c3b0979deaa4ef9d198c54ef2eae030f8f0fb860a00","family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117054,"semantic_template":"qf-density11-tends-principal-candidate-mask/[\"obstructs\",\"obstructs\",\"obstructs\"]"},"fields":{"title":"Typed Principal-Filter Obstruction Candidates","stem":"The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[], ordered candidates=[0, 1, 2]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.","answer":{"kind":"expression","value":"[Yes, Yes, Yes]","explanation":"Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [Yes, Yes, Yes]."},"skills":["Typed Principal-Filter Obstruction Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0055","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54eddc6dd87da281bc58fbd686d1cc8f7df5e1098b9af782084c5135613f8a03","family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117055,"semantic_template":"qf-density11-tends-principal-candidate-mask/[\"outside_generator\",\"outside_generator\",\"outside_type\"]"},"fields":{"title":"Typed Principal-Filter Obstruction Candidates","stem":"The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[3, 4, 102]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.","answer":{"kind":"expression","value":"[No, No, No]","explanation":"Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No]."},"skills":["Typed Principal-Filter Obstruction Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0056","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2164cec25a26cab0820043c0babeaabaaa10526405c19e999f983274b455b7c8","family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117056,"semantic_template":"qf-density11-tends-principal-candidate-mask/[\"maps_inside\",\"maps_inside\",\"outside_type\"]"},"fields":{"title":"Typed Principal-Filter Obstruction Candidates","stem":"The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[0, 0, 102]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.","answer":{"kind":"expression","value":"[No, No, No]","explanation":"Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No]."},"skills":["Typed Principal-Filter Obstruction Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0057","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83b53a5c136cb191aaf2f795ac76d0eeb658d98be30e2495cc7bf1b823f6ff88","family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117057,"semantic_template":"qf-density11-tends-principal-candidate-mask/[\"obstructs\",\"obstructs\",\"outside_type\"]"},"fields":{"title":"Typed Principal-Filter Obstruction Candidates","stem":"The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[1, 2, 102]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.","answer":{"kind":"expression","value":"[Yes, Yes, No]","explanation":"Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [Yes, Yes, No]."},"skills":["Typed Principal-Filter Obstruction Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0058","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0eea6a9f093b65881f13dc380f4787cfb909607a22d5b609c60ed92b06437c38","family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117058,"semantic_template":"qf-density11-tends-principal-candidate-mask/[\"outside_type\",\"outside_generator\",\"outside_generator\"]"},"fields":{"title":"Typed Principal-Filter Obstruction Candidates","stem":"The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[100, 4, 5]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.","answer":{"kind":"expression","value":"[No, No, No]","explanation":"Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No]."},"skills":["Typed Principal-Filter Obstruction Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0059","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"195f0c9752b778da604d27361a7f4490a76b8e21864fa2bb6f0ae58667a5d03b","family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117059,"semantic_template":"qf-density11-tends-principal-candidate-mask/[\"outside_generator\",\"maps_inside\",\"outside_generator\"]"},"fields":{"title":"Typed Principal-Filter Obstruction Candidates","stem":"The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[3, 0, 5]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.","answer":{"kind":"expression","value":"[No, No, No]","explanation":"Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No]."},"skills":["Typed Principal-Filter Obstruction Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0060","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c97d92df4f9fc4f144725a9c10b6b734cad2e404e6ec408f0c7d646d63bfb67a","family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117060,"semantic_template":"qf-density11-tends-principal-candidate-mask/[\"maps_inside\",\"obstructs\",\"outside_generator\"]"},"fields":{"title":"Typed Principal-Filter Obstruction Candidates","stem":"The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[0, 2, 5]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.","answer":{"kind":"expression","value":"[No, Yes, No]","explanation":"Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, Yes, No]."},"skills":["Typed Principal-Filter Obstruction Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0061","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9bfb1c4f1f4c9424bb71a3dfde7df905d65f0935917162f4a8cdd980a9822a3","family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117061,"semantic_template":"qf-density11-tends-principal-candidate-mask/[\"obstructs\",\"outside_type\",\"maps_inside\"]"},"fields":{"title":"Typed Principal-Filter Obstruction Candidates","stem":"The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[1, 101, 0]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.","answer":{"kind":"expression","value":"[Yes, No, No]","explanation":"Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [Yes, No, No]."},"skills":["Typed Principal-Filter Obstruction Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0062","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df8feb5fba796fcf31747af0dfd00c34189517220f324371d7e7eb786cf5f9dd","family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117062,"semantic_template":"qf-density11-tends-principal-candidate-mask/[\"outside_type\",\"maps_inside\",\"maps_inside\"]"},"fields":{"title":"Typed Principal-Filter Obstruction Candidates","stem":"The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[100, 0, 0]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.","answer":{"kind":"expression","value":"[No, No, No]","explanation":"Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No]."},"skills":["Typed Principal-Filter Obstruction Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0063","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c1f59d6b6d9100be4958a06d2f94856ab640cb5fe90817897c5459c7186d885","family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117063,"semantic_template":"qf-density11-tends-principal-candidate-mask/[\"outside_generator\",\"obstructs\",\"maps_inside\"]"},"fields":{"title":"Typed Principal-Filter Obstruction Candidates","stem":"The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[3, 2, 0]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.","answer":{"kind":"expression","value":"[No, Yes, No]","explanation":"Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, Yes, No]."},"skills":["Typed Principal-Filter Obstruction Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0064","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41b6da4e8cbe5235b0f1deb92bfca1d14835d11404d3ec2c99c742371e7a2590","family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117064,"semantic_template":"qf-density11-tends-principal-candidate-mask/[\"maps_inside\",\"outside_type\",\"obstructs\"]"},"fields":{"title":"Typed Principal-Filter Obstruction Candidates","stem":"The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[0, 101, 1]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.","answer":{"kind":"expression","value":"[No, No, Yes]","explanation":"Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, Yes]."},"skills":["Typed Principal-Filter Obstruction Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0065","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4471f77d68694ca26002dcb196811fdba28f4da41e7a0b17ee9bfca5f4f5c650","family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117065,"semantic_template":"qf-density11-tends-principal-candidate-mask/[\"obstructs\",\"outside_generator\",\"obstructs\"]"},"fields":{"title":"Typed Principal-Filter Obstruction Candidates","stem":"The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[1, 4, 1]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.","answer":{"kind":"expression","value":"[Yes, No, Yes]","explanation":"Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [Yes, No, Yes]."},"skills":["Typed Principal-Filter Obstruction Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0066","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27ee22bd969d2cb699e4a5a14eb301a37e5b3d1b494106c62eedb4b918f306c0","family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117066,"semantic_template":"qf-density11-tends-principal-candidate-mask/[\"outside_type\",\"obstructs\",\"obstructs\"]"},"fields":{"title":"Typed Principal-Filter Obstruction Candidates","stem":"The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[100, 2, 1]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.","answer":{"kind":"expression","value":"[No, Yes, Yes]","explanation":"Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, Yes, Yes]."},"skills":["Typed Principal-Filter Obstruction Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0067","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91829126ab9d6004496db9ffae7457b9b02462139fd938024164f066f10b0171","family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117067,"semantic_template":"qf-density11-tends-principal-candidate-mask/[\"outside_generator\",\"outside_type\",\"outside_type\"]"},"fields":{"title":"Typed Principal-Filter Obstruction Candidates","stem":"The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[3, 101, 102]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.","answer":{"kind":"expression","value":"[No, No, No]","explanation":"Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No]."},"skills":["Typed Principal-Filter Obstruction Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0068","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8de69522aa12d5fda17888565b99710f10daad206282950f6f7436e850a34d05","family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117068,"semantic_template":"qf-density11-tends-principal-candidate-mask/[\"maps_inside\",\"outside_generator\",\"outside_type\"]"},"fields":{"title":"Typed Principal-Filter Obstruction Candidates","stem":"The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[0, 4, 102]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.","answer":{"kind":"expression","value":"[No, No, No]","explanation":"Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No]."},"skills":["Typed Principal-Filter Obstruction Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0069","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5922e728a7722b1108cff07deae86a36a64a6d686350e7ebd8d18bf246ffa25","family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117069,"semantic_template":"qf-density11-tends-principal-candidate-mask/[\"obstructs\",\"maps_inside\",\"outside_type\"]"},"fields":{"title":"Typed Principal-Filter Obstruction Candidates","stem":"The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[1, 0, 102]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.","answer":{"kind":"expression","value":"[Yes, No, No]","explanation":"Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [Yes, No, No]."},"skills":["Typed Principal-Filter Obstruction Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0070","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"889116e8cd9a962221bf08bcf39e1fd6c580fb16d1494ded366e9321894c9ed8","family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117070,"semantic_template":"qf-density11-tends-principal-candidate-mask/[\"outside_type\",\"outside_type\",\"outside_generator\"]"},"fields":{"title":"Typed Principal-Filter Obstruction Candidates","stem":"The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[100, 101, 5]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.","answer":{"kind":"expression","value":"[No, No, No]","explanation":"Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No]."},"skills":["Typed Principal-Filter Obstruction Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0071","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"98e713d396933359a0d5d39b06f18576665d62a7e899a634781df8790203f1a2","family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117071,"semantic_template":"qf-density11-tends-principal-candidate-mask/[\"maps_inside\",\"maps_inside\",\"outside_generator\"]"},"fields":{"title":"Typed Principal-Filter Obstruction Candidates","stem":"The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[0, 0, 5]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.","answer":{"kind":"expression","value":"[No, No, No]","explanation":"Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No]."},"skills":["Typed Principal-Filter Obstruction Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0072","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e585942a87eff877bd66d4b82803232559d2aeaf1b507e79d1715d3f2fb5812","family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117072,"semantic_template":"qf-density11-tends-principal-candidate-mask/[\"obstructs\",\"obstructs\",\"outside_generator\"]"},"fields":{"title":"Typed Principal-Filter Obstruction Candidates","stem":"The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[1, 2, 5]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.","answer":{"kind":"expression","value":"[Yes, Yes, No]","explanation":"Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [Yes, Yes, No]."},"skills":["Typed Principal-Filter Obstruction Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0073","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"067d78c20b4864e7b51ff0a68907eeba99149c8c2d625ef02bfdca7e356d224e","family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117073,"semantic_template":"qf-density11-tends-principal-candidate-mask/[\"outside_type\",\"outside_generator\",\"maps_inside\"]"},"fields":{"title":"Typed Principal-Filter Obstruction Candidates","stem":"The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[100, 4, 0]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.","answer":{"kind":"expression","value":"[No, No, No]","explanation":"Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No]."},"skills":["Typed Principal-Filter Obstruction Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0074","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3670af96929b5bb65035fd9fcd1deaacadf47f4fbf30cd953a5bfcce4901da9","family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117074,"semantic_template":"qf-density11-tends-principal-candidate-mask/[\"outside_generator\",\"maps_inside\",\"maps_inside\"]"},"fields":{"title":"Typed Principal-Filter Obstruction Candidates","stem":"The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[3, 0, 0]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.","answer":{"kind":"expression","value":"[No, No, No]","explanation":"Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No]."},"skills":["Typed Principal-Filter Obstruction Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0075","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"acf1a74fb7a2e42bacba1e53f8dab7bf8ea580014f93accfbbbe309e944897fa","family_id":"qf-density11-tends-principal-candidate-mask","seed":202610117075,"semantic_template":"qf-density11-tends-principal-candidate-mask/[\"maps_inside\",\"obstructs\",\"maps_inside\"]"},"fields":{"title":"Typed Principal-Filter Obstruction Candidates","stem":"The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[0, 2, 0]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.","answer":{"kind":"expression","value":"[No, Yes, No]","explanation":"Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, Yes, No]."},"skills":["Typed Principal-Filter Obstruction Candidates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0076","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be9fa2fbc415cc20f9c5526ef5fa361bebe973564e321ff2ee4701c7c8a60a2e","family_id":"qf-density11-tends-principal-composition-images","seed":202610117076,"semantic_template":"qf-density11-tends-principal-composition-images/[]"},"fields":{"title":"Two Finite Principal-Filter Composition Images","stem":"The complete total map f:Fin(0)→Fin(0) has images=[] indexed from 0; Fin(0)=∅. Every table entry must lie in the target type. S=[]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(0)→Fin(0) has after=[]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.","answer":{"kind":"expression","value":"([], [])","explanation":"Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([], [])."},"skills":["Two Finite Principal-Filter Composition Images"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0077","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8fdd13039d24d2325e525fbdc8e4d9626edf9de8bb5622da37c330cbcee279b4","family_id":"qf-density11-tends-principal-composition-images","seed":202610117077,"semantic_template":"qf-density11-tends-principal-composition-images/[0]"},"fields":{"title":"Two Finite Principal-Filter Composition Images","stem":"The complete total map f:Fin(4)→Fin(3) has images=[0, 0, 1, 2] indexed from 0; Fin(4)={0,…,3}. Every table entry must lie in the target type. S=[0, 1, 2, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(3)→Fin(8) has after=[0, 0, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.","answer":{"kind":"expression","value":"([0], [0])","explanation":"Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([0], [0])."},"skills":["Two Finite Principal-Filter Composition Images"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0078","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a20ddbca7009b1eb8dd6eac5ad51885360fc41f8bfd45d111b1b69ecc40c011","family_id":"qf-density11-tends-principal-composition-images","seed":202610117078,"semantic_template":"qf-density11-tends-principal-composition-images/[1]"},"fields":{"title":"Two Finite Principal-Filter Composition Images","stem":"The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[1]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.","answer":{"kind":"expression","value":"([1], [1])","explanation":"Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([1], [1])."},"skills":["Two Finite Principal-Filter Composition Images"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0079","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77f405e97a63383cabcb4c6c894558c466fac79872258ebe54a7c0118fda9f60","family_id":"qf-density11-tends-principal-composition-images","seed":202610117079,"semantic_template":"qf-density11-tends-principal-composition-images/[0,1]"},"fields":{"title":"Two Finite Principal-Filter Composition Images","stem":"The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.","answer":{"kind":"expression","value":"([0, 1], [0, 1])","explanation":"Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([0, 1], [0, 1])."},"skills":["Two Finite Principal-Filter Composition Images"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0080","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee3d9881d22208e48290d2a032c7249ca988a60cd475b27a9b0ee26c0236b27e","family_id":"qf-density11-tends-principal-composition-images","seed":202610117080,"semantic_template":"qf-density11-tends-principal-composition-images/[2]"},"fields":{"title":"Two Finite Principal-Filter Composition Images","stem":"The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.","answer":{"kind":"expression","value":"([2], [2])","explanation":"Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([2], [2])."},"skills":["Two Finite Principal-Filter Composition Images"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0081","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b115585a58fe9f2712854170a42206c8b685af51bac89ef62b85560f848012de","family_id":"qf-density11-tends-principal-composition-images","seed":202610117081,"semantic_template":"qf-density11-tends-principal-composition-images/[0,2]"},"fields":{"title":"Two Finite Principal-Filter Composition Images","stem":"The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.","answer":{"kind":"expression","value":"([0, 2], [0, 2])","explanation":"Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([0, 2], [0, 2])."},"skills":["Two Finite Principal-Filter Composition Images"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0082","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6fa8139962fd61f899ae6c116f3e637d0176177d5dab03a95ddb520be1ffcae6","family_id":"qf-density11-tends-principal-composition-images","seed":202610117082,"semantic_template":"qf-density11-tends-principal-composition-images/[1,2]"},"fields":{"title":"Two Finite Principal-Filter Composition Images","stem":"The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.","answer":{"kind":"expression","value":"([1, 2], [1, 2])","explanation":"Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([1, 2], [1, 2])."},"skills":["Two Finite Principal-Filter Composition Images"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0083","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0bce5c9c0ee6204122346057910cdb321f05b62178cc463ae37d982e7674ad83","family_id":"qf-density11-tends-principal-composition-images","seed":202610117083,"semantic_template":"qf-density11-tends-principal-composition-images/[0,1,2]"},"fields":{"title":"Two Finite Principal-Filter Composition Images","stem":"The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.","answer":{"kind":"expression","value":"([0, 1, 2], [0, 1, 2])","explanation":"Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([0, 1, 2], [0, 1, 2])."},"skills":["Two Finite Principal-Filter Composition Images"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0084","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3317f8f5a53f89370d4949781cf5491843816a4457a0ebe228a0cbfd7dfda6f8","family_id":"qf-density11-tends-principal-composition-images","seed":202610117084,"semantic_template":"qf-density11-tends-principal-composition-images/[3]"},"fields":{"title":"Two Finite Principal-Filter Composition Images","stem":"The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.","answer":{"kind":"expression","value":"([3], [3])","explanation":"Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([3], [3])."},"skills":["Two Finite Principal-Filter Composition Images"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0085","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf6a2805b12739b052da2e5d2478970f0cf056376dc6a94b61e7506cc44a9615","family_id":"qf-density11-tends-principal-composition-images","seed":202610117085,"semantic_template":"qf-density11-tends-principal-composition-images/[0,3]"},"fields":{"title":"Two Finite Principal-Filter Composition Images","stem":"The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.","answer":{"kind":"expression","value":"([0, 3], [0, 3])","explanation":"Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([0, 3], [0, 3])."},"skills":["Two Finite Principal-Filter Composition Images"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0086","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f8c1cbaf7f3506e24431da49385e4e19e8f8ba7af2f6ee770cba224de297edf","family_id":"qf-density11-tends-principal-composition-images","seed":202610117086,"semantic_template":"qf-density11-tends-principal-composition-images/[1,3]"},"fields":{"title":"Two Finite Principal-Filter Composition Images","stem":"The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[1, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.","answer":{"kind":"expression","value":"([1, 3], [1, 3])","explanation":"Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([1, 3], [1, 3])."},"skills":["Two Finite Principal-Filter Composition Images"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0087","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3915121c0fd48d16ca6944aaf5835b33b8ca8eebbeac5b1303341cd4990983c","family_id":"qf-density11-tends-principal-composition-images","seed":202610117087,"semantic_template":"qf-density11-tends-principal-composition-images/[0,1,3]"},"fields":{"title":"Two Finite Principal-Filter Composition Images","stem":"The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.","answer":{"kind":"expression","value":"([0, 1, 3], [0, 1, 3])","explanation":"Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([0, 1, 3], [0, 1, 3])."},"skills":["Two Finite Principal-Filter Composition Images"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0088","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f566b2e889ac450fa4f19493421c357394f51eec7476ebc0571e548fa0640b88","family_id":"qf-density11-tends-principal-composition-images","seed":202610117088,"semantic_template":"qf-density11-tends-principal-composition-images/[2,3]"},"fields":{"title":"Two Finite Principal-Filter Composition Images","stem":"The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[2, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.","answer":{"kind":"expression","value":"([2, 3], [2, 3])","explanation":"Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([2, 3], [2, 3])."},"skills":["Two Finite Principal-Filter Composition Images"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0089","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"360133609285242c9f122f4e90eafaef073d950de60b5b60bb67d67384fde382","family_id":"qf-density11-tends-principal-composition-images","seed":202610117089,"semantic_template":"qf-density11-tends-principal-composition-images/[0,2,3]"},"fields":{"title":"Two Finite Principal-Filter Composition Images","stem":"The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 2, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.","answer":{"kind":"expression","value":"([0, 2, 3], [0, 2, 3])","explanation":"Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([0, 2, 3], [0, 2, 3])."},"skills":["Two Finite Principal-Filter Composition Images"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0090","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4b37a4ba317895d373a838694efaab1ff3a07e5457274f4f3dc99aef813d183","family_id":"qf-density11-tends-principal-composition-images","seed":202610117090,"semantic_template":"qf-density11-tends-principal-composition-images/[1,2,3]"},"fields":{"title":"Two Finite Principal-Filter Composition Images","stem":"The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[1, 2, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.","answer":{"kind":"expression","value":"([1, 2, 3], [1, 2, 3])","explanation":"Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([1, 2, 3], [1, 2, 3])."},"skills":["Two Finite Principal-Filter Composition Images"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0091","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28897f5b125a541a3c07db3da3597e42481d0fb88d887281a05f462500a05265","family_id":"qf-density11-tends-principal-composition-images","seed":202610117091,"semantic_template":"qf-density11-tends-principal-composition-images/[0,1,2,3]"},"fields":{"title":"Two Finite Principal-Filter Composition Images","stem":"The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.","answer":{"kind":"expression","value":"([0, 1, 2, 3], [0, 1, 2, 3])","explanation":"Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([0, 1, 2, 3], [0, 1, 2, 3])."},"skills":["Two Finite Principal-Filter Composition Images"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0092","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a85d406fc9492e67586668cbf4c60c0a94627e8644cbd5851c6494862ab7138","family_id":"qf-density11-tends-principal-composition-images","seed":202610117092,"semantic_template":"qf-density11-tends-principal-composition-images/[4]"},"fields":{"title":"Two Finite Principal-Filter Composition Images","stem":"The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.","answer":{"kind":"expression","value":"([4], [4])","explanation":"Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([4], [4])."},"skills":["Two Finite Principal-Filter Composition Images"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0093","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b521a7e54133df91167de729c0104ae506cae4ea3d1a4bcc7bef1f0e71ab213","family_id":"qf-density11-tends-principal-composition-images","seed":202610117093,"semantic_template":"qf-density11-tends-principal-composition-images/[0,4]"},"fields":{"title":"Two Finite Principal-Filter Composition Images","stem":"The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.","answer":{"kind":"expression","value":"([0, 4], [0, 4])","explanation":"Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([0, 4], [0, 4])."},"skills":["Two Finite Principal-Filter Composition Images"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0094","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f208375cc326e45cc67803ad7e214d875305ecee24e531bad1c2c43f10c315fb","family_id":"qf-density11-tends-principal-composition-images","seed":202610117094,"semantic_template":"qf-density11-tends-principal-composition-images/[1,4]"},"fields":{"title":"Two Finite Principal-Filter Composition Images","stem":"The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[1, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.","answer":{"kind":"expression","value":"([1, 4], [1, 4])","explanation":"Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([1, 4], [1, 4])."},"skills":["Two Finite Principal-Filter Composition Images"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0095","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4cfe977d2d94a40174c662c69cb21b8e2c2133ed5a7d798a27d3fc7e787eafc","family_id":"qf-density11-tends-principal-composition-images","seed":202610117095,"semantic_template":"qf-density11-tends-principal-composition-images/[0,1,4]"},"fields":{"title":"Two Finite Principal-Filter Composition Images","stem":"The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.","answer":{"kind":"expression","value":"([0, 1, 4], [0, 1, 4])","explanation":"Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([0, 1, 4], [0, 1, 4])."},"skills":["Two Finite Principal-Filter Composition Images"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0096","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"755bbb9c55e232437d48c1e8694b1cff3e282bbf570b8ec5498b7a8a07d55411","family_id":"qf-density11-tends-principal-composition-images","seed":202610117096,"semantic_template":"qf-density11-tends-principal-composition-images/[2,4]"},"fields":{"title":"Two Finite Principal-Filter Composition Images","stem":"The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[2, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.","answer":{"kind":"expression","value":"([2, 4], [2, 4])","explanation":"Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([2, 4], [2, 4])."},"skills":["Two Finite Principal-Filter Composition Images"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0097","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81940d438a0a22bc78b168c77a12978071d9004d2827e1f8d005821c67058ae7","family_id":"qf-density11-tends-principal-composition-images","seed":202610117097,"semantic_template":"qf-density11-tends-principal-composition-images/[0,2,4]"},"fields":{"title":"Two Finite Principal-Filter Composition Images","stem":"The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 2, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.","answer":{"kind":"expression","value":"([0, 2, 4], [0, 2, 4])","explanation":"Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([0, 2, 4], [0, 2, 4])."},"skills":["Two Finite Principal-Filter Composition Images"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0098","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24c4614b70c8874fa7522f8a7bf06195b8ab6cf59bd060e7308343242a182187","family_id":"qf-density11-tends-principal-composition-images","seed":202610117098,"semantic_template":"qf-density11-tends-principal-composition-images/[1,2,4]"},"fields":{"title":"Two Finite Principal-Filter Composition Images","stem":"The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[1, 2, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.","answer":{"kind":"expression","value":"([1, 2, 4], [1, 2, 4])","explanation":"Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([1, 2, 4], [1, 2, 4])."},"skills":["Two Finite Principal-Filter Composition Images"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0099","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7dba53338edbcae34a87f363f89d6eaa276da6c8a96c0d53f82d04a1df5cebb9","family_id":"qf-density11-tends-principal-composition-images","seed":202610117099,"semantic_template":"qf-density11-tends-principal-composition-images/[0,1,2,4]"},"fields":{"title":"Two Finite Principal-Filter Composition Images","stem":"The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.","answer":{"kind":"expression","value":"([0, 1, 2, 4], [0, 1, 2, 4])","explanation":"Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([0, 1, 2, 4], [0, 1, 2, 4])."},"skills":["Two Finite Principal-Filter Composition Images"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r07-tends-to-relation-0100","batch_id":"batch-20261008-advance-20261011-r07-tends-to-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"44ce11190d219dcf02ef6877a3f8e01bd63094df92edb2d1405d496943f2dc5a","family_id":"qf-density11-tends-principal-composition-images","seed":202610117100,"semantic_template":"qf-density11-tends-principal-composition-images/[3,4]"},"fields":{"title":"Two Finite Principal-Filter Composition Images","stem":"The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[3, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.","answer":{"kind":"expression","value":"([3, 4], [3, 4])","explanation":"Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([3, 4], [3, 4])."},"skills":["Two Finite Principal-Filter Composition Images"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}