{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0001","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f37c71612766a3239acc78544e58f1e50891f5da3295c9794a7c5cb58a351928","family_id":"qf-mod46-mreduction-total-map-qualification","seed":202611157001,"semantic_template":"qf-mod46-mreduction-total-map-qualification/{\"mapping\":[0,1,null],\"source_carrier\":[0,1,2],\"target_carrier\":[0,1,2]}"},"fields":{"title":"Complete Total Mapping Qualification Precedes Membership Equivalence","stem":"Use source carrier=[0, 1, 2], target carrier=[0, 1, 2], aligned mapping=[0, 1, null]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).","answer":{"kind":"expression","value":"([0, 1, 2], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, null, No]], No, [2, null, No], many_one_reduction_requires_a_complete_total_map_before_membership_audit)","explanation":"For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, null, No]], No, [2, null, No], many_one_reduction_requires_a_complete_total_map_before_membership_audit)."},"skills":["Complete Total Mapping Qualification Precedes Membership Equivalence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0002","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e627125826b23834db28e03182009383ec53c41a8bbbe71e3d1934dec2882e0e","family_id":"qf-mod46-mreduction-total-map-qualification","seed":202611157002,"semantic_template":"qf-mod46-mreduction-total-map-qualification/{\"mapping\":[1,2,0,1],\"source_carrier\":[0,1,2,3],\"target_carrier\":[0,1,2]}"},"fields":{"title":"Complete Total Mapping Qualification Precedes Membership Equivalence","stem":"Use source carrier=[0, 1, 2, 3], target carrier=[0, 1, 2], aligned mapping=[1, 2, 0, 1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).","answer":{"kind":"expression","value":"([0, 1, 2, 3], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)","explanation":"For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)."},"skills":["Complete Total Mapping Qualification Precedes Membership Equivalence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0003","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c8a15e04c3ef865c30035f0077b1d661c44e7aecc5546d473a4f0e8b1cdbe1c","family_id":"qf-mod46-mreduction-total-map-qualification","seed":202611157003,"semantic_template":"qf-mod46-mreduction-total-map-qualification/{\"mapping\":[2,0,1,2,0],\"source_carrier\":[0,1,2,3,4],\"target_carrier\":[0,1,2]}"},"fields":{"title":"Complete Total Mapping Qualification Precedes Membership Equivalence","stem":"Use source carrier=[0, 1, 2, 3, 4], target carrier=[0, 1, 2], aligned mapping=[2, 0, 1, 2, 0]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)","explanation":"For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)."},"skills":["Complete Total Mapping Qualification Precedes Membership Equivalence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0004","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77487e5f72450a6c014cea573b19663663d2339183361faf0c4d4d88aeb017d7","family_id":"qf-mod46-mreduction-total-map-qualification","seed":202611157004,"semantic_template":"qf-mod46-mreduction-total-map-qualification/{\"mapping\":[0,1,2,0,1,2],\"source_carrier\":[0,1,2,3,4,5],\"target_carrier\":[0,1,2]}"},"fields":{"title":"Complete Total Mapping Qualification Precedes Membership Equivalence","stem":"Use source carrier=[0, 1, 2, 3, 4, 5], target carrier=[0, 1, 2], aligned mapping=[0, 1, 2, 0, 1, 2]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)","explanation":"For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)."},"skills":["Complete Total Mapping Qualification Precedes Membership Equivalence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0005","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c01161c003cd8a2917e4ed1b49aa7227d675bcf6630f001038c4f676e06edeb6","family_id":"qf-mod46-mreduction-total-map-qualification","seed":202611157005,"semantic_template":"qf-mod46-mreduction-total-map-qualification/{\"mapping\":[1,2,0,1,2,0,1],\"source_carrier\":[0,1,2,3,4,5,6],\"target_carrier\":[0,1,2]}"},"fields":{"title":"Complete Total Mapping Qualification Precedes Membership Equivalence","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6], target carrier=[0, 1, 2], aligned mapping=[1, 2, 0, 1, 2, 0, 1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)","explanation":"For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)."},"skills":["Complete Total Mapping Qualification Precedes Membership Equivalence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0006","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e80a30e6a5d7d80ddb27155ba53a8478855b828f8aa19df1d3105044f1a84a49","family_id":"qf-mod46-mreduction-total-map-qualification","seed":202611157006,"semantic_template":"qf-mod46-mreduction-total-map-qualification/{\"mapping\":[2,0,1,2,0,1,2,null],\"source_carrier\":[0,1,2,3,4,5,6,7],\"target_carrier\":[0,1,2]}"},"fields":{"title":"Complete Total Mapping Qualification Precedes Membership Equivalence","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7], target carrier=[0, 1, 2], aligned mapping=[2, 0, 1, 2, 0, 1, 2, null]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, null, No]], No, [7, null, No], many_one_reduction_requires_a_complete_total_map_before_membership_audit)","explanation":"For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, null, No]], No, [7, null, No], many_one_reduction_requires_a_complete_total_map_before_membership_audit)."},"skills":["Complete Total Mapping Qualification Precedes Membership Equivalence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0007","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6cf6aa50559c0820509d319d8c3a269915889f74a80a9a6c388622232fa30070","family_id":"qf-mod46-mreduction-total-map-qualification","seed":202611157007,"semantic_template":"qf-mod46-mreduction-total-map-qualification/{\"mapping\":[0,1,2,0,1,2,0,1,2],\"source_carrier\":[0,1,2,3,4,5,6,7,8],\"target_carrier\":[0,1,2]}"},"fields":{"title":"Complete Total Mapping Qualification Precedes Membership Equivalence","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8], target carrier=[0, 1, 2], aligned mapping=[0, 1, 2, 0, 1, 2, 0, 1, 2]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)","explanation":"For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)."},"skills":["Complete Total Mapping Qualification Precedes Membership Equivalence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0008","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b67dd204b6f013ddc4cda3715ce7c3024464db947a9c2bd12ae9864a36903da","family_id":"qf-mod46-mreduction-total-map-qualification","seed":202611157008,"semantic_template":"qf-mod46-mreduction-total-map-qualification/{\"mapping\":[1,2,0,1,2,0,1,2,0,1],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9],\"target_carrier\":[0,1,2]}"},"fields":{"title":"Complete Total Mapping Qualification Precedes Membership Equivalence","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], target carrier=[0, 1, 2], aligned mapping=[1, 2, 0, 1, 2, 0, 1, 2, 0, 1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes], [7, 2, Yes], [8, 0, Yes], [9, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)","explanation":"For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes], [7, 2, Yes], [8, 0, Yes], [9, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)."},"skills":["Complete Total Mapping Qualification Precedes Membership Equivalence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0009","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7eb3bae87371f9373c3af91b4b0d2128872540fe7955063913dccb679eb0ea7d","family_id":"qf-mod46-mreduction-total-map-qualification","seed":202611157009,"semantic_template":"qf-mod46-mreduction-total-map-qualification/{\"mapping\":[2,0,1,2,0,1,2,0,1,2,0],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10],\"target_carrier\":[0,1,2]}"},"fields":{"title":"Complete Total Mapping Qualification Precedes Membership Equivalence","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], target carrier=[0, 1, 2], aligned mapping=[2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, 0, Yes], [8, 1, Yes], [9, 2, Yes], [10, 0, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)","explanation":"For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, 0, Yes], [8, 1, Yes], [9, 2, Yes], [10, 0, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)."},"skills":["Complete Total Mapping Qualification Precedes Membership Equivalence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0010","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"312afcda7ba8ca846ad95e1254b0a47114b60dce2c7dca6181ddaee2ca7e75f4","family_id":"qf-mod46-mreduction-total-map-qualification","seed":202611157010,"semantic_template":"qf-mod46-mreduction-total-map-qualification/{\"mapping\":[0,1,2,0,1,2,0,1,2,0,1,2],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11],\"target_carrier\":[0,1,2]}"},"fields":{"title":"Complete Total Mapping Qualification Precedes Membership Equivalence","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], target carrier=[0, 1, 2], aligned mapping=[0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes], [9, 0, Yes], [10, 1, Yes], [11, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)","explanation":"For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes], [9, 0, Yes], [10, 1, Yes], [11, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)."},"skills":["Complete Total Mapping Qualification Precedes Membership Equivalence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0011","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76e58fce0a3160908e4135e068197b3a20e745ad506a0b3f597beba76d21eb13","family_id":"qf-mod46-mreduction-total-map-qualification","seed":202611157011,"semantic_template":"qf-mod46-mreduction-total-map-qualification/{\"mapping\":[1,2,0,1,2,0,1,2,0,1,2,0,null],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12],\"target_carrier\":[0,1,2]}"},"fields":{"title":"Complete Total Mapping Qualification Precedes Membership Equivalence","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], target carrier=[0, 1, 2], aligned mapping=[1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, null]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes], [7, 2, Yes], [8, 0, Yes], [9, 1, Yes], [10, 2, Yes], [11, 0, Yes], [12, null, No]], No, [12, null, No], many_one_reduction_requires_a_complete_total_map_before_membership_audit)","explanation":"For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes], [7, 2, Yes], [8, 0, Yes], [9, 1, Yes], [10, 2, Yes], [11, 0, Yes], [12, null, No]], No, [12, null, No], many_one_reduction_requires_a_complete_total_map_before_membership_audit)."},"skills":["Complete Total Mapping Qualification Precedes Membership Equivalence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0012","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ffa86c27f0e60e396afd77de089eb047127bfcf67a73d4ef42ea9b01bf91f5d2","family_id":"qf-mod46-mreduction-total-map-qualification","seed":202611157012,"semantic_template":"qf-mod46-mreduction-total-map-qualification/{\"mapping\":[2,0,1,2,0,1,2,0,1,2,0,1,2,0],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13],\"target_carrier\":[0,1,2]}"},"fields":{"title":"Complete Total Mapping Qualification Precedes Membership Equivalence","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], target carrier=[0, 1, 2], aligned mapping=[2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, 0, Yes], [8, 1, Yes], [9, 2, Yes], [10, 0, Yes], [11, 1, Yes], [12, 2, Yes], [13, 0, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)","explanation":"For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, 0, Yes], [8, 1, Yes], [9, 2, Yes], [10, 0, Yes], [11, 1, Yes], [12, 2, Yes], [13, 0, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)."},"skills":["Complete Total Mapping Qualification Precedes Membership Equivalence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0013","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"308a2fe689c970f00ae7faf1f066ecf93ffab9472febe923756e74fe8de88b4f","family_id":"qf-mod46-mreduction-total-map-qualification","seed":202611157013,"semantic_template":"qf-mod46-mreduction-total-map-qualification/{\"mapping\":[0,1,2,0,1,2,0,1,2,0,1,2,0,1,2],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],\"target_carrier\":[0,1,2]}"},"fields":{"title":"Complete Total Mapping Qualification Precedes Membership Equivalence","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], target carrier=[0, 1, 2], aligned mapping=[0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes], [9, 0, Yes], [10, 1, Yes], [11, 2, Yes], [12, 0, Yes], [13, 1, Yes], [14, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)","explanation":"For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes], [9, 0, Yes], [10, 1, Yes], [11, 2, Yes], [12, 0, Yes], [13, 1, Yes], [14, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)."},"skills":["Complete Total Mapping Qualification Precedes Membership Equivalence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0014","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"80f7fa783e0e3766749525b91d2fa1fbd9dd55a3e5197dd89e9b350c3889d308","family_id":"qf-mod46-mreduction-total-map-qualification","seed":202611157014,"semantic_template":"qf-mod46-mreduction-total-map-qualification/{\"mapping\":[1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],\"target_carrier\":[0,1,2]}"},"fields":{"title":"Complete Total Mapping Qualification Precedes Membership Equivalence","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], target carrier=[0, 1, 2], aligned mapping=[1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes], [7, 2, Yes], [8, 0, Yes], [9, 1, Yes], [10, 2, Yes], [11, 0, Yes], [12, 1, Yes], [13, 2, Yes], [14, 0, Yes], [15, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)","explanation":"For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes], [7, 2, Yes], [8, 0, Yes], [9, 1, Yes], [10, 2, Yes], [11, 0, Yes], [12, 1, Yes], [13, 2, Yes], [14, 0, Yes], [15, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)."},"skills":["Complete Total Mapping Qualification Precedes Membership Equivalence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0015","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da2b18d5f69e020e49e770f151adda6fafdfa0bb78935794b3cd33a9daca623f","family_id":"qf-mod46-mreduction-total-map-qualification","seed":202611157015,"semantic_template":"qf-mod46-mreduction-total-map-qualification/{\"mapping\":[2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],\"target_carrier\":[0,1,2]}"},"fields":{"title":"Complete Total Mapping Qualification Precedes Membership Equivalence","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], target carrier=[0, 1, 2], aligned mapping=[2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, 0, Yes], [8, 1, Yes], [9, 2, Yes], [10, 0, Yes], [11, 1, Yes], [12, 2, Yes], [13, 0, Yes], [14, 1, Yes], [15, 2, Yes], [16, 0, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)","explanation":"For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, 0, Yes], [8, 1, Yes], [9, 2, Yes], [10, 0, Yes], [11, 1, Yes], [12, 2, Yes], [13, 0, Yes], [14, 1, Yes], [15, 2, Yes], [16, 0, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)."},"skills":["Complete Total Mapping Qualification Precedes Membership Equivalence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0016","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c1e7242c9a40cdefa46d5ec51784e413b11ecc1218c7cc3f3105395944d671f","family_id":"qf-mod46-mreduction-total-map-qualification","seed":202611157016,"semantic_template":"qf-mod46-mreduction-total-map-qualification/{\"mapping\":[0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,null],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],\"target_carrier\":[0,1,2]}"},"fields":{"title":"Complete Total Mapping Qualification Precedes Membership Equivalence","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], target carrier=[0, 1, 2], aligned mapping=[0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, null]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes], [9, 0, Yes], [10, 1, Yes], [11, 2, Yes], [12, 0, Yes], [13, 1, Yes], [14, 2, Yes], [15, 0, Yes], [16, 1, Yes], [17, null, No]], No, [17, null, No], many_one_reduction_requires_a_complete_total_map_before_membership_audit)","explanation":"For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes], [9, 0, Yes], [10, 1, Yes], [11, 2, Yes], [12, 0, Yes], [13, 1, Yes], [14, 2, Yes], [15, 0, Yes], [16, 1, Yes], [17, null, No]], No, [17, null, No], many_one_reduction_requires_a_complete_total_map_before_membership_audit)."},"skills":["Complete Total Mapping Qualification Precedes Membership Equivalence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0017","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b75d7ee56a70ea93e1f79da48629c4571e7600ef2258aaeadacad544893ea6ec","family_id":"qf-mod46-mreduction-total-map-qualification","seed":202611157017,"semantic_template":"qf-mod46-mreduction-total-map-qualification/{\"mapping\":[1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],\"target_carrier\":[0,1,2]}"},"fields":{"title":"Complete Total Mapping Qualification Precedes Membership Equivalence","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], target carrier=[0, 1, 2], aligned mapping=[1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes], [7, 2, Yes], [8, 0, Yes], [9, 1, Yes], [10, 2, Yes], [11, 0, Yes], [12, 1, Yes], [13, 2, Yes], [14, 0, Yes], [15, 1, Yes], [16, 2, Yes], [17, 0, Yes], [18, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)","explanation":"For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes], [7, 2, Yes], [8, 0, Yes], [9, 1, Yes], [10, 2, Yes], [11, 0, Yes], [12, 1, Yes], [13, 2, Yes], [14, 0, Yes], [15, 1, Yes], [16, 2, Yes], [17, 0, Yes], [18, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)."},"skills":["Complete Total Mapping Qualification Precedes Membership Equivalence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0018","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f0f1663196983c902fede9820766a26e990a7f2fcbac2eadb4a80338c3e9f96","family_id":"qf-mod46-mreduction-total-map-qualification","seed":202611157018,"semantic_template":"qf-mod46-mreduction-total-map-qualification/{\"mapping\":[2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],\"target_carrier\":[0,1,2]}"},"fields":{"title":"Complete Total Mapping Qualification Precedes Membership Equivalence","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], target carrier=[0, 1, 2], aligned mapping=[2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, 0, Yes], [8, 1, Yes], [9, 2, Yes], [10, 0, Yes], [11, 1, Yes], [12, 2, Yes], [13, 0, Yes], [14, 1, Yes], [15, 2, Yes], [16, 0, Yes], [17, 1, Yes], [18, 2, Yes], [19, 0, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)","explanation":"For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, 0, Yes], [8, 1, Yes], [9, 2, Yes], [10, 0, Yes], [11, 1, Yes], [12, 2, Yes], [13, 0, Yes], [14, 1, Yes], [15, 2, Yes], [16, 0, Yes], [17, 1, Yes], [18, 2, Yes], [19, 0, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)."},"skills":["Complete Total Mapping Qualification Precedes Membership Equivalence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0019","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be0ad56906229a9cb0e29a3b625ae63ebed5ac9e2397f12bdb49be57c1463591","family_id":"qf-mod46-mreduction-total-map-qualification","seed":202611157019,"semantic_template":"qf-mod46-mreduction-total-map-qualification/{\"mapping\":[0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],\"target_carrier\":[0,1,2]}"},"fields":{"title":"Complete Total Mapping Qualification Precedes Membership Equivalence","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], target carrier=[0, 1, 2], aligned mapping=[0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes], [9, 0, Yes], [10, 1, Yes], [11, 2, Yes], [12, 0, Yes], [13, 1, Yes], [14, 2, Yes], [15, 0, Yes], [16, 1, Yes], [17, 2, Yes], [18, 0, Yes], [19, 1, Yes], [20, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)","explanation":"For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes], [9, 0, Yes], [10, 1, Yes], [11, 2, Yes], [12, 0, Yes], [13, 1, Yes], [14, 2, Yes], [15, 0, Yes], [16, 1, Yes], [17, 2, Yes], [18, 0, Yes], [19, 1, Yes], [20, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)."},"skills":["Complete Total Mapping Qualification Precedes Membership Equivalence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0020","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0aebfd660718b433e6f29691196f7dc4415118507c0bb4796355b15b5c47e74b","family_id":"qf-mod46-mreduction-total-map-qualification","seed":202611157020,"semantic_template":"qf-mod46-mreduction-total-map-qualification/{\"mapping\":[1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],\"target_carrier\":[0,1,2]}"},"fields":{"title":"Complete Total Mapping Qualification Precedes Membership Equivalence","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], target carrier=[0, 1, 2], aligned mapping=[1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes], [7, 2, Yes], [8, 0, Yes], [9, 1, Yes], [10, 2, Yes], [11, 0, Yes], [12, 1, Yes], [13, 2, Yes], [14, 0, Yes], [15, 1, Yes], [16, 2, Yes], [17, 0, Yes], [18, 1, Yes], [19, 2, Yes], [20, 0, Yes], [21, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)","explanation":"For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes], [7, 2, Yes], [8, 0, Yes], [9, 1, Yes], [10, 2, Yes], [11, 0, Yes], [12, 1, Yes], [13, 2, Yes], [14, 0, Yes], [15, 1, Yes], [16, 2, Yes], [17, 0, Yes], [18, 1, Yes], [19, 2, Yes], [20, 0, Yes], [21, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)."},"skills":["Complete Total Mapping Qualification Precedes Membership Equivalence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0021","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0f409cacb6267b608a35d0c6bf05572c20dac823fe5ecb8e704d82d324ac650","family_id":"qf-mod46-mreduction-total-map-qualification","seed":202611157021,"semantic_template":"qf-mod46-mreduction-total-map-qualification/{\"mapping\":[2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,null],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],\"target_carrier\":[0,1,2]}"},"fields":{"title":"Complete Total Mapping Qualification Precedes Membership Equivalence","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], target carrier=[0, 1, 2], aligned mapping=[2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, null]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, 0, Yes], [8, 1, Yes], [9, 2, Yes], [10, 0, Yes], [11, 1, Yes], [12, 2, Yes], [13, 0, Yes], [14, 1, Yes], [15, 2, Yes], [16, 0, Yes], [17, 1, Yes], [18, 2, Yes], [19, 0, Yes], [20, 1, Yes], [21, 2, Yes], [22, null, No]], No, [22, null, No], many_one_reduction_requires_a_complete_total_map_before_membership_audit)","explanation":"For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, 0, Yes], [8, 1, Yes], [9, 2, Yes], [10, 0, Yes], [11, 1, Yes], [12, 2, Yes], [13, 0, Yes], [14, 1, Yes], [15, 2, Yes], [16, 0, Yes], [17, 1, Yes], [18, 2, Yes], [19, 0, Yes], [20, 1, Yes], [21, 2, Yes], [22, null, No]], No, [22, null, No], many_one_reduction_requires_a_complete_total_map_before_membership_audit)."},"skills":["Complete Total Mapping Qualification Precedes Membership Equivalence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0022","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1ee4d956ac96f6d92f97da9f4a63b43d02d60eaee8edbb0219094f0de1bc275","family_id":"qf-mod46-mreduction-total-map-qualification","seed":202611157022,"semantic_template":"qf-mod46-mreduction-total-map-qualification/{\"mapping\":[0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],\"target_carrier\":[0,1,2]}"},"fields":{"title":"Complete Total Mapping Qualification Precedes Membership Equivalence","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], target carrier=[0, 1, 2], aligned mapping=[0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes], [9, 0, Yes], [10, 1, Yes], [11, 2, Yes], [12, 0, Yes], [13, 1, Yes], [14, 2, Yes], [15, 0, Yes], [16, 1, Yes], [17, 2, Yes], [18, 0, Yes], [19, 1, Yes], [20, 2, Yes], [21, 0, Yes], [22, 1, Yes], [23, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)","explanation":"For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes], [9, 0, Yes], [10, 1, Yes], [11, 2, Yes], [12, 0, Yes], [13, 1, Yes], [14, 2, Yes], [15, 0, Yes], [16, 1, Yes], [17, 2, Yes], [18, 0, Yes], [19, 1, Yes], [20, 2, Yes], [21, 0, Yes], [22, 1, Yes], [23, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)."},"skills":["Complete Total Mapping Qualification Precedes Membership Equivalence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0023","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57d941e48542f15743e9dc05f56cd07ef6d8cec76452740b9deb024fb65b4ed8","family_id":"qf-mod46-mreduction-total-map-qualification","seed":202611157023,"semantic_template":"qf-mod46-mreduction-total-map-qualification/{\"mapping\":[1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],\"target_carrier\":[0,1,2]}"},"fields":{"title":"Complete Total Mapping Qualification Precedes Membership Equivalence","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], target carrier=[0, 1, 2], aligned mapping=[1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes], [7, 2, Yes], [8, 0, Yes], [9, 1, Yes], [10, 2, Yes], [11, 0, Yes], [12, 1, Yes], [13, 2, Yes], [14, 0, Yes], [15, 1, Yes], [16, 2, Yes], [17, 0, Yes], [18, 1, Yes], [19, 2, Yes], [20, 0, Yes], [21, 1, Yes], [22, 2, Yes], [23, 0, Yes], [24, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)","explanation":"For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes], [7, 2, Yes], [8, 0, Yes], [9, 1, Yes], [10, 2, Yes], [11, 0, Yes], [12, 1, Yes], [13, 2, Yes], [14, 0, Yes], [15, 1, Yes], [16, 2, Yes], [17, 0, Yes], [18, 1, Yes], [19, 2, Yes], [20, 0, Yes], [21, 1, Yes], [22, 2, Yes], [23, 0, Yes], [24, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)."},"skills":["Complete Total Mapping Qualification Precedes Membership Equivalence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0024","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fca24afad729960ea67b8167e3155d0f3671fe4fabfa9ab0bdb2b28fa9a8fd57","family_id":"qf-mod46-mreduction-total-map-qualification","seed":202611157024,"semantic_template":"qf-mod46-mreduction-total-map-qualification/{\"mapping\":[2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],\"target_carrier\":[0,1,2]}"},"fields":{"title":"Complete Total Mapping Qualification Precedes Membership Equivalence","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], target carrier=[0, 1, 2], aligned mapping=[2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, 0, Yes], [8, 1, Yes], [9, 2, Yes], [10, 0, Yes], [11, 1, Yes], [12, 2, Yes], [13, 0, Yes], [14, 1, Yes], [15, 2, Yes], [16, 0, Yes], [17, 1, Yes], [18, 2, Yes], [19, 0, Yes], [20, 1, Yes], [21, 2, Yes], [22, 0, Yes], [23, 1, Yes], [24, 2, Yes], [25, 0, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)","explanation":"For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, 0, Yes], [8, 1, Yes], [9, 2, Yes], [10, 0, Yes], [11, 1, Yes], [12, 2, Yes], [13, 0, Yes], [14, 1, Yes], [15, 2, Yes], [16, 0, Yes], [17, 1, Yes], [18, 2, Yes], [19, 0, Yes], [20, 1, Yes], [21, 2, Yes], [22, 0, Yes], [23, 1, Yes], [24, 2, Yes], [25, 0, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)."},"skills":["Complete Total Mapping Qualification Precedes Membership Equivalence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0025","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dedadeb3c70a7b8263f3147a6d3cb788ea363d2c7ee4b8da0f0035b7762f1d10","family_id":"qf-mod46-mreduction-total-map-qualification","seed":202611157025,"semantic_template":"qf-mod46-mreduction-total-map-qualification/{\"mapping\":[0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],\"target_carrier\":[0,1,2]}"},"fields":{"title":"Complete Total Mapping Qualification Precedes Membership Equivalence","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], target carrier=[0, 1, 2], aligned mapping=[0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).","answer":{"kind":"expression","value":"([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes], [9, 0, Yes], [10, 1, Yes], [11, 2, Yes], [12, 0, Yes], [13, 1, Yes], [14, 2, Yes], [15, 0, Yes], [16, 1, Yes], [17, 2, Yes], [18, 0, Yes], [19, 1, Yes], [20, 2, Yes], [21, 0, Yes], [22, 1, Yes], [23, 2, Yes], [24, 0, Yes], [25, 1, Yes], [26, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)","explanation":"For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes], [9, 0, Yes], [10, 1, Yes], [11, 2, Yes], [12, 0, Yes], [13, 1, Yes], [14, 2, Yes], [15, 0, Yes], [16, 1, Yes], [17, 2, Yes], [18, 0, Yes], [19, 1, Yes], [20, 2, Yes], [21, 0, Yes], [22, 1, Yes], [23, 2, Yes], [24, 0, Yes], [25, 1, Yes], [26, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)."},"skills":["Complete Total Mapping Qualification Precedes Membership Equivalence"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0026","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff1f0002c4fb6888e2c895adecc4b84abe12216d29f202689bdfa3935a1ad2ec","family_id":"qf-mod46-mreduction-iff-audit","seed":202611157026,"semantic_template":"qf-mod46-mreduction-iff-audit/{\"mapping\":[0,1,0],\"source_carrier\":[0,1,2],\"source_yes\":[1],\"target_carrier\":[0,1],\"target_yes\":[1]}"},"fields":{"title":"A Many-One Reduction Preserves Both Yes and No Membership","stem":"Use source carrier=[0, 1, 2], target carrier=[0, 1], aligned mapping=[0, 1, 0], source yes-set=[1], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).","answer":{"kind":"expression","value":"([1], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)","explanation":"Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)."},"skills":["A Many-One Reduction Preserves Both Yes and No Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0027","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0865512b70b2709444f514b7ac3d38a0672e8506724077382ed1faf3a43dae9d","family_id":"qf-mod46-mreduction-iff-audit","seed":202611157027,"semantic_template":"qf-mod46-mreduction-iff-audit/{\"mapping\":[0,1,0,1],\"source_carrier\":[0,1,2,3],\"source_yes\":[1,3],\"target_carrier\":[0,1],\"target_yes\":[1]}"},"fields":{"title":"A Many-One Reduction Preserves Both Yes and No Membership","stem":"Use source carrier=[0, 1, 2, 3], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1], source yes-set=[1, 3], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).","answer":{"kind":"expression","value":"([1, 3], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)","explanation":"Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)."},"skills":["A Many-One Reduction Preserves Both Yes and No Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0028","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5de9af0a80d4639369f7526c0ebd4dd3494efa6899d749a161c8d56a90e5f3e6","family_id":"qf-mod46-mreduction-iff-audit","seed":202611157028,"semantic_template":"qf-mod46-mreduction-iff-audit/{\"mapping\":[0,1,0,1,0],\"source_carrier\":[0,1,2,3,4],\"source_yes\":[1,3],\"target_carrier\":[0,1],\"target_yes\":[1]}"},"fields":{"title":"A Many-One Reduction Preserves Both Yes and No Membership","stem":"Use source carrier=[0, 1, 2, 3, 4], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0], source yes-set=[1, 3], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).","answer":{"kind":"expression","value":"([1, 3], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)","explanation":"Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)."},"skills":["A Many-One Reduction Preserves Both Yes and No Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0029","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c398dfea79d5be9a2662fb57f64b5321505293be32bb633d4a84383a2ea6dd9","family_id":"qf-mod46-mreduction-iff-audit","seed":202611157029,"semantic_template":"qf-mod46-mreduction-iff-audit/{\"mapping\":[0,1,0,1,0,1],\"source_carrier\":[0,1,2,3,4,5],\"source_yes\":[1,3,5],\"target_carrier\":[0,1],\"target_yes\":[1]}"},"fields":{"title":"A Many-One Reduction Preserves Both Yes and No Membership","stem":"Use source carrier=[0, 1, 2, 3, 4, 5], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).","answer":{"kind":"expression","value":"([1, 3, 5], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)","explanation":"Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)."},"skills":["A Many-One Reduction Preserves Both Yes and No Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0030","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a43c35ec1141b6da1f447dd34f73087b012ac2dcea0c5a81beb8f7faa6e00be","family_id":"qf-mod46-mreduction-iff-audit","seed":202611157030,"semantic_template":"qf-mod46-mreduction-iff-audit/{\"mapping\":[0,1,0,1,0,1,0],\"source_carrier\":[0,1,2,3,4,5,6],\"source_yes\":[1,3,5],\"target_carrier\":[0,1],\"target_yes\":[1]}"},"fields":{"title":"A Many-One Reduction Preserves Both Yes and No Membership","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).","answer":{"kind":"expression","value":"([1, 3, 5], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)","explanation":"Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)."},"skills":["A Many-One Reduction Preserves Both Yes and No Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0031","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b1566996febc069ca781308864157d360f1d8679a686ff99c9a2bc31ad3fdfe","family_id":"qf-mod46-mreduction-iff-audit","seed":202611157031,"semantic_template":"qf-mod46-mreduction-iff-audit/{\"mapping\":[0,1,0,1,0,1,0,1],\"source_carrier\":[0,1,2,3,4,5,6,7],\"source_yes\":[1,3,5,7],\"target_carrier\":[0,1],\"target_yes\":[1]}"},"fields":{"title":"A Many-One Reduction Preserves Both Yes and No Membership","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).","answer":{"kind":"expression","value":"([1, 3, 5, 7], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)","explanation":"Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)."},"skills":["A Many-One Reduction Preserves Both Yes and No Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0032","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d100d3b27fca2a2fe4282a5b3f342fbb62e24c30e83ce0a51691b8ec0e55ea9","family_id":"qf-mod46-mreduction-iff-audit","seed":202611157032,"semantic_template":"qf-mod46-mreduction-iff-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0],\"source_carrier\":[0,1,2,3,4,5,6,7,8],\"source_yes\":[1,3,5,7],\"target_carrier\":[0,1],\"target_yes\":[1]}"},"fields":{"title":"A Many-One Reduction Preserves Both Yes and No Membership","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).","answer":{"kind":"expression","value":"([1, 3, 5, 7], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)","explanation":"Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)."},"skills":["A Many-One Reduction Preserves Both Yes and No Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0033","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b7c80fac0374208ecca9f59ea4d535de91de698bd1d5bf17e7df1a066dcdf55","family_id":"qf-mod46-mreduction-iff-audit","seed":202611157033,"semantic_template":"qf-mod46-mreduction-iff-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9],\"source_yes\":[1,3,5,7,9],\"target_carrier\":[0,1],\"target_yes\":[1]}"},"fields":{"title":"A Many-One Reduction Preserves Both Yes and No Membership","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).","answer":{"kind":"expression","value":"([1, 3, 5, 7, 9], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)","explanation":"Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)."},"skills":["A Many-One Reduction Preserves Both Yes and No Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0034","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d32390bc46ba05ff32a33a5ad7bcb9b7487937ef8637837e97cacd7b1d2ecec8","family_id":"qf-mod46-mreduction-iff-audit","seed":202611157034,"semantic_template":"qf-mod46-mreduction-iff-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10],\"source_yes\":[1,3,5,7,9],\"target_carrier\":[0,1],\"target_yes\":[1]}"},"fields":{"title":"A Many-One Reduction Preserves Both Yes and No Membership","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).","answer":{"kind":"expression","value":"([1, 3, 5, 7, 9], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)","explanation":"Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)."},"skills":["A Many-One Reduction Preserves Both Yes and No Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0035","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fabb92e77866742ad95083881374d22afb696267139c1a29af4b12eac63fdf9e","family_id":"qf-mod46-mreduction-iff-audit","seed":202611157035,"semantic_template":"qf-mod46-mreduction-iff-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11],\"source_yes\":[1,11,3,5,7,9],\"target_carrier\":[0,1],\"target_yes\":[1]}"},"fields":{"title":"A Many-One Reduction Preserves Both Yes and No Membership","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9, 11], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).","answer":{"kind":"expression","value":"([1, 3, 5, 7, 9, 11], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)","explanation":"Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)."},"skills":["A Many-One Reduction Preserves Both Yes and No Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0036","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1f40b33214b814ee9f98952f36a2ca30220f719d04e0f59fea078ebe8b37dc2","family_id":"qf-mod46-mreduction-iff-audit","seed":202611157036,"semantic_template":"qf-mod46-mreduction-iff-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12],\"source_yes\":[1,11,3,5,7,9],\"target_carrier\":[0,1],\"target_yes\":[1]}"},"fields":{"title":"A Many-One Reduction Preserves Both Yes and No Membership","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9, 11], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).","answer":{"kind":"expression","value":"([1, 3, 5, 7, 9, 11], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)","explanation":"Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)."},"skills":["A Many-One Reduction Preserves Both Yes and No Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0037","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a2c421e5e27db9d712cb25a52fe96e0e7419946fce67db1f2dc07636c97edee","family_id":"qf-mod46-mreduction-iff-audit","seed":202611157037,"semantic_template":"qf-mod46-mreduction-iff-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13],\"source_yes\":[1,11,13,3,5,7,9],\"target_carrier\":[0,1],\"target_yes\":[1]}"},"fields":{"title":"A Many-One Reduction Preserves Both Yes and No Membership","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9, 11, 13], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).","answer":{"kind":"expression","value":"([1, 3, 5, 7, 9, 11, 13], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)","explanation":"Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)."},"skills":["A Many-One Reduction Preserves Both Yes and No Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0038","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e28b5815188bd3e715be01574d21681dd4ca6eb7ccbddffa01cf82c50465e448","family_id":"qf-mod46-mreduction-iff-audit","seed":202611157038,"semantic_template":"qf-mod46-mreduction-iff-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],\"source_yes\":[1,11,13,3,5,7,9],\"target_carrier\":[0,1],\"target_yes\":[1]}"},"fields":{"title":"A Many-One Reduction Preserves Both Yes and No Membership","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9, 11, 13], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).","answer":{"kind":"expression","value":"([1, 3, 5, 7, 9, 11, 13], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)","explanation":"Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)."},"skills":["A Many-One Reduction Preserves Both Yes and No Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0039","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62abbb7a16ce84696a26a842b2ffb1f65797a35befadaf6b6744b5c295dca579","family_id":"qf-mod46-mreduction-iff-audit","seed":202611157039,"semantic_template":"qf-mod46-mreduction-iff-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],\"source_yes\":[1,11,13,15,3,5,7,9],\"target_carrier\":[0,1],\"target_yes\":[1]}"},"fields":{"title":"A Many-One Reduction Preserves Both Yes and No Membership","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).","answer":{"kind":"expression","value":"([1, 3, 5, 7, 9, 11, 13, 15], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)","explanation":"Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13, 15], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)."},"skills":["A Many-One Reduction Preserves Both Yes and No Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0040","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69dd22a3642b6916fa4a546fc8b05ed5a85855e6564da86f6121ca8976b1de17","family_id":"qf-mod46-mreduction-iff-audit","seed":202611157040,"semantic_template":"qf-mod46-mreduction-iff-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],\"source_yes\":[1,11,13,15,3,5,7,9],\"target_carrier\":[0,1],\"target_yes\":[1]}"},"fields":{"title":"A Many-One Reduction Preserves Both Yes and No Membership","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).","answer":{"kind":"expression","value":"([1, 3, 5, 7, 9, 11, 13, 15], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)","explanation":"Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13, 15], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)."},"skills":["A Many-One Reduction Preserves Both Yes and No Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0041","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0611c82465887a6c14e2e2b207e9a0aa0ce528828e8a90fb0da01fc57dad661a","family_id":"qf-mod46-mreduction-iff-audit","seed":202611157041,"semantic_template":"qf-mod46-mreduction-iff-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],\"source_yes\":[1,11,13,15,17,3,5,7,9],\"target_carrier\":[0,1],\"target_yes\":[1]}"},"fields":{"title":"A Many-One Reduction Preserves Both Yes and No Membership","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).","answer":{"kind":"expression","value":"([1, 3, 5, 7, 9, 11, 13, 15, 17], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)","explanation":"Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13, 15, 17], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)."},"skills":["A Many-One Reduction Preserves Both Yes and No Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0042","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b400775ad8ca26df32cd97cd2650bee2314df19784d4f48dec4b61fdd23d049","family_id":"qf-mod46-mreduction-iff-audit","seed":202611157042,"semantic_template":"qf-mod46-mreduction-iff-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],\"source_yes\":[1,11,13,15,17,3,5,7,9],\"target_carrier\":[0,1],\"target_yes\":[1]}"},"fields":{"title":"A Many-One Reduction Preserves Both Yes and No Membership","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).","answer":{"kind":"expression","value":"([1, 3, 5, 7, 9, 11, 13, 15, 17], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)","explanation":"Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13, 15, 17], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)."},"skills":["A Many-One Reduction Preserves Both Yes and No Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0043","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e62b1a82cca437d11b82734e3ecf05fd9096faaac456ec2e4edb17ed68db7c8","family_id":"qf-mod46-mreduction-iff-audit","seed":202611157043,"semantic_template":"qf-mod46-mreduction-iff-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],\"source_yes\":[1,11,13,15,17,19,3,5,7,9],\"target_carrier\":[0,1],\"target_yes\":[1]}"},"fields":{"title":"A Many-One Reduction Preserves Both Yes and No Membership","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).","answer":{"kind":"expression","value":"([1, 3, 5, 7, 9, 11, 13, 15, 17, 19], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)","explanation":"Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13, 15, 17, 19], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)."},"skills":["A Many-One Reduction Preserves Both Yes and No Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0044","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4626b1a81e800a909f038cde7c18d499a0c5357b9003500c9d9bda9d432503a8","family_id":"qf-mod46-mreduction-iff-audit","seed":202611157044,"semantic_template":"qf-mod46-mreduction-iff-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],\"source_yes\":[1,11,13,15,17,19,3,5,7,9],\"target_carrier\":[0,1],\"target_yes\":[1]}"},"fields":{"title":"A Many-One Reduction Preserves Both Yes and No Membership","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).","answer":{"kind":"expression","value":"([1, 3, 5, 7, 9, 11, 13, 15, 17, 19], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)","explanation":"Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13, 15, 17, 19], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)."},"skills":["A Many-One Reduction Preserves Both Yes and No Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0045","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19e33c62b58a836176f16276f4105450a00b34f697821fe081d4f6d30160e33d","family_id":"qf-mod46-mreduction-iff-audit","seed":202611157045,"semantic_template":"qf-mod46-mreduction-iff-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],\"source_yes\":[1,11,13,15,17,19,21,3,5,7,9],\"target_carrier\":[0,1],\"target_yes\":[1]}"},"fields":{"title":"A Many-One Reduction Preserves Both Yes and No Membership","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).","answer":{"kind":"expression","value":"([1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)","explanation":"Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)."},"skills":["A Many-One Reduction Preserves Both Yes and No Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0046","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02b3e4f007a09fe87a6d6bccaa148beb145e72b2186936aa5f17cddda607c9c4","family_id":"qf-mod46-mreduction-iff-audit","seed":202611157046,"semantic_template":"qf-mod46-mreduction-iff-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],\"source_yes\":[1,11,13,15,17,19,21,3,5,7,9],\"target_carrier\":[0,1],\"target_yes\":[1]}"},"fields":{"title":"A Many-One Reduction Preserves Both Yes and No Membership","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).","answer":{"kind":"expression","value":"([1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)","explanation":"Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)."},"skills":["A Many-One Reduction Preserves Both Yes and No Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0047","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d9e4f8171c561aee942d5f999842e0f652fa53bcc293c5d3b43d7c9ae723beb","family_id":"qf-mod46-mreduction-iff-audit","seed":202611157047,"semantic_template":"qf-mod46-mreduction-iff-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],\"source_yes\":[1,11,13,15,17,19,21,23,3,5,7,9],\"target_carrier\":[0,1],\"target_yes\":[1]}"},"fields":{"title":"A Many-One Reduction Preserves Both Yes and No Membership","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).","answer":{"kind":"expression","value":"([1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)","explanation":"Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)."},"skills":["A Many-One Reduction Preserves Both Yes and No Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0048","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1da59aa358760d8471ca9f356d004e1d940f755e4d47a62ae0bd089641e847c","family_id":"qf-mod46-mreduction-iff-audit","seed":202611157048,"semantic_template":"qf-mod46-mreduction-iff-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],\"source_yes\":[1,11,13,15,17,19,21,23,3,5,7,9],\"target_carrier\":[0,1],\"target_yes\":[1]}"},"fields":{"title":"A Many-One Reduction Preserves Both Yes and No Membership","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).","answer":{"kind":"expression","value":"([1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)","explanation":"Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)."},"skills":["A Many-One Reduction Preserves Both Yes and No Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0049","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dbd267253326cc81192291a3464d49cefb20c7c2c1d24e552c87a1ff07b49d25","family_id":"qf-mod46-mreduction-iff-audit","seed":202611157049,"semantic_template":"qf-mod46-mreduction-iff-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],\"source_yes\":[1,11,13,15,17,19,21,23,25,3,5,7,9],\"target_carrier\":[0,1],\"target_yes\":[1]}"},"fields":{"title":"A Many-One Reduction Preserves Both Yes and No Membership","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).","answer":{"kind":"expression","value":"([1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, 0, No, Yes], [25, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)","explanation":"Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, 0, No, Yes], [25, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)."},"skills":["A Many-One Reduction Preserves Both Yes and No Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0050","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1cd303477873c17e5981cf6ca961f1dd45298bb8a87b4d87483f8a936643b54e","family_id":"qf-mod46-mreduction-iff-audit","seed":202611157050,"semantic_template":"qf-mod46-mreduction-iff-audit/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],\"source_yes\":[1,11,13,15,17,19,21,23,25,3,5,7,9],\"target_carrier\":[0,1],\"target_yes\":[1]}"},"fields":{"title":"A Many-One Reduction Preserves Both Yes and No Membership","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).","answer":{"kind":"expression","value":"([1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, 0, No, Yes], [25, Yes, 1, Yes, Yes], [26, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)","explanation":"Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, 0, No, Yes], [25, Yes, 1, Yes, Yes], [26, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)."},"skills":["A Many-One Reduction Preserves Both Yes and No Membership"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0051","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2d99d1876da30bef3a29b40d00465725ce5fc7d5a58fc051023b100f5b16b15","family_id":"qf-mod46-mreduction-composition-direction","seed":202611157051,"semantic_template":"qf-mod46-mreduction-composition-direction/{\"mapping_ab\":[0,1,0],\"mapping_bc\":[10,11],\"middle_carrier\":[0,1],\"middle_yes\":[1],\"source_carrier\":[0,1,2],\"source_yes\":[1],\"target_carrier\":[10,11],\"target_yes\":[11]}"},"fields":{"title":"Compose A-to-B then B-to-C without Reversing the Reduction","stem":"Use source carrier=[0, 1, 2], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1], [1], [11]], aligned maps A-to-B=[0, 1, 0], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).","answer":{"kind":"expression","value":"([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)","explanation":"Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)."},"skills":["Compose A-to-B then B-to-C without Reversing the Reduction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0052","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3923974c5041dad7f0c71c9db5579eaf14693d1b7d01654f0229361d78b6c044","family_id":"qf-mod46-mreduction-composition-direction","seed":202611157052,"semantic_template":"qf-mod46-mreduction-composition-direction/{\"mapping_ab\":[0,1,0,1],\"mapping_bc\":[10,11],\"middle_carrier\":[0,1],\"middle_yes\":[1],\"source_carrier\":[0,1,2,3],\"source_yes\":[1,3],\"target_carrier\":[10,11],\"target_yes\":[11]}"},"fields":{"title":"Compose A-to-B then B-to-C without Reversing the Reduction","stem":"Use source carrier=[0, 1, 2, 3], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).","answer":{"kind":"expression","value":"([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)","explanation":"Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)."},"skills":["Compose A-to-B then B-to-C without Reversing the Reduction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0053","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea1113adebf419d6f69eb1a94f2cd0827545e9a7c728bf31ea17a7ae269fd5a9","family_id":"qf-mod46-mreduction-composition-direction","seed":202611157053,"semantic_template":"qf-mod46-mreduction-composition-direction/{\"mapping_ab\":[0,1,0,1,0],\"mapping_bc\":[10,11],\"middle_carrier\":[0,1],\"middle_yes\":[1],\"source_carrier\":[0,1,2,3,4],\"source_yes\":[1,3],\"target_carrier\":[10,11],\"target_yes\":[11]}"},"fields":{"title":"Compose A-to-B then B-to-C without Reversing the Reduction","stem":"Use source carrier=[0, 1, 2, 3, 4], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).","answer":{"kind":"expression","value":"([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)","explanation":"Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)."},"skills":["Compose A-to-B then B-to-C without Reversing the Reduction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0054","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d6c3e9154e451c19e9793a0784903006628b4c2409073c5a3a335dc4e65b5c7","family_id":"qf-mod46-mreduction-composition-direction","seed":202611157054,"semantic_template":"qf-mod46-mreduction-composition-direction/{\"mapping_ab\":[0,1,0,1,0,1],\"mapping_bc\":[10,11],\"middle_carrier\":[0,1],\"middle_yes\":[1],\"source_carrier\":[0,1,2,3,4,5],\"source_yes\":[1,3,5],\"target_carrier\":[10,11],\"target_yes\":[11]}"},"fields":{"title":"Compose A-to-B then B-to-C without Reversing the Reduction","stem":"Use source carrier=[0, 1, 2, 3, 4, 5], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).","answer":{"kind":"expression","value":"([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)","explanation":"Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)."},"skills":["Compose A-to-B then B-to-C without Reversing the Reduction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0055","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea6ca8ce0f7af691b77ea2a688282747c069a442dde47bfbf98fa722ab6f4958","family_id":"qf-mod46-mreduction-composition-direction","seed":202611157055,"semantic_template":"qf-mod46-mreduction-composition-direction/{\"mapping_ab\":[0,1,0,1,0,1,0],\"mapping_bc\":[10,11],\"middle_carrier\":[0,1],\"middle_yes\":[1],\"source_carrier\":[0,1,2,3,4,5,6],\"source_yes\":[1,3,5],\"target_carrier\":[10,11],\"target_yes\":[11]}"},"fields":{"title":"Compose A-to-B then B-to-C without Reversing the Reduction","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).","answer":{"kind":"expression","value":"([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)","explanation":"Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)."},"skills":["Compose A-to-B then B-to-C without Reversing the Reduction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0056","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1893686434c72a7604a9935ced5d1c90dea649e1560282c54a47ced31acefac","family_id":"qf-mod46-mreduction-composition-direction","seed":202611157056,"semantic_template":"qf-mod46-mreduction-composition-direction/{\"mapping_ab\":[0,1,0,1,0,1,0,1],\"mapping_bc\":[10,11],\"middle_carrier\":[0,1],\"middle_yes\":[1],\"source_carrier\":[0,1,2,3,4,5,6,7],\"source_yes\":[1,3,5,7],\"target_carrier\":[10,11],\"target_yes\":[11]}"},"fields":{"title":"Compose A-to-B then B-to-C without Reversing the Reduction","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).","answer":{"kind":"expression","value":"([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)","explanation":"Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)."},"skills":["Compose A-to-B then B-to-C without Reversing the Reduction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0057","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3dfda9449eb9759ebe4d45961d468664d405ca51f3bcfdf0cadc0708242311ca","family_id":"qf-mod46-mreduction-composition-direction","seed":202611157057,"semantic_template":"qf-mod46-mreduction-composition-direction/{\"mapping_ab\":[0,1,0,1,0,1,0,1,0],\"mapping_bc\":[10,11],\"middle_carrier\":[0,1],\"middle_yes\":[1],\"source_carrier\":[0,1,2,3,4,5,6,7,8],\"source_yes\":[1,3,5,7],\"target_carrier\":[10,11],\"target_yes\":[11]}"},"fields":{"title":"Compose A-to-B then B-to-C without Reversing the Reduction","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).","answer":{"kind":"expression","value":"([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)","explanation":"Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)."},"skills":["Compose A-to-B then B-to-C without Reversing the Reduction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0058","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"209b09704c94a5b90b68346630b31e5056ac1d0eab26ee05a319326c073a06ef","family_id":"qf-mod46-mreduction-composition-direction","seed":202611157058,"semantic_template":"qf-mod46-mreduction-composition-direction/{\"mapping_ab\":[0,1,0,1,0,1,0,1,0,1],\"mapping_bc\":[10,11],\"middle_carrier\":[0,1],\"middle_yes\":[1],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9],\"source_yes\":[1,3,5,7,9],\"target_carrier\":[10,11],\"target_yes\":[11]}"},"fields":{"title":"Compose A-to-B then B-to-C without Reversing the Reduction","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).","answer":{"kind":"expression","value":"([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)","explanation":"Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)."},"skills":["Compose A-to-B then B-to-C without Reversing the Reduction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0059","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"82c3640063a666fcf6991cd19fb1e970afb64e750ae53daeebc9444dda10cc51","family_id":"qf-mod46-mreduction-composition-direction","seed":202611157059,"semantic_template":"qf-mod46-mreduction-composition-direction/{\"mapping_ab\":[0,1,0,1,0,1,0,1,0,1,0],\"mapping_bc\":[10,11],\"middle_carrier\":[0,1],\"middle_yes\":[1],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10],\"source_yes\":[1,3,5,7,9],\"target_carrier\":[10,11],\"target_yes\":[11]}"},"fields":{"title":"Compose A-to-B then B-to-C without Reversing the Reduction","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).","answer":{"kind":"expression","value":"([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)","explanation":"Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)."},"skills":["Compose A-to-B then B-to-C without Reversing the Reduction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0060","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65fe58b63e6eacfacb9e8313b8bb776a38eadd834839819addb70bf2bcfdaebc","family_id":"qf-mod46-mreduction-composition-direction","seed":202611157060,"semantic_template":"qf-mod46-mreduction-composition-direction/{\"mapping_ab\":[0,1,0,1,0,1,0,1,0,1,0,1],\"mapping_bc\":[10,11],\"middle_carrier\":[0,1],\"middle_yes\":[1],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11],\"source_yes\":[1,11,3,5,7,9],\"target_carrier\":[10,11],\"target_yes\":[11]}"},"fields":{"title":"Compose A-to-B then B-to-C without Reversing the Reduction","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).","answer":{"kind":"expression","value":"([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)","explanation":"Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)."},"skills":["Compose A-to-B then B-to-C without Reversing the Reduction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0061","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0009c502d8dbc419f5a4321d2731b32a059e1f2907a87d022d1f0b4fb4f0a78","family_id":"qf-mod46-mreduction-composition-direction","seed":202611157061,"semantic_template":"qf-mod46-mreduction-composition-direction/{\"mapping_ab\":[0,1,0,1,0,1,0,1,0,1,0,1,0],\"mapping_bc\":[10,11],\"middle_carrier\":[0,1],\"middle_yes\":[1],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12],\"source_yes\":[1,11,3,5,7,9],\"target_carrier\":[10,11],\"target_yes\":[11]}"},"fields":{"title":"Compose A-to-B then B-to-C without Reversing the Reduction","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).","answer":{"kind":"expression","value":"([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)","explanation":"Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)."},"skills":["Compose A-to-B then B-to-C without Reversing the Reduction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0062","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da4947cd5a9214ff68537967230966ac9bdf3339fdb45d9f6daeb22fcf03dc5a","family_id":"qf-mod46-mreduction-composition-direction","seed":202611157062,"semantic_template":"qf-mod46-mreduction-composition-direction/{\"mapping_ab\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"mapping_bc\":[10,11],\"middle_carrier\":[0,1],\"middle_yes\":[1],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13],\"source_yes\":[1,11,13,3,5,7,9],\"target_carrier\":[10,11],\"target_yes\":[11]}"},"fields":{"title":"Compose A-to-B then B-to-C without Reversing the Reduction","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).","answer":{"kind":"expression","value":"([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)","explanation":"Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)."},"skills":["Compose A-to-B then B-to-C without Reversing the Reduction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0063","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5afe1cea2a3096c227d8ec603dc4b201712012de1ce1d51dfd7f7c08606cf5be","family_id":"qf-mod46-mreduction-composition-direction","seed":202611157063,"semantic_template":"qf-mod46-mreduction-composition-direction/{\"mapping_ab\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0],\"mapping_bc\":[10,11],\"middle_carrier\":[0,1],\"middle_yes\":[1],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],\"source_yes\":[1,11,13,3,5,7,9],\"target_carrier\":[10,11],\"target_yes\":[11]}"},"fields":{"title":"Compose A-to-B then B-to-C without Reversing the Reduction","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).","answer":{"kind":"expression","value":"([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)","explanation":"Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)."},"skills":["Compose A-to-B then B-to-C without Reversing the Reduction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0064","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55a0a9f8616db80f631039792839c9918e7c8251b099a353579b776c47eef680","family_id":"qf-mod46-mreduction-composition-direction","seed":202611157064,"semantic_template":"qf-mod46-mreduction-composition-direction/{\"mapping_ab\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"mapping_bc\":[10,11],\"middle_carrier\":[0,1],\"middle_yes\":[1],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],\"source_yes\":[1,11,13,15,3,5,7,9],\"target_carrier\":[10,11],\"target_yes\":[11]}"},"fields":{"title":"Compose A-to-B then B-to-C without Reversing the Reduction","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13, 15], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).","answer":{"kind":"expression","value":"([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)","explanation":"Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)."},"skills":["Compose A-to-B then B-to-C without Reversing the Reduction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0065","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40991f45e8d1bf2449abf237592e525bc82a0244e82da7f7ec096fa89381cf21","family_id":"qf-mod46-mreduction-composition-direction","seed":202611157065,"semantic_template":"qf-mod46-mreduction-composition-direction/{\"mapping_ab\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0],\"mapping_bc\":[10,11],\"middle_carrier\":[0,1],\"middle_yes\":[1],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],\"source_yes\":[1,11,13,15,3,5,7,9],\"target_carrier\":[10,11],\"target_yes\":[11]}"},"fields":{"title":"Compose A-to-B then B-to-C without Reversing the Reduction","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13, 15], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).","answer":{"kind":"expression","value":"([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)","explanation":"Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)."},"skills":["Compose A-to-B then B-to-C without Reversing the Reduction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0066","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d61fdac76b123a3361c9720308bc70142409acd3dde6082a751d477c2eed4b5","family_id":"qf-mod46-mreduction-composition-direction","seed":202611157066,"semantic_template":"qf-mod46-mreduction-composition-direction/{\"mapping_ab\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"mapping_bc\":[10,11],\"middle_carrier\":[0,1],\"middle_yes\":[1],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],\"source_yes\":[1,11,13,15,17,3,5,7,9],\"target_carrier\":[10,11],\"target_yes\":[11]}"},"fields":{"title":"Compose A-to-B then B-to-C without Reversing the Reduction","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13, 15, 17], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).","answer":{"kind":"expression","value":"([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)","explanation":"Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)."},"skills":["Compose A-to-B then B-to-C without Reversing the Reduction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0067","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2269f24e095106a55b47eb012b9e9fccd3dc095c5ae1f1fb76506c13821636c6","family_id":"qf-mod46-mreduction-composition-direction","seed":202611157067,"semantic_template":"qf-mod46-mreduction-composition-direction/{\"mapping_ab\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0],\"mapping_bc\":[10,11],\"middle_carrier\":[0,1],\"middle_yes\":[1],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],\"source_yes\":[1,11,13,15,17,3,5,7,9],\"target_carrier\":[10,11],\"target_yes\":[11]}"},"fields":{"title":"Compose A-to-B then B-to-C without Reversing the Reduction","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13, 15, 17], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).","answer":{"kind":"expression","value":"([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)","explanation":"Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)."},"skills":["Compose A-to-B then B-to-C without Reversing the Reduction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0068","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7f82e3d40b6d531a2e66cb0c54090c1178aff766c90cd6deb06858c9166b123","family_id":"qf-mod46-mreduction-composition-direction","seed":202611157068,"semantic_template":"qf-mod46-mreduction-composition-direction/{\"mapping_ab\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"mapping_bc\":[10,11],\"middle_carrier\":[0,1],\"middle_yes\":[1],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],\"source_yes\":[1,11,13,15,17,19,3,5,7,9],\"target_carrier\":[10,11],\"target_yes\":[11]}"},"fields":{"title":"Compose A-to-B then B-to-C without Reversing the Reduction","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13, 15, 17, 19], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).","answer":{"kind":"expression","value":"([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)","explanation":"Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)."},"skills":["Compose A-to-B then B-to-C without Reversing the Reduction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0069","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eaf7771946a63e712af98722ab5554dc243aa0fcd60dc0fcc2f60ccf89c68cd1","family_id":"qf-mod46-mreduction-composition-direction","seed":202611157069,"semantic_template":"qf-mod46-mreduction-composition-direction/{\"mapping_ab\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0],\"mapping_bc\":[10,11],\"middle_carrier\":[0,1],\"middle_yes\":[1],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],\"source_yes\":[1,11,13,15,17,19,3,5,7,9],\"target_carrier\":[10,11],\"target_yes\":[11]}"},"fields":{"title":"Compose A-to-B then B-to-C without Reversing the Reduction","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13, 15, 17, 19], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).","answer":{"kind":"expression","value":"([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)","explanation":"Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)."},"skills":["Compose A-to-B then B-to-C without Reversing the Reduction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0070","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df77cfd6810f17005038979684af396c4b270cca7fd1523420dedd5990a3294e","family_id":"qf-mod46-mreduction-composition-direction","seed":202611157070,"semantic_template":"qf-mod46-mreduction-composition-direction/{\"mapping_ab\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"mapping_bc\":[10,11],\"middle_carrier\":[0,1],\"middle_yes\":[1],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],\"source_yes\":[1,11,13,15,17,19,21,3,5,7,9],\"target_carrier\":[10,11],\"target_yes\":[11]}"},"fields":{"title":"Compose A-to-B then B-to-C without Reversing the Reduction","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).","answer":{"kind":"expression","value":"([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes], [21, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)","explanation":"Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes], [21, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)."},"skills":["Compose A-to-B then B-to-C without Reversing the Reduction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0071","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a49eed1cc124d808db68e1ca1285961c1828c1ad55ebb6bb273fadff854a465d","family_id":"qf-mod46-mreduction-composition-direction","seed":202611157071,"semantic_template":"qf-mod46-mreduction-composition-direction/{\"mapping_ab\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0],\"mapping_bc\":[10,11],\"middle_carrier\":[0,1],\"middle_yes\":[1],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],\"source_yes\":[1,11,13,15,17,19,21,3,5,7,9],\"target_carrier\":[10,11],\"target_yes\":[11]}"},"fields":{"title":"Compose A-to-B then B-to-C without Reversing the Reduction","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).","answer":{"kind":"expression","value":"([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes], [21, Yes, 11, Yes, Yes], [22, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)","explanation":"Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes], [21, Yes, 11, Yes, Yes], [22, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)."},"skills":["Compose A-to-B then B-to-C without Reversing the Reduction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0072","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cfde33a7f9ddb466b947ec92e61a088ca29afd0634b18b6f3d11f8f4345b98ba","family_id":"qf-mod46-mreduction-composition-direction","seed":202611157072,"semantic_template":"qf-mod46-mreduction-composition-direction/{\"mapping_ab\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"mapping_bc\":[10,11],\"middle_carrier\":[0,1],\"middle_yes\":[1],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],\"source_yes\":[1,11,13,15,17,19,21,23,3,5,7,9],\"target_carrier\":[10,11],\"target_yes\":[11]}"},"fields":{"title":"Compose A-to-B then B-to-C without Reversing the Reduction","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).","answer":{"kind":"expression","value":"([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes], [21, Yes, 11, Yes, Yes], [22, No, 10, No, Yes], [23, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)","explanation":"Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes], [21, Yes, 11, Yes, Yes], [22, No, 10, No, Yes], [23, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)."},"skills":["Compose A-to-B then B-to-C without Reversing the Reduction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0073","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1bdbe5d8fc8c7aab5cf0574b09b80476b6654929e287d60badfefde2ffe61ed","family_id":"qf-mod46-mreduction-composition-direction","seed":202611157073,"semantic_template":"qf-mod46-mreduction-composition-direction/{\"mapping_ab\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0],\"mapping_bc\":[10,11],\"middle_carrier\":[0,1],\"middle_yes\":[1],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],\"source_yes\":[1,11,13,15,17,19,21,23,3,5,7,9],\"target_carrier\":[10,11],\"target_yes\":[11]}"},"fields":{"title":"Compose A-to-B then B-to-C without Reversing the Reduction","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).","answer":{"kind":"expression","value":"([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes], [21, Yes, 11, Yes, Yes], [22, No, 10, No, Yes], [23, Yes, 11, Yes, Yes], [24, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)","explanation":"Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes], [21, Yes, 11, Yes, Yes], [22, No, 10, No, Yes], [23, Yes, 11, Yes, Yes], [24, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)."},"skills":["Compose A-to-B then B-to-C without Reversing the Reduction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0074","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"25d42cdddde01cdb612437c4ca994429ceae8182cc8cfd87be04baca845a0f04","family_id":"qf-mod46-mreduction-composition-direction","seed":202611157074,"semantic_template":"qf-mod46-mreduction-composition-direction/{\"mapping_ab\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"mapping_bc\":[10,11],\"middle_carrier\":[0,1],\"middle_yes\":[1],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],\"source_yes\":[1,11,13,15,17,19,21,23,25,3,5,7,9],\"target_carrier\":[10,11],\"target_yes\":[11]}"},"fields":{"title":"Compose A-to-B then B-to-C without Reversing the Reduction","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).","answer":{"kind":"expression","value":"([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, 0, No, Yes], [25, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes], [21, Yes, 11, Yes, Yes], [22, No, 10, No, Yes], [23, Yes, 11, Yes, Yes], [24, No, 10, No, Yes], [25, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)","explanation":"Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, 0, No, Yes], [25, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes], [21, Yes, 11, Yes, Yes], [22, No, 10, No, Yes], [23, Yes, 11, Yes, Yes], [24, No, 10, No, Yes], [25, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)."},"skills":["Compose A-to-B then B-to-C without Reversing the Reduction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0075","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1df0d8beaf858a882c3bdd11c102760233705b2d265eba8ec8ad538513f2c0e","family_id":"qf-mod46-mreduction-composition-direction","seed":202611157075,"semantic_template":"qf-mod46-mreduction-composition-direction/{\"mapping_ab\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0],\"mapping_bc\":[10,11],\"middle_carrier\":[0,1],\"middle_yes\":[1],\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],\"source_yes\":[1,11,13,15,17,19,21,23,25,3,5,7,9],\"target_carrier\":[10,11],\"target_yes\":[11]}"},"fields":{"title":"Compose A-to-B then B-to-C without Reversing the Reduction","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).","answer":{"kind":"expression","value":"([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, 0, No, Yes], [25, Yes, 1, Yes, Yes], [26, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes], [21, Yes, 11, Yes, Yes], [22, No, 10, No, Yes], [23, Yes, 11, Yes, Yes], [24, No, 10, No, Yes], [25, Yes, 11, Yes, Yes], [26, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)","explanation":"Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, 0, No, Yes], [25, Yes, 1, Yes, Yes], [26, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes], [21, Yes, 11, Yes, Yes], [22, No, 10, No, Yes], [23, Yes, 11, Yes, Yes], [24, No, 10, No, Yes], [25, Yes, 11, Yes, Yes], [26, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)."},"skills":["Compose A-to-B then B-to-C without Reversing the Reduction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0076","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7eb14c4356553453c8d065b942f66185e6ef85ae3391dd034d6add2a0bf5da51","family_id":"qf-mod46-mreduction-decider-transfer-boundary","seed":202611157076,"semantic_template":"qf-mod46-mreduction-decider-transfer-boundary/{\"mapping\":[0,1,0],\"mode\":\"valid_transfer\",\"source_carrier\":[0,1,2],\"source_yes\":[1],\"target_carrier\":[0,1],\"target_decider\":[false,true],\"target_yes\":[1]}"},"fields":{"title":"Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider","stem":"Use source carrier=[0, 1, 2], target carrier=[0, 1], aligned mapping=[0, 1, 0], source yes-set=[1], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=valid_transfer. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).","answer":{"kind":"expression","value":"(valid_transfer, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No], Yes, Yes, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)","explanation":"Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (valid_transfer, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No], Yes, Yes, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)."},"skills":["Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0077","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b50de37e300751ffc54556bc35c7c67b09c5c3e5e01b31a5f94e909d3fb05d5","family_id":"qf-mod46-mreduction-decider-transfer-boundary","seed":202611157077,"semantic_template":"qf-mod46-mreduction-decider-transfer-boundary/{\"mapping\":[0,1,0,1],\"mode\":\"reverse_misuse\",\"source_carrier\":[0,1,2,3],\"source_yes\":[1,3],\"target_carrier\":[0,1],\"target_decider\":[false,true],\"target_yes\":[1]}"},"fields":{"title":"Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider","stem":"Use source carrier=[0, 1, 2, 3], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1], source yes-set=[1, 3], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=reverse_misuse. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).","answer":{"kind":"expression","value":"(reverse_misuse, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes], Yes, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)","explanation":"Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (reverse_misuse, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes], Yes, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)."},"skills":["Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0078","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5fbe256af33617644c9df03d106f63394a2f8645702e165e3594c2dd38436cd5","family_id":"qf-mod46-mreduction-decider-transfer-boundary","seed":202611157078,"semantic_template":"qf-mod46-mreduction-decider-transfer-boundary/{\"mapping\":[0,1,0,1,null],\"mode\":\"partial_map\",\"source_carrier\":[0,1,2,3,4],\"source_yes\":[1,3],\"target_carrier\":[0,1],\"target_decider\":[false,true],\"target_yes\":[1]}"},"fields":{"title":"Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider","stem":"Use source carrier=[0, 1, 2, 3, 4], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, null], source yes-set=[1, 3], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=partial_map. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).","answer":{"kind":"expression","value":"(partial_map, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, null, null, No]], No, No, [No, Yes], Yes, null, No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)","explanation":"Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (partial_map, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, null, null, No]], No, No, [No, Yes], Yes, null, No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)."},"skills":["Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0079","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c917e42f90f5a45e1a3438c6f66e282be656f80f5cdb2be1cc83f200accb2f22","family_id":"qf-mod46-mreduction-decider-transfer-boundary","seed":202611157079,"semantic_template":"qf-mod46-mreduction-decider-transfer-boundary/{\"mapping\":[0,0,0,0,0,0],\"mode\":\"one_way_yes_only\",\"source_carrier\":[0,1,2,3,4,5],\"source_yes\":[0],\"target_carrier\":[0,1],\"target_decider\":[true,false],\"target_yes\":[0]}"},"fields":{"title":"Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider","stem":"Use source carrier=[0, 1, 2, 3, 4, 5], target carrier=[0, 1], aligned mapping=[0, 0, 0, 0, 0, 0], source yes-set=[0], target yes-set=[0]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[true, false], aligned with the supplied target carrier. Mode=one_way_yes_only. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).","answer":{"kind":"expression","value":"(one_way_yes_only, [[0, Yes, 0, Yes, Yes], [1, No, 0, Yes, No], [2, No, 0, Yes, No], [3, No, 0, Yes, No], [4, No, 0, Yes, No], [5, No, 0, Yes, No]], Yes, No, [Yes, No], Yes, [Yes, Yes, Yes, Yes, Yes, Yes], No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)","explanation":"Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (one_way_yes_only, [[0, Yes, 0, Yes, Yes], [1, No, 0, Yes, No], [2, No, 0, Yes, No], [3, No, 0, Yes, No], [4, No, 0, Yes, No], [5, No, 0, Yes, No]], Yes, No, [Yes, No], Yes, [Yes, Yes, Yes, Yes, Yes, Yes], No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)."},"skills":["Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0080","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ee750d324c8c3dea4f2a73c0ac89fa8c455f1816ea5f34c5788c7ddce775a9c","family_id":"qf-mod46-mreduction-decider-transfer-boundary","seed":202611157080,"semantic_template":"qf-mod46-mreduction-decider-transfer-boundary/{\"mapping\":[0,1,0,1,0,1,0],\"mode\":\"valid_transfer\",\"source_carrier\":[0,1,2,3,4,5,6],\"source_yes\":[1,3,5],\"target_carrier\":[0,1],\"target_decider\":[false,true],\"target_yes\":[1]}"},"fields":{"title":"Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=valid_transfer. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).","answer":{"kind":"expression","value":"(valid_transfer, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No], Yes, Yes, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)","explanation":"Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (valid_transfer, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No], Yes, Yes, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)."},"skills":["Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0081","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64ddf330c0637f211a2ec39f7c4da88ed90e15025c9b904d18bdc57e3211ab6b","family_id":"qf-mod46-mreduction-decider-transfer-boundary","seed":202611157081,"semantic_template":"qf-mod46-mreduction-decider-transfer-boundary/{\"mapping\":[0,1,0,1,0,1,0,1],\"mode\":\"reverse_misuse\",\"source_carrier\":[0,1,2,3,4,5,6,7],\"source_yes\":[1,3,5,7],\"target_carrier\":[0,1],\"target_decider\":[false,true],\"target_yes\":[1]}"},"fields":{"title":"Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=reverse_misuse. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).","answer":{"kind":"expression","value":"(reverse_misuse, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes], Yes, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)","explanation":"Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (reverse_misuse, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes], Yes, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)."},"skills":["Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0082","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bab035973a9861f47c9f77fdafd79da6941e4ca93bed7c600c9f59f206e5b342","family_id":"qf-mod46-mreduction-decider-transfer-boundary","seed":202611157082,"semantic_template":"qf-mod46-mreduction-decider-transfer-boundary/{\"mapping\":[0,1,0,1,0,1,0,1,null],\"mode\":\"partial_map\",\"source_carrier\":[0,1,2,3,4,5,6,7,8],\"source_yes\":[1,3,5,7],\"target_carrier\":[0,1],\"target_decider\":[false,true],\"target_yes\":[1]}"},"fields":{"title":"Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, null], source yes-set=[1, 3, 5, 7], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=partial_map. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).","answer":{"kind":"expression","value":"(partial_map, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, null, null, No]], No, No, [No, Yes], Yes, null, No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)","explanation":"Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (partial_map, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, null, null, No]], No, No, [No, Yes], Yes, null, No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)."},"skills":["Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0083","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e28e9992baa3e05f10caee1559b2ed5dbb4e5245c3eb5a4870ad448c0890a17","family_id":"qf-mod46-mreduction-decider-transfer-boundary","seed":202611157083,"semantic_template":"qf-mod46-mreduction-decider-transfer-boundary/{\"mapping\":[0,0,0,0,0,0,0,0,0,0],\"mode\":\"one_way_yes_only\",\"source_carrier\":[0,1,2,3,4,5,6,7,8,9],\"source_yes\":[0],\"target_carrier\":[0,1],\"target_decider\":[true,false],\"target_yes\":[0]}"},"fields":{"title":"Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], target carrier=[0, 1], aligned mapping=[0, 0, 0, 0, 0, 0, 0, 0, 0, 0], source yes-set=[0], target yes-set=[0]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[true, false], aligned with the supplied target carrier. Mode=one_way_yes_only. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).","answer":{"kind":"expression","value":"(one_way_yes_only, [[0, Yes, 0, Yes, Yes], [1, No, 0, Yes, No], [2, No, 0, Yes, No], [3, No, 0, Yes, No], [4, No, 0, Yes, No], [5, No, 0, Yes, No], [6, No, 0, Yes, No], [7, No, 0, Yes, No], [8, No, 0, Yes, No], [9, No, 0, Yes, No]], Yes, No, [Yes, No], Yes, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)","explanation":"Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (one_way_yes_only, [[0, Yes, 0, Yes, Yes], [1, No, 0, Yes, No], [2, No, 0, Yes, No], [3, No, 0, Yes, No], [4, No, 0, Yes, No], [5, No, 0, Yes, No], [6, No, 0, Yes, No], [7, No, 0, Yes, No], [8, No, 0, Yes, No], [9, No, 0, Yes, No]], Yes, No, [Yes, No], Yes, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)."},"skills":["Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0084","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06820ab7140b0c74b2e3ad3130fa26c61137f583caaf4911da30ac83fdf4c54a","family_id":"qf-mod46-mreduction-decider-transfer-boundary","seed":202611157084,"semantic_template":"qf-mod46-mreduction-decider-transfer-boundary/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0],\"mode\":\"valid_transfer\",\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10],\"source_yes\":[1,3,5,7,9],\"target_carrier\":[0,1],\"target_decider\":[false,true],\"target_yes\":[1]}"},"fields":{"title":"Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=valid_transfer. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).","answer":{"kind":"expression","value":"(valid_transfer, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No], Yes, Yes, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)","explanation":"Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (valid_transfer, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No], Yes, Yes, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)."},"skills":["Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0085","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6506b7711d55b1c0d34e86bc2a46cae8968a45c0fc9cb611075f5ecc6bc3349f","family_id":"qf-mod46-mreduction-decider-transfer-boundary","seed":202611157085,"semantic_template":"qf-mod46-mreduction-decider-transfer-boundary/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1],\"mode\":\"reverse_misuse\",\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11],\"source_yes\":[1,11,3,5,7,9],\"target_carrier\":[0,1],\"target_decider\":[false,true],\"target_yes\":[1]}"},"fields":{"title":"Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9, 11], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=reverse_misuse. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).","answer":{"kind":"expression","value":"(reverse_misuse, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes], Yes, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)","explanation":"Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (reverse_misuse, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes], Yes, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)."},"skills":["Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0086","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c54de883d8bf368793360451c8f3ab058ca2c2f24e91f11244f1e312a9b7fb65","family_id":"qf-mod46-mreduction-decider-transfer-boundary","seed":202611157086,"semantic_template":"qf-mod46-mreduction-decider-transfer-boundary/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,null],\"mode\":\"partial_map\",\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12],\"source_yes\":[1,11,3,5,7,9],\"target_carrier\":[0,1],\"target_decider\":[false,true],\"target_yes\":[1]}"},"fields":{"title":"Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, null], source yes-set=[1, 3, 5, 7, 9, 11], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=partial_map. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).","answer":{"kind":"expression","value":"(partial_map, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, null, null, No]], No, No, [No, Yes], Yes, null, No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)","explanation":"Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (partial_map, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, null, null, No]], No, No, [No, Yes], Yes, null, No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)."},"skills":["Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0087","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ee950bd74d337279d1b78f64c58d5ad559e585fb0984f51ab814fa5804754d5","family_id":"qf-mod46-mreduction-decider-transfer-boundary","seed":202611157087,"semantic_template":"qf-mod46-mreduction-decider-transfer-boundary/{\"mapping\":[0,0,0,0,0,0,0,0,0,0,0,0,0,0],\"mode\":\"one_way_yes_only\",\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13],\"source_yes\":[0],\"target_carrier\":[0,1],\"target_decider\":[true,false],\"target_yes\":[0]}"},"fields":{"title":"Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], target carrier=[0, 1], aligned mapping=[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], source yes-set=[0], target yes-set=[0]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[true, false], aligned with the supplied target carrier. Mode=one_way_yes_only. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).","answer":{"kind":"expression","value":"(one_way_yes_only, [[0, Yes, 0, Yes, Yes], [1, No, 0, Yes, No], [2, No, 0, Yes, No], [3, No, 0, Yes, No], [4, No, 0, Yes, No], [5, No, 0, Yes, No], [6, No, 0, Yes, No], [7, No, 0, Yes, No], [8, No, 0, Yes, No], [9, No, 0, Yes, No], [10, No, 0, Yes, No], [11, No, 0, Yes, No], [12, No, 0, Yes, No], [13, No, 0, Yes, No]], Yes, No, [Yes, No], Yes, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)","explanation":"Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (one_way_yes_only, [[0, Yes, 0, Yes, Yes], [1, No, 0, Yes, No], [2, No, 0, Yes, No], [3, No, 0, Yes, No], [4, No, 0, Yes, No], [5, No, 0, Yes, No], [6, No, 0, Yes, No], [7, No, 0, Yes, No], [8, No, 0, Yes, No], [9, No, 0, Yes, No], [10, No, 0, Yes, No], [11, No, 0, Yes, No], [12, No, 0, Yes, No], [13, No, 0, Yes, No]], Yes, No, [Yes, No], Yes, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)."},"skills":["Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0088","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"601bff2c862b4d381b9c55e9f5aebccfdc222bd2dae493d44780fdf093407716","family_id":"qf-mod46-mreduction-decider-transfer-boundary","seed":202611157088,"semantic_template":"qf-mod46-mreduction-decider-transfer-boundary/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0],\"mode\":\"valid_transfer\",\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],\"source_yes\":[1,11,13,3,5,7,9],\"target_carrier\":[0,1],\"target_decider\":[false,true],\"target_yes\":[1]}"},"fields":{"title":"Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9, 11, 13], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=valid_transfer. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).","answer":{"kind":"expression","value":"(valid_transfer, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No], Yes, Yes, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)","explanation":"Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (valid_transfer, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No], Yes, Yes, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)."},"skills":["Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0089","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ff216b39d723e608364bcb04d8fa21a9f2636052587aee62b604a1fe882d55e","family_id":"qf-mod46-mreduction-decider-transfer-boundary","seed":202611157089,"semantic_template":"qf-mod46-mreduction-decider-transfer-boundary/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"mode\":\"reverse_misuse\",\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],\"source_yes\":[1,11,13,15,3,5,7,9],\"target_carrier\":[0,1],\"target_decider\":[false,true],\"target_yes\":[1]}"},"fields":{"title":"Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=reverse_misuse. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).","answer":{"kind":"expression","value":"(reverse_misuse, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes], Yes, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)","explanation":"Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (reverse_misuse, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes], Yes, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)."},"skills":["Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0090","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a7a63f9fd1ae04ab999b215f2ed913dcdfd2080fff1c331f191d9a66b818705","family_id":"qf-mod46-mreduction-decider-transfer-boundary","seed":202611157090,"semantic_template":"qf-mod46-mreduction-decider-transfer-boundary/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,null],\"mode\":\"partial_map\",\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],\"source_yes\":[1,11,13,15,3,5,7,9],\"target_carrier\":[0,1],\"target_decider\":[false,true],\"target_yes\":[1]}"},"fields":{"title":"Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, null], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=partial_map. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).","answer":{"kind":"expression","value":"(partial_map, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, null, null, No]], No, No, [No, Yes], Yes, null, No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)","explanation":"Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (partial_map, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, null, null, No]], No, No, [No, Yes], Yes, null, No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)."},"skills":["Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0091","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ca7cf166bd07ae82f74f9dad8ac6a66272909e5e1dcddbd02ff5fcd4bda0062","family_id":"qf-mod46-mreduction-decider-transfer-boundary","seed":202611157091,"semantic_template":"qf-mod46-mreduction-decider-transfer-boundary/{\"mapping\":[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],\"mode\":\"one_way_yes_only\",\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],\"source_yes\":[0],\"target_carrier\":[0,1],\"target_decider\":[true,false],\"target_yes\":[0]}"},"fields":{"title":"Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], target carrier=[0, 1], aligned mapping=[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], source yes-set=[0], target yes-set=[0]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[true, false], aligned with the supplied target carrier. Mode=one_way_yes_only. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).","answer":{"kind":"expression","value":"(one_way_yes_only, [[0, Yes, 0, Yes, Yes], [1, No, 0, Yes, No], [2, No, 0, Yes, No], [3, No, 0, Yes, No], [4, No, 0, Yes, No], [5, No, 0, Yes, No], [6, No, 0, Yes, No], [7, No, 0, Yes, No], [8, No, 0, Yes, No], [9, No, 0, Yes, No], [10, No, 0, Yes, No], [11, No, 0, Yes, No], [12, No, 0, Yes, No], [13, No, 0, Yes, No], [14, No, 0, Yes, No], [15, No, 0, Yes, No], [16, No, 0, Yes, No], [17, No, 0, Yes, No]], Yes, No, [Yes, No], Yes, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)","explanation":"Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (one_way_yes_only, [[0, Yes, 0, Yes, Yes], [1, No, 0, Yes, No], [2, No, 0, Yes, No], [3, No, 0, Yes, No], [4, No, 0, Yes, No], [5, No, 0, Yes, No], [6, No, 0, Yes, No], [7, No, 0, Yes, No], [8, No, 0, Yes, No], [9, No, 0, Yes, No], [10, No, 0, Yes, No], [11, No, 0, Yes, No], [12, No, 0, Yes, No], [13, No, 0, Yes, No], [14, No, 0, Yes, No], [15, No, 0, Yes, No], [16, No, 0, Yes, No], [17, No, 0, Yes, No]], Yes, No, [Yes, No], Yes, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)."},"skills":["Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0092","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6430ebcd2397446b2fc8fad85c80e75f8f74c46bc0c05ce26015ab9c9c47bdf","family_id":"qf-mod46-mreduction-decider-transfer-boundary","seed":202611157092,"semantic_template":"qf-mod46-mreduction-decider-transfer-boundary/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0],\"mode\":\"valid_transfer\",\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],\"source_yes\":[1,11,13,15,17,3,5,7,9],\"target_carrier\":[0,1],\"target_decider\":[false,true],\"target_yes\":[1]}"},"fields":{"title":"Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=valid_transfer. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).","answer":{"kind":"expression","value":"(valid_transfer, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No], Yes, Yes, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)","explanation":"Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (valid_transfer, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No], Yes, Yes, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)."},"skills":["Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0093","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6aa938bb9d609e3d9116aa058c2188358fa313d951c8d75c0d606525efe5d353","family_id":"qf-mod46-mreduction-decider-transfer-boundary","seed":202611157093,"semantic_template":"qf-mod46-mreduction-decider-transfer-boundary/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"mode\":\"reverse_misuse\",\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],\"source_yes\":[1,11,13,15,17,19,3,5,7,9],\"target_carrier\":[0,1],\"target_decider\":[false,true],\"target_yes\":[1]}"},"fields":{"title":"Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=reverse_misuse. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).","answer":{"kind":"expression","value":"(reverse_misuse, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes], Yes, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)","explanation":"Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (reverse_misuse, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes], Yes, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)."},"skills":["Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0094","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"915188ff81e8677f8a64eab2d648174769c0eb91f8c97e5554fe24877332e8c9","family_id":"qf-mod46-mreduction-decider-transfer-boundary","seed":202611157094,"semantic_template":"qf-mod46-mreduction-decider-transfer-boundary/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,null],\"mode\":\"partial_map\",\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],\"source_yes\":[1,11,13,15,17,19,3,5,7,9],\"target_carrier\":[0,1],\"target_decider\":[false,true],\"target_yes\":[1]}"},"fields":{"title":"Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, null], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=partial_map. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).","answer":{"kind":"expression","value":"(partial_map, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, null, null, No]], No, No, [No, Yes], Yes, null, No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)","explanation":"Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (partial_map, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, null, null, No]], No, No, [No, Yes], Yes, null, No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)."},"skills":["Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0095","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"662974e04af653be4af6f5079094cc663e20aff63b0c2be4024054f8f884f068","family_id":"qf-mod46-mreduction-decider-transfer-boundary","seed":202611157095,"semantic_template":"qf-mod46-mreduction-decider-transfer-boundary/{\"mapping\":[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],\"mode\":\"one_way_yes_only\",\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21],\"source_yes\":[0],\"target_carrier\":[0,1],\"target_decider\":[true,false],\"target_yes\":[0]}"},"fields":{"title":"Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], target carrier=[0, 1], aligned mapping=[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], source yes-set=[0], target yes-set=[0]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[true, false], aligned with the supplied target carrier. Mode=one_way_yes_only. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).","answer":{"kind":"expression","value":"(one_way_yes_only, [[0, Yes, 0, Yes, Yes], [1, No, 0, Yes, No], [2, No, 0, Yes, No], [3, No, 0, Yes, No], [4, No, 0, Yes, No], [5, No, 0, Yes, No], [6, No, 0, Yes, No], [7, No, 0, Yes, No], [8, No, 0, Yes, No], [9, No, 0, Yes, No], [10, No, 0, Yes, No], [11, No, 0, Yes, No], [12, No, 0, Yes, No], [13, No, 0, Yes, No], [14, No, 0, Yes, No], [15, No, 0, Yes, No], [16, No, 0, Yes, No], [17, No, 0, Yes, No], [18, No, 0, Yes, No], [19, No, 0, Yes, No], [20, No, 0, Yes, No], [21, No, 0, Yes, No]], Yes, No, [Yes, No], Yes, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)","explanation":"Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (one_way_yes_only, [[0, Yes, 0, Yes, Yes], [1, No, 0, Yes, No], [2, No, 0, Yes, No], [3, No, 0, Yes, No], [4, No, 0, Yes, No], [5, No, 0, Yes, No], [6, No, 0, Yes, No], [7, No, 0, Yes, No], [8, No, 0, Yes, No], [9, No, 0, Yes, No], [10, No, 0, Yes, No], [11, No, 0, Yes, No], [12, No, 0, Yes, No], [13, No, 0, Yes, No], [14, No, 0, Yes, No], [15, No, 0, Yes, No], [16, No, 0, Yes, No], [17, No, 0, Yes, No], [18, No, 0, Yes, No], [19, No, 0, Yes, No], [20, No, 0, Yes, No], [21, No, 0, Yes, No]], Yes, No, [Yes, No], Yes, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)."},"skills":["Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0096","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2be5737039ae2c16f1982893dbd19ea694b1862f8dd9b5e57654a4e5cd48043","family_id":"qf-mod46-mreduction-decider-transfer-boundary","seed":202611157096,"semantic_template":"qf-mod46-mreduction-decider-transfer-boundary/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0],\"mode\":\"valid_transfer\",\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],\"source_yes\":[1,11,13,15,17,19,21,3,5,7,9],\"target_carrier\":[0,1],\"target_decider\":[false,true],\"target_yes\":[1]}"},"fields":{"title":"Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=valid_transfer. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).","answer":{"kind":"expression","value":"(valid_transfer, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No], Yes, Yes, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)","explanation":"Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (valid_transfer, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No], Yes, Yes, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)."},"skills":["Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0097","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ef4e726f21d63ffb4d8bc2acd387643eea11a8dfa4eda1a93f8f0dc1c15ca89","family_id":"qf-mod46-mreduction-decider-transfer-boundary","seed":202611157097,"semantic_template":"qf-mod46-mreduction-decider-transfer-boundary/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1],\"mode\":\"reverse_misuse\",\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],\"source_yes\":[1,11,13,15,17,19,21,23,3,5,7,9],\"target_carrier\":[0,1],\"target_decider\":[false,true],\"target_yes\":[1]}"},"fields":{"title":"Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=reverse_misuse. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).","answer":{"kind":"expression","value":"(reverse_misuse, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes], Yes, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)","explanation":"Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (reverse_misuse, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes], Yes, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)."},"skills":["Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0098","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21ef079ad289fc1981e1007d9c16a154b6c23dcd4ec6efeb9fd01574d71acf54","family_id":"qf-mod46-mreduction-decider-transfer-boundary","seed":202611157098,"semantic_template":"qf-mod46-mreduction-decider-transfer-boundary/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,null],\"mode\":\"partial_map\",\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],\"source_yes\":[1,11,13,15,17,19,21,23,3,5,7,9],\"target_carrier\":[0,1],\"target_decider\":[false,true],\"target_yes\":[1]}"},"fields":{"title":"Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, null], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=partial_map. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).","answer":{"kind":"expression","value":"(partial_map, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, null, null, No]], No, No, [No, Yes], Yes, null, No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)","explanation":"Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (partial_map, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, null, null, No]], No, No, [No, Yes], Yes, null, No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)."},"skills":["Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0099","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"527479585ad1a91418ed5100add9da20a9703fd466fa687cae3b0dfef9f3b512","family_id":"qf-mod46-mreduction-decider-transfer-boundary","seed":202611157099,"semantic_template":"qf-mod46-mreduction-decider-transfer-boundary/{\"mapping\":[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],\"mode\":\"one_way_yes_only\",\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25],\"source_yes\":[0],\"target_carrier\":[0,1],\"target_decider\":[true,false],\"target_yes\":[0]}"},"fields":{"title":"Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], target carrier=[0, 1], aligned mapping=[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], source yes-set=[0], target yes-set=[0]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[true, false], aligned with the supplied target carrier. Mode=one_way_yes_only. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).","answer":{"kind":"expression","value":"(one_way_yes_only, [[0, Yes, 0, Yes, Yes], [1, No, 0, Yes, No], [2, No, 0, Yes, No], [3, No, 0, Yes, No], [4, No, 0, Yes, No], [5, No, 0, Yes, No], [6, No, 0, Yes, No], [7, No, 0, Yes, No], [8, No, 0, Yes, No], [9, No, 0, Yes, No], [10, No, 0, Yes, No], [11, No, 0, Yes, No], [12, No, 0, Yes, No], [13, No, 0, Yes, No], [14, No, 0, Yes, No], [15, No, 0, Yes, No], [16, No, 0, Yes, No], [17, No, 0, Yes, No], [18, No, 0, Yes, No], [19, No, 0, Yes, No], [20, No, 0, Yes, No], [21, No, 0, Yes, No], [22, No, 0, Yes, No], [23, No, 0, Yes, No], [24, No, 0, Yes, No], [25, No, 0, Yes, No]], Yes, No, [Yes, No], Yes, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)","explanation":"Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (one_way_yes_only, [[0, Yes, 0, Yes, Yes], [1, No, 0, Yes, No], [2, No, 0, Yes, No], [3, No, 0, Yes, No], [4, No, 0, Yes, No], [5, No, 0, Yes, No], [6, No, 0, Yes, No], [7, No, 0, Yes, No], [8, No, 0, Yes, No], [9, No, 0, Yes, No], [10, No, 0, Yes, No], [11, No, 0, Yes, No], [12, No, 0, Yes, No], [13, No, 0, Yes, No], [14, No, 0, Yes, No], [15, No, 0, Yes, No], [16, No, 0, Yes, No], [17, No, 0, Yes, No], [18, No, 0, Yes, No], [19, No, 0, Yes, No], [20, No, 0, Yes, No], [21, No, 0, Yes, No], [22, No, 0, Yes, No], [23, No, 0, Yes, No], [24, No, 0, Yes, No], [25, No, 0, Yes, No]], Yes, No, [Yes, No], Yes, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)."},"skills":["Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r07-many-one-reduction-0100","batch_id":"batch-20261015-advance-20261115-r07-many-one-reduction-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5d2011b3245bcbe3d8d209680f714575b24416cd03d84acefa1ad3100502ce3","family_id":"qf-mod46-mreduction-decider-transfer-boundary","seed":202611157100,"semantic_template":"qf-mod46-mreduction-decider-transfer-boundary/{\"mapping\":[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0],\"mode\":\"valid_transfer\",\"source_carrier\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],\"source_yes\":[1,11,13,15,17,19,21,23,25,3,5,7,9],\"target_carrier\":[0,1],\"target_decider\":[false,true],\"target_yes\":[1]}"},"fields":{"title":"Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider","stem":"Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=valid_transfer. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).","answer":{"kind":"expression","value":"(valid_transfer, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, 0, No, Yes], [25, Yes, 1, Yes, Yes], [26, No, 0, No, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No], Yes, Yes, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)","explanation":"Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (valid_transfer, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, 0, No, Yes], [25, Yes, 1, Yes, Yes], [26, No, 0, No, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No], Yes, Yes, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)."},"skills":["Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}