{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0001","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad4d75386977e5b4a481ac8a51acb3cc84975fb24f85d3f330f81b3204075a83","family_id":"qf-mod38-pigeon-collision-witness","seed":202611075001,"semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":3,\"mapping\":[0,1,2,0]}"},"fields":{"title":"A Complete Larger-Domain Map Has Two Distinct Sources with One Image","stem":"Use the complete total mapping=[0, 1, 2, 0] from source indices Fin(4)={0,…,3} into Fin(3)={0,…,2}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).","answer":{"kind":"expression","value":"(4, 3, [0, 3, 0], Yes, complete_total_map_collision)","explanation":"Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (4, 3, [0, 3, 0], Yes, complete_total_map_collision)."},"skills":["A Complete Larger-Domain Map Has Two Distinct Sources with One Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0002","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b3900c045e990ed67fd09d59b90a360a41299ae6a4f82036ae445b410f5f075","family_id":"qf-mod38-pigeon-collision-witness","seed":202611075002,"semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":4,\"mapping\":[1,3,1,3,1]}"},"fields":{"title":"A Complete Larger-Domain Map Has Two Distinct Sources with One Image","stem":"Use the complete total mapping=[1, 3, 1, 3, 1] from source indices Fin(5)={0,…,4} into Fin(4)={0,…,3}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).","answer":{"kind":"expression","value":"(5, 4, [0, 2, 1], Yes, complete_total_map_collision)","explanation":"Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (5, 4, [0, 2, 1], Yes, complete_total_map_collision)."},"skills":["A Complete Larger-Domain Map Has Two Distinct Sources with One Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0003","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"121e90a1329c59551e8b2c96129c210dcb24d6d6927d50881ea790ec285184b1","family_id":"qf-mod38-pigeon-collision-witness","seed":202611075003,"semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":5,\"mapping\":[2,0,3,1,4,2]}"},"fields":{"title":"A Complete Larger-Domain Map Has Two Distinct Sources with One Image","stem":"Use the complete total mapping=[2, 0, 3, 1, 4, 2] from source indices Fin(6)={0,…,5} into Fin(5)={0,…,4}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).","answer":{"kind":"expression","value":"(6, 5, [0, 5, 2], Yes, complete_total_map_collision)","explanation":"Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (6, 5, [0, 5, 2], Yes, complete_total_map_collision)."},"skills":["A Complete Larger-Domain Map Has Two Distinct Sources with One Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0004","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"070e39bc6622e892cc8df2334bfb4eadfc1858f8c5c46f5b58e2c8180f53ccdd","family_id":"qf-mod38-pigeon-collision-witness","seed":202611075004,"semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":6,\"mapping\":[3,4,5,0,1,2,3]}"},"fields":{"title":"A Complete Larger-Domain Map Has Two Distinct Sources with One Image","stem":"Use the complete total mapping=[3, 4, 5, 0, 1, 2, 3] from source indices Fin(7)={0,…,6} into Fin(6)={0,…,5}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).","answer":{"kind":"expression","value":"(7, 6, [0, 6, 3], Yes, complete_total_map_collision)","explanation":"Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (7, 6, [0, 6, 3], Yes, complete_total_map_collision)."},"skills":["A Complete Larger-Domain Map Has Two Distinct Sources with One Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0005","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5d841a5a1c82966e020fcafe534c8e3c9c2652196306ba3d6622b1eda643d80","family_id":"qf-mod38-pigeon-collision-witness","seed":202611075005,"semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":7,\"mapping\":[4,6,1,3,5,0,2,4]}"},"fields":{"title":"A Complete Larger-Domain Map Has Two Distinct Sources with One Image","stem":"Use the complete total mapping=[4, 6, 1, 3, 5, 0, 2, 4] from source indices Fin(8)={0,…,7} into Fin(7)={0,…,6}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).","answer":{"kind":"expression","value":"(8, 7, [0, 7, 4], Yes, complete_total_map_collision)","explanation":"Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (8, 7, [0, 7, 4], Yes, complete_total_map_collision)."},"skills":["A Complete Larger-Domain Map Has Two Distinct Sources with One Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0006","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"905731a5e72c3fb1347ca01c4fe40d2b8b24f6ce4d58a0ef862cb17f3fdd265e","family_id":"qf-mod38-pigeon-collision-witness","seed":202611075006,"semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":3,\"mapping\":[2,2,2,2,2]}"},"fields":{"title":"A Complete Larger-Domain Map Has Two Distinct Sources with One Image","stem":"Use the complete total mapping=[2, 2, 2, 2, 2] from source indices Fin(5)={0,…,4} into Fin(3)={0,…,2}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).","answer":{"kind":"expression","value":"(5, 3, [0, 1, 2], Yes, complete_total_map_collision)","explanation":"Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (5, 3, [0, 1, 2], Yes, complete_total_map_collision)."},"skills":["A Complete Larger-Domain Map Has Two Distinct Sources with One Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0007","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4eff9f93bbabd3d968fb53779211a6dfb2334a28dc034b4c329d0a481e4c8c8","family_id":"qf-mod38-pigeon-collision-witness","seed":202611075007,"semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":4,\"mapping\":[2,3,0,1,2,3]}"},"fields":{"title":"A Complete Larger-Domain Map Has Two Distinct Sources with One Image","stem":"Use the complete total mapping=[2, 3, 0, 1, 2, 3] from source indices Fin(6)={0,…,5} into Fin(4)={0,…,3}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).","answer":{"kind":"expression","value":"(6, 4, [0, 4, 2], Yes, complete_total_map_collision)","explanation":"Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (6, 4, [0, 4, 2], Yes, complete_total_map_collision)."},"skills":["A Complete Larger-Domain Map Has Two Distinct Sources with One Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0008","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36687f4db043da543e83b3af621cf0860d8b19cea3e80703ea1452ad6dfb2c64","family_id":"qf-mod38-pigeon-collision-witness","seed":202611075008,"semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":5,\"mapping\":[2,4,1,3,0,2,4]}"},"fields":{"title":"A Complete Larger-Domain Map Has Two Distinct Sources with One Image","stem":"Use the complete total mapping=[2, 4, 1, 3, 0, 2, 4] from source indices Fin(7)={0,…,6} into Fin(5)={0,…,4}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).","answer":{"kind":"expression","value":"(7, 5, [0, 5, 2], Yes, complete_total_map_collision)","explanation":"Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (7, 5, [0, 5, 2], Yes, complete_total_map_collision)."},"skills":["A Complete Larger-Domain Map Has Two Distinct Sources with One Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0009","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"feca3de7103ee5e9b7b3a32132e4dfa1d708c8497ebfcf7ff5d3471b7513d194","family_id":"qf-mod38-pigeon-collision-witness","seed":202611075009,"semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":6,\"mapping\":[2,5,2,5,2,5,2,5]}"},"fields":{"title":"A Complete Larger-Domain Map Has Two Distinct Sources with One Image","stem":"Use the complete total mapping=[2, 5, 2, 5, 2, 5, 2, 5] from source indices Fin(8)={0,…,7} into Fin(6)={0,…,5}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).","answer":{"kind":"expression","value":"(8, 6, [0, 2, 2], Yes, complete_total_map_collision)","explanation":"Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (8, 6, [0, 2, 2], Yes, complete_total_map_collision)."},"skills":["A Complete Larger-Domain Map Has Two Distinct Sources with One Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0010","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e3001dec8e2b7ebedc878edb7a5388dbdfe59425b7189545af5e124042030ccf","family_id":"qf-mod38-pigeon-collision-witness","seed":202611075010,"semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":7,\"mapping\":[2,3,4,5,6,0,1,2,3]}"},"fields":{"title":"A Complete Larger-Domain Map Has Two Distinct Sources with One Image","stem":"Use the complete total mapping=[2, 3, 4, 5, 6, 0, 1, 2, 3] from source indices Fin(9)={0,…,8} into Fin(7)={0,…,6}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).","answer":{"kind":"expression","value":"(9, 7, [0, 7, 2], Yes, complete_total_map_collision)","explanation":"Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (9, 7, [0, 7, 2], Yes, complete_total_map_collision)."},"skills":["A Complete Larger-Domain Map Has Two Distinct Sources with One Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0011","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c55d9ae62be28ed039cd14bc4c3756e4a6f95773f8632e224bc37dc0250e870","family_id":"qf-mod38-pigeon-collision-witness","seed":202611075011,"semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":3,\"mapping\":[1,0,2,1,0,2]}"},"fields":{"title":"A Complete Larger-Domain Map Has Two Distinct Sources with One Image","stem":"Use the complete total mapping=[1, 0, 2, 1, 0, 2] from source indices Fin(6)={0,…,5} into Fin(3)={0,…,2}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).","answer":{"kind":"expression","value":"(6, 3, [0, 3, 1], Yes, complete_total_map_collision)","explanation":"Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (6, 3, [0, 3, 1], Yes, complete_total_map_collision)."},"skills":["A Complete Larger-Domain Map Has Two Distinct Sources with One Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0012","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2caba322a6bbe5ceebd436f6ede97b60f8c6c5d48d8999e142053b2ae323802","family_id":"qf-mod38-pigeon-collision-witness","seed":202611075012,"semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":4,\"mapping\":[3,2,1,0,3,2,1]}"},"fields":{"title":"A Complete Larger-Domain Map Has Two Distinct Sources with One Image","stem":"Use the complete total mapping=[3, 2, 1, 0, 3, 2, 1] from source indices Fin(7)={0,…,6} into Fin(4)={0,…,3}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).","answer":{"kind":"expression","value":"(7, 4, [0, 4, 3], Yes, complete_total_map_collision)","explanation":"Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (7, 4, [0, 4, 3], Yes, complete_total_map_collision)."},"skills":["A Complete Larger-Domain Map Has Two Distinct Sources with One Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0013","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66610a299a8c666b545e987cebfb5433c886f0c70065ea63acbfc01b95ba847b","family_id":"qf-mod38-pigeon-collision-witness","seed":202611075013,"semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":5,\"mapping\":[2,3,4,0,1,2,3,4]}"},"fields":{"title":"A Complete Larger-Domain Map Has Two Distinct Sources with One Image","stem":"Use the complete total mapping=[2, 3, 4, 0, 1, 2, 3, 4] from source indices Fin(8)={0,…,7} into Fin(5)={0,…,4}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).","answer":{"kind":"expression","value":"(8, 5, [0, 5, 2], Yes, complete_total_map_collision)","explanation":"Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (8, 5, [0, 5, 2], Yes, complete_total_map_collision)."},"skills":["A Complete Larger-Domain Map Has Two Distinct Sources with One Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0014","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ae82d8efbe63d1b55652d2947503f51658bf9b35436a13e61265b2fe2c59f9e","family_id":"qf-mod38-pigeon-collision-witness","seed":202611075014,"semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":6,\"mapping\":[1,3,5,1,3,5,1,3,5]}"},"fields":{"title":"A Complete Larger-Domain Map Has Two Distinct Sources with One Image","stem":"Use the complete total mapping=[1, 3, 5, 1, 3, 5, 1, 3, 5] from source indices Fin(9)={0,…,8} into Fin(6)={0,…,5}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).","answer":{"kind":"expression","value":"(9, 6, [0, 3, 1], Yes, complete_total_map_collision)","explanation":"Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (9, 6, [0, 3, 1], Yes, complete_total_map_collision)."},"skills":["A Complete Larger-Domain Map Has Two Distinct Sources with One Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0015","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f7ea0718df002b1c9890eeb0387150071448ad7f6b721c728a6b3c8119d83b2","family_id":"qf-mod38-pigeon-collision-witness","seed":202611075015,"semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":7,\"mapping\":[0,3,6,2,5,1,4,0,3,6]}"},"fields":{"title":"A Complete Larger-Domain Map Has Two Distinct Sources with One Image","stem":"Use the complete total mapping=[0, 3, 6, 2, 5, 1, 4, 0, 3, 6] from source indices Fin(10)={0,…,9} into Fin(7)={0,…,6}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).","answer":{"kind":"expression","value":"(10, 7, [0, 7, 0], Yes, complete_total_map_collision)","explanation":"Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (10, 7, [0, 7, 0], Yes, complete_total_map_collision)."},"skills":["A Complete Larger-Domain Map Has Two Distinct Sources with One Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0016","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c16d2e8d1ab17760261b31fbd5783cc973dca61f1b37855c150df3c97413860","family_id":"qf-mod38-pigeon-collision-witness","seed":202611075016,"semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":3,\"mapping\":[0,1,2,0,1,2,0]}"},"fields":{"title":"A Complete Larger-Domain Map Has Two Distinct Sources with One Image","stem":"Use the complete total mapping=[0, 1, 2, 0, 1, 2, 0] from source indices Fin(7)={0,…,6} into Fin(3)={0,…,2}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).","answer":{"kind":"expression","value":"(7, 3, [0, 3, 0], Yes, complete_total_map_collision)","explanation":"Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (7, 3, [0, 3, 0], Yes, complete_total_map_collision)."},"skills":["A Complete Larger-Domain Map Has Two Distinct Sources with One Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0017","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57512521e31db10a8a9700c127d2480687aee84308dfc000015326db702c732f","family_id":"qf-mod38-pigeon-collision-witness","seed":202611075017,"semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":4,\"mapping\":[0,2,0,2,0,2,0,2]}"},"fields":{"title":"A Complete Larger-Domain Map Has Two Distinct Sources with One Image","stem":"Use the complete total mapping=[0, 2, 0, 2, 0, 2, 0, 2] from source indices Fin(8)={0,…,7} into Fin(4)={0,…,3}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).","answer":{"kind":"expression","value":"(8, 4, [0, 2, 0], Yes, complete_total_map_collision)","explanation":"Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (8, 4, [0, 2, 0], Yes, complete_total_map_collision)."},"skills":["A Complete Larger-Domain Map Has Two Distinct Sources with One Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0018","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9ca735c77b0ca97658165009f34b71e8d9511e8f929304a515146dc89b90462","family_id":"qf-mod38-pigeon-collision-witness","seed":202611075018,"semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":5,\"mapping\":[2,0,3,1,4,2,0,3,1]}"},"fields":{"title":"A Complete Larger-Domain Map Has Two Distinct Sources with One Image","stem":"Use the complete total mapping=[2, 0, 3, 1, 4, 2, 0, 3, 1] from source indices Fin(9)={0,…,8} into Fin(5)={0,…,4}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).","answer":{"kind":"expression","value":"(9, 5, [0, 5, 2], Yes, complete_total_map_collision)","explanation":"Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (9, 5, [0, 5, 2], Yes, complete_total_map_collision)."},"skills":["A Complete Larger-Domain Map Has Two Distinct Sources with One Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0019","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5731fbedae4c4de1fe250fb8a830a0a7922f66db21d9893c017c4534e775b1df","family_id":"qf-mod38-pigeon-collision-witness","seed":202611075019,"semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":6,\"mapping\":[0,1,2,3,4,5,0,1,2,3]}"},"fields":{"title":"A Complete Larger-Domain Map Has Two Distinct Sources with One Image","stem":"Use the complete total mapping=[0, 1, 2, 3, 4, 5, 0, 1, 2, 3] from source indices Fin(10)={0,…,9} into Fin(6)={0,…,5}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).","answer":{"kind":"expression","value":"(10, 6, [0, 6, 0], Yes, complete_total_map_collision)","explanation":"Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (10, 6, [0, 6, 0], Yes, complete_total_map_collision)."},"skills":["A Complete Larger-Domain Map Has Two Distinct Sources with One Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0020","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0e08d830d8ff42a5417bdc0cd37e35b6ef081a4eb1c13bf69038117f30bdf2d","family_id":"qf-mod38-pigeon-collision-witness","seed":202611075020,"semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":7,\"mapping\":[5,0,2,4,6,1,3,5,0,2,4]}"},"fields":{"title":"A Complete Larger-Domain Map Has Two Distinct Sources with One Image","stem":"Use the complete total mapping=[5, 0, 2, 4, 6, 1, 3, 5, 0, 2, 4] from source indices Fin(11)={0,…,10} into Fin(7)={0,…,6}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).","answer":{"kind":"expression","value":"(11, 7, [0, 7, 5], Yes, complete_total_map_collision)","explanation":"Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (11, 7, [0, 7, 5], Yes, complete_total_map_collision)."},"skills":["A Complete Larger-Domain Map Has Two Distinct Sources with One Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0021","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d58dbf08a509af18f0c45335d95ae2e84cf34232993f880cb3bb875bf7924e91","family_id":"qf-mod38-pigeon-collision-witness","seed":202611075021,"semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":3,\"mapping\":[2,2,2,2,2,2,2,2]}"},"fields":{"title":"A Complete Larger-Domain Map Has Two Distinct Sources with One Image","stem":"Use the complete total mapping=[2, 2, 2, 2, 2, 2, 2, 2] from source indices Fin(8)={0,…,7} into Fin(3)={0,…,2}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).","answer":{"kind":"expression","value":"(8, 3, [0, 1, 2], Yes, complete_total_map_collision)","explanation":"Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (8, 3, [0, 1, 2], Yes, complete_total_map_collision)."},"skills":["A Complete Larger-Domain Map Has Two Distinct Sources with One Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0022","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb182659f1462114f06001a28b60a29921f47d4bf4b655482cf6e0d72a95a902","family_id":"qf-mod38-pigeon-collision-witness","seed":202611075022,"semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":4,\"mapping\":[1,2,3,0,1,2,3,0,1]}"},"fields":{"title":"A Complete Larger-Domain Map Has Two Distinct Sources with One Image","stem":"Use the complete total mapping=[1, 2, 3, 0, 1, 2, 3, 0, 1] from source indices Fin(9)={0,…,8} into Fin(4)={0,…,3}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).","answer":{"kind":"expression","value":"(9, 4, [0, 4, 1], Yes, complete_total_map_collision)","explanation":"Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (9, 4, [0, 4, 1], Yes, complete_total_map_collision)."},"skills":["A Complete Larger-Domain Map Has Two Distinct Sources with One Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0023","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a90026755b3bfb114572b1a29bffd390bc37490939360a3dbd7c66107f01b7c","family_id":"qf-mod38-pigeon-collision-witness","seed":202611075023,"semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":5,\"mapping\":[2,4,1,3,0,2,4,1,3,0]}"},"fields":{"title":"A Complete Larger-Domain Map Has Two Distinct Sources with One Image","stem":"Use the complete total mapping=[2, 4, 1, 3, 0, 2, 4, 1, 3, 0] from source indices Fin(10)={0,…,9} into Fin(5)={0,…,4}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).","answer":{"kind":"expression","value":"(10, 5, [0, 5, 2], Yes, complete_total_map_collision)","explanation":"Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (10, 5, [0, 5, 2], Yes, complete_total_map_collision)."},"skills":["A Complete Larger-Domain Map Has Two Distinct Sources with One Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0024","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95262857712a13fd1828caeb1370211151105e60c8fe76d9df74d52854d412d9","family_id":"qf-mod38-pigeon-collision-witness","seed":202611075024,"semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":6,\"mapping\":[5,2,5,2,5,2,5,2,5,2,5]}"},"fields":{"title":"A Complete Larger-Domain Map Has Two Distinct Sources with One Image","stem":"Use the complete total mapping=[5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5] from source indices Fin(11)={0,…,10} into Fin(6)={0,…,5}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).","answer":{"kind":"expression","value":"(11, 6, [0, 2, 5], Yes, complete_total_map_collision)","explanation":"Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (11, 6, [0, 2, 5], Yes, complete_total_map_collision)."},"skills":["A Complete Larger-Domain Map Has Two Distinct Sources with One Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0025","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5bb4a145053e88beb384408899378967f7ae3ed4469f533c55c0afd36ac13ed5","family_id":"qf-mod38-pigeon-collision-witness","seed":202611075025,"semantic_template":"qf-mod38-pigeon-collision-witness/{\"codomain_count\":7,\"mapping\":[3,4,5,6,0,1,2,3,4,5,6,0]}"},"fields":{"title":"A Complete Larger-Domain Map Has Two Distinct Sources with One Image","stem":"Use the complete total mapping=[3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6, 0] from source indices Fin(12)={0,…,11} into Fin(7)={0,…,6}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).","answer":{"kind":"expression","value":"(12, 7, [0, 7, 3], Yes, complete_total_map_collision)","explanation":"Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (12, 7, [0, 7, 3], Yes, complete_total_map_collision)."},"skills":["A Complete Larger-Domain Map Has Two Distinct Sources with One Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0026","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e2a240366cf9d61e1f310abcc9828e338af4bbf0926287d208b9b0d4090e9d8","family_id":"qf-mod38-pigeon-generalized-load","seed":202611075026,"semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":3,\"mapping\":[0,1,2,0]}"},"fields":{"title":"The Largest Fibre Is at Least the Exact Ceiling Average","stem":"Use the complete total mapping=[0, 1, 2, 0] from source indices Fin(4)={0,…,3} into Fin(3)={0,…,2}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([2, 1, 1], 2, 2, Yes, generalized_pigeonhole_load_bound)","explanation":"List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([2, 1, 1], 2, 2, Yes, generalized_pigeonhole_load_bound)."},"skills":["The Largest Fibre Is at Least the Exact Ceiling Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0027","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b0eeca63224be0d36b6dddfa4ae976388e14a3442ecfe05867acc3b41e72f97","family_id":"qf-mod38-pigeon-generalized-load","seed":202611075027,"semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":4,\"mapping\":[1,3,1,3,1]}"},"fields":{"title":"The Largest Fibre Is at Least the Exact Ceiling Average","stem":"Use the complete total mapping=[1, 3, 1, 3, 1] from source indices Fin(5)={0,…,4} into Fin(4)={0,…,3}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([0, 3, 0, 2], 3, 2, Yes, generalized_pigeonhole_load_bound)","explanation":"List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([0, 3, 0, 2], 3, 2, Yes, generalized_pigeonhole_load_bound)."},"skills":["The Largest Fibre Is at Least the Exact Ceiling Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0028","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8a64832b747d5576743b622fec2e55ff4eec18274cd8518d6f210cadec62b90","family_id":"qf-mod38-pigeon-generalized-load","seed":202611075028,"semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":5,\"mapping\":[2,0,3,1,4,2]}"},"fields":{"title":"The Largest Fibre Is at Least the Exact Ceiling Average","stem":"Use the complete total mapping=[2, 0, 3, 1, 4, 2] from source indices Fin(6)={0,…,5} into Fin(5)={0,…,4}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([1, 1, 2, 1, 1], 2, 2, Yes, generalized_pigeonhole_load_bound)","explanation":"List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([1, 1, 2, 1, 1], 2, 2, Yes, generalized_pigeonhole_load_bound)."},"skills":["The Largest Fibre Is at Least the Exact Ceiling Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0029","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a1b08ca0f411dd5ee6696605df79f26154d13e997b3220f001501080f3f5335","family_id":"qf-mod38-pigeon-generalized-load","seed":202611075029,"semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":6,\"mapping\":[3,4,5,0,1,2,3]}"},"fields":{"title":"The Largest Fibre Is at Least the Exact Ceiling Average","stem":"Use the complete total mapping=[3, 4, 5, 0, 1, 2, 3] from source indices Fin(7)={0,…,6} into Fin(6)={0,…,5}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([1, 1, 1, 2, 1, 1], 2, 2, Yes, generalized_pigeonhole_load_bound)","explanation":"List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([1, 1, 1, 2, 1, 1], 2, 2, Yes, generalized_pigeonhole_load_bound)."},"skills":["The Largest Fibre Is at Least the Exact Ceiling Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0030","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11189b68213e0624339bc5359e3040e03319b442db85910e2368d353b29dea10","family_id":"qf-mod38-pigeon-generalized-load","seed":202611075030,"semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":7,\"mapping\":[4,6,1,3,5,0,2,4]}"},"fields":{"title":"The Largest Fibre Is at Least the Exact Ceiling Average","stem":"Use the complete total mapping=[4, 6, 1, 3, 5, 0, 2, 4] from source indices Fin(8)={0,…,7} into Fin(7)={0,…,6}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([1, 1, 1, 1, 2, 1, 1], 2, 2, Yes, generalized_pigeonhole_load_bound)","explanation":"List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([1, 1, 1, 1, 2, 1, 1], 2, 2, Yes, generalized_pigeonhole_load_bound)."},"skills":["The Largest Fibre Is at Least the Exact Ceiling Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0031","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e4af17f898f1c7a3dad647ff18c56e76c63ee99ca3b02fefe692e19e59d1075","family_id":"qf-mod38-pigeon-generalized-load","seed":202611075031,"semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":3,\"mapping\":[2,2,2,2,2]}"},"fields":{"title":"The Largest Fibre Is at Least the Exact Ceiling Average","stem":"Use the complete total mapping=[2, 2, 2, 2, 2] from source indices Fin(5)={0,…,4} into Fin(3)={0,…,2}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([0, 0, 5], 5, 2, Yes, generalized_pigeonhole_load_bound)","explanation":"List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([0, 0, 5], 5, 2, Yes, generalized_pigeonhole_load_bound)."},"skills":["The Largest Fibre Is at Least the Exact Ceiling Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0032","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ceea97172043c3475530279a29fa2901dc5aeabfe4e6ce6e1c1e0a93cbfe51bf","family_id":"qf-mod38-pigeon-generalized-load","seed":202611075032,"semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":4,\"mapping\":[2,3,0,1,2,3]}"},"fields":{"title":"The Largest Fibre Is at Least the Exact Ceiling Average","stem":"Use the complete total mapping=[2, 3, 0, 1, 2, 3] from source indices Fin(6)={0,…,5} into Fin(4)={0,…,3}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([1, 1, 2, 2], 2, 2, Yes, generalized_pigeonhole_load_bound)","explanation":"List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([1, 1, 2, 2], 2, 2, Yes, generalized_pigeonhole_load_bound)."},"skills":["The Largest Fibre Is at Least the Exact Ceiling Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0033","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad3d086ef8991484882805d5dfd2b09a9893c346dbfdb5b2e5bb2201ad39fe56","family_id":"qf-mod38-pigeon-generalized-load","seed":202611075033,"semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":5,\"mapping\":[2,4,1,3,0,2,4]}"},"fields":{"title":"The Largest Fibre Is at Least the Exact Ceiling Average","stem":"Use the complete total mapping=[2, 4, 1, 3, 0, 2, 4] from source indices Fin(7)={0,…,6} into Fin(5)={0,…,4}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([1, 1, 2, 1, 2], 2, 2, Yes, generalized_pigeonhole_load_bound)","explanation":"List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([1, 1, 2, 1, 2], 2, 2, Yes, generalized_pigeonhole_load_bound)."},"skills":["The Largest Fibre Is at Least the Exact Ceiling Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0034","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30464e67672d056e1c7effbadefc43118d5fd5d6657bd72ef7de8cd4c5cce6a3","family_id":"qf-mod38-pigeon-generalized-load","seed":202611075034,"semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":6,\"mapping\":[2,5,2,5,2,5,2,5]}"},"fields":{"title":"The Largest Fibre Is at Least the Exact Ceiling Average","stem":"Use the complete total mapping=[2, 5, 2, 5, 2, 5, 2, 5] from source indices Fin(8)={0,…,7} into Fin(6)={0,…,5}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([0, 0, 4, 0, 0, 4], 4, 2, Yes, generalized_pigeonhole_load_bound)","explanation":"List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([0, 0, 4, 0, 0, 4], 4, 2, Yes, generalized_pigeonhole_load_bound)."},"skills":["The Largest Fibre Is at Least the Exact Ceiling Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0035","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7f1ae85f8f434b21aced5583a95d397a4721d31d1082721a86a37cd73fafa72","family_id":"qf-mod38-pigeon-generalized-load","seed":202611075035,"semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":7,\"mapping\":[2,3,4,5,6,0,1,2,3]}"},"fields":{"title":"The Largest Fibre Is at Least the Exact Ceiling Average","stem":"Use the complete total mapping=[2, 3, 4, 5, 6, 0, 1, 2, 3] from source indices Fin(9)={0,…,8} into Fin(7)={0,…,6}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([1, 1, 2, 2, 1, 1, 1], 2, 2, Yes, generalized_pigeonhole_load_bound)","explanation":"List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([1, 1, 2, 2, 1, 1, 1], 2, 2, Yes, generalized_pigeonhole_load_bound)."},"skills":["The Largest Fibre Is at Least the Exact Ceiling Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0036","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e7569ab28face69326428db22ecf0b343da8b117d76eff0a4ac6e0b63433456","family_id":"qf-mod38-pigeon-generalized-load","seed":202611075036,"semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":3,\"mapping\":[1,0,2,1,0,2]}"},"fields":{"title":"The Largest Fibre Is at Least the Exact Ceiling Average","stem":"Use the complete total mapping=[1, 0, 2, 1, 0, 2] from source indices Fin(6)={0,…,5} into Fin(3)={0,…,2}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([2, 2, 2], 2, 2, Yes, generalized_pigeonhole_load_bound)","explanation":"List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([2, 2, 2], 2, 2, Yes, generalized_pigeonhole_load_bound)."},"skills":["The Largest Fibre Is at Least the Exact Ceiling Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0037","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3f22192f2212d4f94d1d1ee9be4591df81128e8f40191d8b35c5735e30e0bf5","family_id":"qf-mod38-pigeon-generalized-load","seed":202611075037,"semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":4,\"mapping\":[3,2,1,0,3,2,1]}"},"fields":{"title":"The Largest Fibre Is at Least the Exact Ceiling Average","stem":"Use the complete total mapping=[3, 2, 1, 0, 3, 2, 1] from source indices Fin(7)={0,…,6} into Fin(4)={0,…,3}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([1, 2, 2, 2], 2, 2, Yes, generalized_pigeonhole_load_bound)","explanation":"List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([1, 2, 2, 2], 2, 2, Yes, generalized_pigeonhole_load_bound)."},"skills":["The Largest Fibre Is at Least the Exact Ceiling Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0038","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aaaf9aba939de89bbb796740fd8ef2880e6c2095da5f7c7d1c98c6c32b5dd4ab","family_id":"qf-mod38-pigeon-generalized-load","seed":202611075038,"semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":5,\"mapping\":[2,3,4,0,1,2,3,4]}"},"fields":{"title":"The Largest Fibre Is at Least the Exact Ceiling Average","stem":"Use the complete total mapping=[2, 3, 4, 0, 1, 2, 3, 4] from source indices Fin(8)={0,…,7} into Fin(5)={0,…,4}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([1, 1, 2, 2, 2], 2, 2, Yes, generalized_pigeonhole_load_bound)","explanation":"List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([1, 1, 2, 2, 2], 2, 2, Yes, generalized_pigeonhole_load_bound)."},"skills":["The Largest Fibre Is at Least the Exact Ceiling Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0039","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"842dc66c80ed99eb2d695f8db074a060efbb4cf28d5fe3691b2e01926adefdf0","family_id":"qf-mod38-pigeon-generalized-load","seed":202611075039,"semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":6,\"mapping\":[1,3,5,1,3,5,1,3,5]}"},"fields":{"title":"The Largest Fibre Is at Least the Exact Ceiling Average","stem":"Use the complete total mapping=[1, 3, 5, 1, 3, 5, 1, 3, 5] from source indices Fin(9)={0,…,8} into Fin(6)={0,…,5}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([0, 3, 0, 3, 0, 3], 3, 2, Yes, generalized_pigeonhole_load_bound)","explanation":"List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([0, 3, 0, 3, 0, 3], 3, 2, Yes, generalized_pigeonhole_load_bound)."},"skills":["The Largest Fibre Is at Least the Exact Ceiling Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0040","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41f8d19be1d19dd113ebd470dbf451e27a91c1d02d5c15361ad938596c83eab8","family_id":"qf-mod38-pigeon-generalized-load","seed":202611075040,"semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":7,\"mapping\":[0,3,6,2,5,1,4,0,3,6]}"},"fields":{"title":"The Largest Fibre Is at Least the Exact Ceiling Average","stem":"Use the complete total mapping=[0, 3, 6, 2, 5, 1, 4, 0, 3, 6] from source indices Fin(10)={0,…,9} into Fin(7)={0,…,6}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([2, 1, 1, 2, 1, 1, 2], 2, 2, Yes, generalized_pigeonhole_load_bound)","explanation":"List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([2, 1, 1, 2, 1, 1, 2], 2, 2, Yes, generalized_pigeonhole_load_bound)."},"skills":["The Largest Fibre Is at Least the Exact Ceiling Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0041","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d366c6e0f03c734fa1d59853fe244180d4ad64ce12c78b43174ed9eff09d242","family_id":"qf-mod38-pigeon-generalized-load","seed":202611075041,"semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":3,\"mapping\":[0,1,2,0,1,2,0]}"},"fields":{"title":"The Largest Fibre Is at Least the Exact Ceiling Average","stem":"Use the complete total mapping=[0, 1, 2, 0, 1, 2, 0] from source indices Fin(7)={0,…,6} into Fin(3)={0,…,2}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([3, 2, 2], 3, 3, Yes, generalized_pigeonhole_load_bound)","explanation":"List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([3, 2, 2], 3, 3, Yes, generalized_pigeonhole_load_bound)."},"skills":["The Largest Fibre Is at Least the Exact Ceiling Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0042","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e679e788f797e8e929ee1e046b909ff850d8e59da9fa6ba5eba24e3b574cf9c","family_id":"qf-mod38-pigeon-generalized-load","seed":202611075042,"semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":4,\"mapping\":[0,2,0,2,0,2,0,2]}"},"fields":{"title":"The Largest Fibre Is at Least the Exact Ceiling Average","stem":"Use the complete total mapping=[0, 2, 0, 2, 0, 2, 0, 2] from source indices Fin(8)={0,…,7} into Fin(4)={0,…,3}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([4, 0, 4, 0], 4, 2, Yes, generalized_pigeonhole_load_bound)","explanation":"List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([4, 0, 4, 0], 4, 2, Yes, generalized_pigeonhole_load_bound)."},"skills":["The Largest Fibre Is at Least the Exact Ceiling Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0043","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1cf71bbccead9eea283f9f7a4be526c49bf9df309939751de3db2e8334749ef1","family_id":"qf-mod38-pigeon-generalized-load","seed":202611075043,"semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":5,\"mapping\":[2,0,3,1,4,2,0,3,1]}"},"fields":{"title":"The Largest Fibre Is at Least the Exact Ceiling Average","stem":"Use the complete total mapping=[2, 0, 3, 1, 4, 2, 0, 3, 1] from source indices Fin(9)={0,…,8} into Fin(5)={0,…,4}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([2, 2, 2, 2, 1], 2, 2, Yes, generalized_pigeonhole_load_bound)","explanation":"List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([2, 2, 2, 2, 1], 2, 2, Yes, generalized_pigeonhole_load_bound)."},"skills":["The Largest Fibre Is at Least the Exact Ceiling Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0044","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"992960e47d428ae9e8536a4b1ffd651343a89c0403040244d394f8b4429a207f","family_id":"qf-mod38-pigeon-generalized-load","seed":202611075044,"semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":6,\"mapping\":[0,1,2,3,4,5,0,1,2,3]}"},"fields":{"title":"The Largest Fibre Is at Least the Exact Ceiling Average","stem":"Use the complete total mapping=[0, 1, 2, 3, 4, 5, 0, 1, 2, 3] from source indices Fin(10)={0,…,9} into Fin(6)={0,…,5}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([2, 2, 2, 2, 1, 1], 2, 2, Yes, generalized_pigeonhole_load_bound)","explanation":"List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([2, 2, 2, 2, 1, 1], 2, 2, Yes, generalized_pigeonhole_load_bound)."},"skills":["The Largest Fibre Is at Least the Exact Ceiling Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0045","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a0bfdb1b64fa4bdf74c5c0b1377b3d463b1ba288d82bc98083ba922489b2ce2","family_id":"qf-mod38-pigeon-generalized-load","seed":202611075045,"semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":7,\"mapping\":[5,0,2,4,6,1,3,5,0,2,4]}"},"fields":{"title":"The Largest Fibre Is at Least the Exact Ceiling Average","stem":"Use the complete total mapping=[5, 0, 2, 4, 6, 1, 3, 5, 0, 2, 4] from source indices Fin(11)={0,…,10} into Fin(7)={0,…,6}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([2, 1, 2, 1, 2, 2, 1], 2, 2, Yes, generalized_pigeonhole_load_bound)","explanation":"List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([2, 1, 2, 1, 2, 2, 1], 2, 2, Yes, generalized_pigeonhole_load_bound)."},"skills":["The Largest Fibre Is at Least the Exact Ceiling Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0046","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac590df9b27bd2b9e5d359d685a71008c1ecc124c8e93ff268011b764c76756b","family_id":"qf-mod38-pigeon-generalized-load","seed":202611075046,"semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":3,\"mapping\":[2,2,2,2,2,2,2,2]}"},"fields":{"title":"The Largest Fibre Is at Least the Exact Ceiling Average","stem":"Use the complete total mapping=[2, 2, 2, 2, 2, 2, 2, 2] from source indices Fin(8)={0,…,7} into Fin(3)={0,…,2}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([0, 0, 8], 8, 3, Yes, generalized_pigeonhole_load_bound)","explanation":"List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([0, 0, 8], 8, 3, Yes, generalized_pigeonhole_load_bound)."},"skills":["The Largest Fibre Is at Least the Exact Ceiling Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0047","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be8d5bc8ecac332baa89841a1fc608ed56443049ab0bf5ba1ef29b2d9998afa5","family_id":"qf-mod38-pigeon-generalized-load","seed":202611075047,"semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":4,\"mapping\":[1,2,3,0,1,2,3,0,1]}"},"fields":{"title":"The Largest Fibre Is at Least the Exact Ceiling Average","stem":"Use the complete total mapping=[1, 2, 3, 0, 1, 2, 3, 0, 1] from source indices Fin(9)={0,…,8} into Fin(4)={0,…,3}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([2, 3, 2, 2], 3, 3, Yes, generalized_pigeonhole_load_bound)","explanation":"List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([2, 3, 2, 2], 3, 3, Yes, generalized_pigeonhole_load_bound)."},"skills":["The Largest Fibre Is at Least the Exact Ceiling Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0048","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b01b6e22a077735b56cd3fd66bdea2e6255d8f271c4d581291a75d0b43fe4f5c","family_id":"qf-mod38-pigeon-generalized-load","seed":202611075048,"semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":5,\"mapping\":[2,4,1,3,0,2,4,1,3,0]}"},"fields":{"title":"The Largest Fibre Is at Least the Exact Ceiling Average","stem":"Use the complete total mapping=[2, 4, 1, 3, 0, 2, 4, 1, 3, 0] from source indices Fin(10)={0,…,9} into Fin(5)={0,…,4}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([2, 2, 2, 2, 2], 2, 2, Yes, generalized_pigeonhole_load_bound)","explanation":"List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([2, 2, 2, 2, 2], 2, 2, Yes, generalized_pigeonhole_load_bound)."},"skills":["The Largest Fibre Is at Least the Exact Ceiling Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0049","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2efaf018ee0c5a2836b172911b1bbf03452ca6bf0e641afd20cc097b0bc353ba","family_id":"qf-mod38-pigeon-generalized-load","seed":202611075049,"semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":6,\"mapping\":[5,2,5,2,5,2,5,2,5,2,5]}"},"fields":{"title":"The Largest Fibre Is at Least the Exact Ceiling Average","stem":"Use the complete total mapping=[5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5] from source indices Fin(11)={0,…,10} into Fin(6)={0,…,5}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([0, 0, 5, 0, 0, 6], 6, 2, Yes, generalized_pigeonhole_load_bound)","explanation":"List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([0, 0, 5, 0, 0, 6], 6, 2, Yes, generalized_pigeonhole_load_bound)."},"skills":["The Largest Fibre Is at Least the Exact Ceiling Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0050","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b640190d1e929a849c5f1bc4d114fd225d17377a5b45c025db709d668829988","family_id":"qf-mod38-pigeon-generalized-load","seed":202611075050,"semantic_template":"qf-mod38-pigeon-generalized-load/{\"codomain_count\":7,\"mapping\":[3,4,5,6,0,1,2,3,4,5,6,0]}"},"fields":{"title":"The Largest Fibre Is at Least the Exact Ceiling Average","stem":"Use the complete total mapping=[3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6, 0] from source indices Fin(12)={0,…,11} into Fin(7)={0,…,6}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).","answer":{"kind":"expression","value":"([2, 1, 1, 2, 2, 2, 2], 2, 2, Yes, generalized_pigeonhole_load_bound)","explanation":"List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([2, 1, 1, 2, 2, 2, 2], 2, 2, Yes, generalized_pigeonhole_load_bound)."},"skills":["The Largest Fibre Is at Least the Exact Ceiling Average"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0051","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3dea4ac5f831f1b41f3f634edb08ff5dd36bf7cf309980ee374aa0e519c1f3f","family_id":"qf-mod38-pigeon-premise-audit","seed":202611075051,"semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":3,\"domain_count\":5}"},"fields":{"title":"The Size Premise Forces a Collision but Has No General Reverse Inference","stem":"Use complete finite domain size=5≥0 and codomain size=3≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(5, 3, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation":"Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (5, 3, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)."},"skills":["The Size Premise Forces a Collision but Has No General Reverse Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0052","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"784eda480c1c09ef42ba2fc31a659ee26c40be35bbf600421b8255e18e835e2a","family_id":"qf-mod38-pigeon-premise-audit","seed":202611075052,"semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":4,\"domain_count\":6}"},"fields":{"title":"The Size Premise Forces a Collision but Has No General Reverse Inference","stem":"Use complete finite domain size=6≥0 and codomain size=4≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(6, 4, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation":"Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (6, 4, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)."},"skills":["The Size Premise Forces a Collision but Has No General Reverse Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0053","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54b546eccbde9e2122ccf1b61c888969264853f62312516600004169742af2a8","family_id":"qf-mod38-pigeon-premise-audit","seed":202611075053,"semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":5,\"domain_count\":7}"},"fields":{"title":"The Size Premise Forces a Collision but Has No General Reverse Inference","stem":"Use complete finite domain size=7≥0 and codomain size=5≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(7, 5, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation":"Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (7, 5, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)."},"skills":["The Size Premise Forces a Collision but Has No General Reverse Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0054","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"398522225767d83eca7a988649f6175a95a8288ba5645376960e13abc16b34a3","family_id":"qf-mod38-pigeon-premise-audit","seed":202611075054,"semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":6,\"domain_count\":8}"},"fields":{"title":"The Size Premise Forces a Collision but Has No General Reverse Inference","stem":"Use complete finite domain size=8≥0 and codomain size=6≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(8, 6, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation":"Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (8, 6, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)."},"skills":["The Size Premise Forces a Collision but Has No General Reverse Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0055","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23a037f7c812cb2dd4f819e1deeaa9439e9d740a8d1f59a1e45066fb9e3997e9","family_id":"qf-mod38-pigeon-premise-audit","seed":202611075055,"semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":7,\"domain_count\":9}"},"fields":{"title":"The Size Premise Forces a Collision but Has No General Reverse Inference","stem":"Use complete finite domain size=9≥0 and codomain size=7≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(9, 7, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation":"Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (9, 7, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)."},"skills":["The Size Premise Forces a Collision but Has No General Reverse Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0056","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd609c2a7a1150d9f4e20a5c99d738d30204a5071c6edbd74961e2eb77ae6af1","family_id":"qf-mod38-pigeon-premise-audit","seed":202611075056,"semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":8,\"domain_count\":10}"},"fields":{"title":"The Size Premise Forces a Collision but Has No General Reverse Inference","stem":"Use complete finite domain size=10≥0 and codomain size=8≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(10, 8, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation":"Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (10, 8, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)."},"skills":["The Size Premise Forces a Collision but Has No General Reverse Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0057","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55ceb373302f3b23b4c640686c9336ecc19db49d914323f3bb1247c628c21243","family_id":"qf-mod38-pigeon-premise-audit","seed":202611075057,"semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":9,\"domain_count\":11}"},"fields":{"title":"The Size Premise Forces a Collision but Has No General Reverse Inference","stem":"Use complete finite domain size=11≥0 and codomain size=9≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(11, 9, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation":"Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (11, 9, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)."},"skills":["The Size Premise Forces a Collision but Has No General Reverse Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0058","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c099c3f9c6649c96dd34dc6733d2bfdb6bf2545bdfc3541c6c4a7dcd643819e","family_id":"qf-mod38-pigeon-premise-audit","seed":202611075058,"semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":10,\"domain_count\":12}"},"fields":{"title":"The Size Premise Forces a Collision but Has No General Reverse Inference","stem":"Use complete finite domain size=12≥0 and codomain size=10≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(12, 10, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation":"Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (12, 10, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)."},"skills":["The Size Premise Forces a Collision but Has No General Reverse Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0059","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6522d076975e315a12d328454e585c2c1dfdccb2b4a5f1db0199a5d2b4c0fe1a","family_id":"qf-mod38-pigeon-premise-audit","seed":202611075059,"semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":11,\"domain_count\":13}"},"fields":{"title":"The Size Premise Forces a Collision but Has No General Reverse Inference","stem":"Use complete finite domain size=13≥0 and codomain size=11≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(13, 11, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation":"Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (13, 11, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)."},"skills":["The Size Premise Forces a Collision but Has No General Reverse Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0060","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1b1577d4bc74b312ec0853a8ae0dec26d8d0fc5a8a7c18de6d48de3c0eb7bef","family_id":"qf-mod38-pigeon-premise-audit","seed":202611075060,"semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":12,\"domain_count\":14}"},"fields":{"title":"The Size Premise Forces a Collision but Has No General Reverse Inference","stem":"Use complete finite domain size=14≥0 and codomain size=12≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(14, 12, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation":"Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (14, 12, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)."},"skills":["The Size Premise Forces a Collision but Has No General Reverse Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0061","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7877f3c9a7bb376a6248cbb2114cd62745cf523e3da47535c7d0f8f28da3b1b9","family_id":"qf-mod38-pigeon-premise-audit","seed":202611075061,"semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":13,\"domain_count\":15}"},"fields":{"title":"The Size Premise Forces a Collision but Has No General Reverse Inference","stem":"Use complete finite domain size=15≥0 and codomain size=13≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(15, 13, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation":"Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (15, 13, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)."},"skills":["The Size Premise Forces a Collision but Has No General Reverse Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0062","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"873bf296987475513b9261fb02646a8ef6089eed7040af16b51a70c353f2e992","family_id":"qf-mod38-pigeon-premise-audit","seed":202611075062,"semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":14,\"domain_count\":16}"},"fields":{"title":"The Size Premise Forces a Collision but Has No General Reverse Inference","stem":"Use complete finite domain size=16≥0 and codomain size=14≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(16, 14, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation":"Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (16, 14, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)."},"skills":["The Size Premise Forces a Collision but Has No General Reverse Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0063","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"80919acd1e69d4fbcdd0ab3b56407d63e20b4aa5149ccfbd353812ee8b64e951","family_id":"qf-mod38-pigeon-premise-audit","seed":202611075063,"semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":15,\"domain_count\":17}"},"fields":{"title":"The Size Premise Forces a Collision but Has No General Reverse Inference","stem":"Use complete finite domain size=17≥0 and codomain size=15≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(17, 15, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation":"Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (17, 15, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)."},"skills":["The Size Premise Forces a Collision but Has No General Reverse Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0064","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89cba803f4430ee89ed5a3ccab642f1e73c2a27410325c6aaffbc71e5ef7b06a","family_id":"qf-mod38-pigeon-premise-audit","seed":202611075064,"semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":16,\"domain_count\":18}"},"fields":{"title":"The Size Premise Forces a Collision but Has No General Reverse Inference","stem":"Use complete finite domain size=18≥0 and codomain size=16≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(18, 16, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation":"Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (18, 16, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)."},"skills":["The Size Premise Forces a Collision but Has No General Reverse Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0065","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd96d35555ab8ba4a7b7effdeaed963db49e36935eb26220b99f220aa4e96adf","family_id":"qf-mod38-pigeon-premise-audit","seed":202611075065,"semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":17,\"domain_count\":19}"},"fields":{"title":"The Size Premise Forces a Collision but Has No General Reverse Inference","stem":"Use complete finite domain size=19≥0 and codomain size=17≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(19, 17, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation":"Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (19, 17, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)."},"skills":["The Size Premise Forces a Collision but Has No General Reverse Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0066","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"589c5098b91b8b7c27e0214bd0ed7efb9e4a7039fb2995ee82ab2e15911ebd96","family_id":"qf-mod38-pigeon-premise-audit","seed":202611075066,"semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":18,\"domain_count\":20}"},"fields":{"title":"The Size Premise Forces a Collision but Has No General Reverse Inference","stem":"Use complete finite domain size=20≥0 and codomain size=18≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(20, 18, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation":"Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (20, 18, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)."},"skills":["The Size Premise Forces a Collision but Has No General Reverse Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0067","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be95e8e339132b0d38c6562b8232203925e02b7734004635c93106370d794ffb","family_id":"qf-mod38-pigeon-premise-audit","seed":202611075067,"semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":19,\"domain_count\":21}"},"fields":{"title":"The Size Premise Forces a Collision but Has No General Reverse Inference","stem":"Use complete finite domain size=21≥0 and codomain size=19≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(21, 19, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation":"Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (21, 19, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)."},"skills":["The Size Premise Forces a Collision but Has No General Reverse Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0068","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0991c880b245d153229f0229a76a90b671f49523b388b7d2db850f5671add825","family_id":"qf-mod38-pigeon-premise-audit","seed":202611075068,"semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":20,\"domain_count\":22}"},"fields":{"title":"The Size Premise Forces a Collision but Has No General Reverse Inference","stem":"Use complete finite domain size=22≥0 and codomain size=20≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(22, 20, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation":"Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (22, 20, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)."},"skills":["The Size Premise Forces a Collision but Has No General Reverse Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0069","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ef1bcdea67a07e3c54a6139f94aa6c1ba17b666631cf6fa0721c43827b7eb8e","family_id":"qf-mod38-pigeon-premise-audit","seed":202611075069,"semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":21,\"domain_count\":23}"},"fields":{"title":"The Size Premise Forces a Collision but Has No General Reverse Inference","stem":"Use complete finite domain size=23≥0 and codomain size=21≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(23, 21, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation":"Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (23, 21, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)."},"skills":["The Size Premise Forces a Collision but Has No General Reverse Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0070","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aedcf9c6229ff536fcde42b3722d86170e2445a658b184c318f8cb910933f9ea","family_id":"qf-mod38-pigeon-premise-audit","seed":202611075070,"semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":22,\"domain_count\":24}"},"fields":{"title":"The Size Premise Forces a Collision but Has No General Reverse Inference","stem":"Use complete finite domain size=24≥0 and codomain size=22≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(24, 22, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation":"Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (24, 22, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)."},"skills":["The Size Premise Forces a Collision but Has No General Reverse Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0071","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8aa756c973dcac95304a4e7a76041083948fa821b93adcb61308059605930dbf","family_id":"qf-mod38-pigeon-premise-audit","seed":202611075071,"semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":23,\"domain_count\":25}"},"fields":{"title":"The Size Premise Forces a Collision but Has No General Reverse Inference","stem":"Use complete finite domain size=25≥0 and codomain size=23≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(25, 23, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation":"Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (25, 23, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)."},"skills":["The Size Premise Forces a Collision but Has No General Reverse Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0072","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dabc11110cbe5220e603992d9ed3f272c4cdcd1914988b5f9765084c6c364cbe","family_id":"qf-mod38-pigeon-premise-audit","seed":202611075072,"semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":24,\"domain_count\":26}"},"fields":{"title":"The Size Premise Forces a Collision but Has No General Reverse Inference","stem":"Use complete finite domain size=26≥0 and codomain size=24≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(26, 24, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation":"Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (26, 24, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)."},"skills":["The Size Premise Forces a Collision but Has No General Reverse Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0073","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73564fdc071769fa792f6ca41e06fa4b9ef3705d4947e2a1b17df6db0ec98c3b","family_id":"qf-mod38-pigeon-premise-audit","seed":202611075073,"semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":25,\"domain_count\":27}"},"fields":{"title":"The Size Premise Forces a Collision but Has No General Reverse Inference","stem":"Use complete finite domain size=27≥0 and codomain size=25≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(27, 25, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation":"Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (27, 25, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)."},"skills":["The Size Premise Forces a Collision but Has No General Reverse Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0074","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ffc566b249ec16112b013aabff3f4086670645346cfb9a24952c207aaecda24","family_id":"qf-mod38-pigeon-premise-audit","seed":202611075074,"semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":26,\"domain_count\":28}"},"fields":{"title":"The Size Premise Forces a Collision but Has No General Reverse Inference","stem":"Use complete finite domain size=28≥0 and codomain size=26≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(28, 26, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation":"Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (28, 26, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)."},"skills":["The Size Premise Forces a Collision but Has No General Reverse Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0075","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3d70ff5975d9c08151063ceae0c3587454a81d520d3a24a2e1858dd3f782275","family_id":"qf-mod38-pigeon-premise-audit","seed":202611075075,"semantic_template":"qf-mod38-pigeon-premise-audit/{\"codomain_count\":27,\"domain_count\":29}"},"fields":{"title":"The Size Premise Forces a Collision but Has No General Reverse Inference","stem":"Use complete finite domain size=29≥0 and codomain size=27≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(29, 27, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)","explanation":"Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (29, 27, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)."},"skills":["The Size Premise Forces a Collision but Has No General Reverse Inference"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0076","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d262bb0978840748278c02fdc8c19e2ba3e2b84238920c191176eca14f2d498d","family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075076,"semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":4,\"mapping\":[0,0,2]}"},"fields":{"title":"A Smaller Domain Does Not Generally Decide the Given Map Is Injective","stem":"Use the complete total mapping=[0, 0, 2] from source indices Fin(3)={0,…,2} into Fin(4)={0,…,3}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(3, 4, No, No, size_condition_failure_does_not_imply_given_map_is_injective)","explanation":"Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (3, 4, No, No, size_condition_failure_does_not_imply_given_map_is_injective)."},"skills":["A Smaller Domain Does Not Generally Decide the Given Map Is Injective"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0077","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f9add5eeb9abb7eff03a4d2afe9e2ae0765cb0f8a51f74462fd9d9ace72dcb73","family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075077,"semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":5,\"mapping\":[0,0,2,3]}"},"fields":{"title":"A Smaller Domain Does Not Generally Decide the Given Map Is Injective","stem":"Use the complete total mapping=[0, 0, 2, 3] from source indices Fin(4)={0,…,3} into Fin(5)={0,…,4}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(4, 5, No, No, size_condition_failure_does_not_imply_given_map_is_injective)","explanation":"Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (4, 5, No, No, size_condition_failure_does_not_imply_given_map_is_injective)."},"skills":["A Smaller Domain Does Not Generally Decide the Given Map Is Injective"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0078","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"659e710912b910f7ef0c23528febb3b7ea3953df8c5df09aea2423101abb2f70","family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075078,"semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":6,\"mapping\":[0,0,2,3,4]}"},"fields":{"title":"A Smaller Domain Does Not Generally Decide the Given Map Is Injective","stem":"Use the complete total mapping=[0, 0, 2, 3, 4] from source indices Fin(5)={0,…,4} into Fin(6)={0,…,5}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(5, 6, No, No, size_condition_failure_does_not_imply_given_map_is_injective)","explanation":"Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (5, 6, No, No, size_condition_failure_does_not_imply_given_map_is_injective)."},"skills":["A Smaller Domain Does Not Generally Decide the Given Map Is Injective"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0079","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35e4ce80c4643e30eae38d99718071be3a4b26611838dc9696fe75092c3a36b0","family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075079,"semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":7,\"mapping\":[0,0,2,3,4,5]}"},"fields":{"title":"A Smaller Domain Does Not Generally Decide the Given Map Is Injective","stem":"Use the complete total mapping=[0, 0, 2, 3, 4, 5] from source indices Fin(6)={0,…,5} into Fin(7)={0,…,6}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(6, 7, No, No, size_condition_failure_does_not_imply_given_map_is_injective)","explanation":"Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (6, 7, No, No, size_condition_failure_does_not_imply_given_map_is_injective)."},"skills":["A Smaller Domain Does Not Generally Decide the Given Map Is Injective"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0080","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52ac3933627ef7bad612545b18548aa3c5e2a7840ab78a158eda28e83d996aae","family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075080,"semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":8,\"mapping\":[0,0,2,3,4,5,6]}"},"fields":{"title":"A Smaller Domain Does Not Generally Decide the Given Map Is Injective","stem":"Use the complete total mapping=[0, 0, 2, 3, 4, 5, 6] from source indices Fin(7)={0,…,6} into Fin(8)={0,…,7}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(7, 8, No, No, size_condition_failure_does_not_imply_given_map_is_injective)","explanation":"Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (7, 8, No, No, size_condition_failure_does_not_imply_given_map_is_injective)."},"skills":["A Smaller Domain Does Not Generally Decide the Given Map Is Injective"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0081","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d66508d21f4f3d45f4aa28d24a319e6ac6bb369051e1ce616022b5262b613683","family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075081,"semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":5,\"mapping\":[0,0,2]}"},"fields":{"title":"A Smaller Domain Does Not Generally Decide the Given Map Is Injective","stem":"Use the complete total mapping=[0, 0, 2] from source indices Fin(3)={0,…,2} into Fin(5)={0,…,4}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(3, 5, No, No, size_condition_failure_does_not_imply_given_map_is_injective)","explanation":"Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (3, 5, No, No, size_condition_failure_does_not_imply_given_map_is_injective)."},"skills":["A Smaller Domain Does Not Generally Decide the Given Map Is Injective"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0082","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3742aaa7a84e682dd6c639182e9be74cf6c3644d2a284c6da9c73c20e218d38d","family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075082,"semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":6,\"mapping\":[0,0,2,3]}"},"fields":{"title":"A Smaller Domain Does Not Generally Decide the Given Map Is Injective","stem":"Use the complete total mapping=[0, 0, 2, 3] from source indices Fin(4)={0,…,3} into Fin(6)={0,…,5}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(4, 6, No, No, size_condition_failure_does_not_imply_given_map_is_injective)","explanation":"Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (4, 6, No, No, size_condition_failure_does_not_imply_given_map_is_injective)."},"skills":["A Smaller Domain Does Not Generally Decide the Given Map Is Injective"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0083","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"10b8b87f53fbcf5adedcb5e077931b11d0a6a0cde4069c9980cca833acde9662","family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075083,"semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":7,\"mapping\":[0,0,2,3,4]}"},"fields":{"title":"A Smaller Domain Does Not Generally Decide the Given Map Is Injective","stem":"Use the complete total mapping=[0, 0, 2, 3, 4] from source indices Fin(5)={0,…,4} into Fin(7)={0,…,6}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(5, 7, No, No, size_condition_failure_does_not_imply_given_map_is_injective)","explanation":"Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (5, 7, No, No, size_condition_failure_does_not_imply_given_map_is_injective)."},"skills":["A Smaller Domain Does Not Generally Decide the Given Map Is Injective"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0084","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0672e5842840a5763c902aca2c732f388ba0ab9366b4d96f5239bd7370538d10","family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075084,"semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":8,\"mapping\":[0,0,2,3,4,5]}"},"fields":{"title":"A Smaller Domain Does Not Generally Decide the Given Map Is Injective","stem":"Use the complete total mapping=[0, 0, 2, 3, 4, 5] from source indices Fin(6)={0,…,5} into Fin(8)={0,…,7}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(6, 8, No, No, size_condition_failure_does_not_imply_given_map_is_injective)","explanation":"Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (6, 8, No, No, size_condition_failure_does_not_imply_given_map_is_injective)."},"skills":["A Smaller Domain Does Not Generally Decide the Given Map Is Injective"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0085","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf928246a739835193a3a6af3abd9849ee85a55d97097a83f5f233019962b99b","family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075085,"semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":9,\"mapping\":[0,0,2,3,4,5,6]}"},"fields":{"title":"A Smaller Domain Does Not Generally Decide the Given Map Is Injective","stem":"Use the complete total mapping=[0, 0, 2, 3, 4, 5, 6] from source indices Fin(7)={0,…,6} into Fin(9)={0,…,8}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(7, 9, No, No, size_condition_failure_does_not_imply_given_map_is_injective)","explanation":"Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (7, 9, No, No, size_condition_failure_does_not_imply_given_map_is_injective)."},"skills":["A Smaller Domain Does Not Generally Decide the Given Map Is Injective"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0086","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e477a148ec4a1cc6661b95ea0efa30a25dd1f19dd7c9337b1bedb9b7dd829e40","family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075086,"semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":6,\"mapping\":[0,0,2]}"},"fields":{"title":"A Smaller Domain Does Not Generally Decide the Given Map Is Injective","stem":"Use the complete total mapping=[0, 0, 2] from source indices Fin(3)={0,…,2} into Fin(6)={0,…,5}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(3, 6, No, No, size_condition_failure_does_not_imply_given_map_is_injective)","explanation":"Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (3, 6, No, No, size_condition_failure_does_not_imply_given_map_is_injective)."},"skills":["A Smaller Domain Does Not Generally Decide the Given Map Is Injective"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0087","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d590d943ab41a052d03062ae4682bbdffd390687571881e463ac657439c8a69","family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075087,"semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":7,\"mapping\":[0,0,2,3]}"},"fields":{"title":"A Smaller Domain Does Not Generally Decide the Given Map Is Injective","stem":"Use the complete total mapping=[0, 0, 2, 3] from source indices Fin(4)={0,…,3} into Fin(7)={0,…,6}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(4, 7, No, No, size_condition_failure_does_not_imply_given_map_is_injective)","explanation":"Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (4, 7, No, No, size_condition_failure_does_not_imply_given_map_is_injective)."},"skills":["A Smaller Domain Does Not Generally Decide the Given Map Is Injective"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0088","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d24315117ee2d56e92aa5652542c5afca0ab5e75bcfdae9c47656f0de277d48","family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075088,"semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":8,\"mapping\":[0,0,2,3,4]}"},"fields":{"title":"A Smaller Domain Does Not Generally Decide the Given Map Is Injective","stem":"Use the complete total mapping=[0, 0, 2, 3, 4] from source indices Fin(5)={0,…,4} into Fin(8)={0,…,7}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(5, 8, No, No, size_condition_failure_does_not_imply_given_map_is_injective)","explanation":"Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (5, 8, No, No, size_condition_failure_does_not_imply_given_map_is_injective)."},"skills":["A Smaller Domain Does Not Generally Decide the Given Map Is Injective"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0089","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e76a07758278ab8683d63b11e76645b9285b4e66b3201ff419cb954aa163c3f1","family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075089,"semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":9,\"mapping\":[0,0,2,3,4,5]}"},"fields":{"title":"A Smaller Domain Does Not Generally Decide the Given Map Is Injective","stem":"Use the complete total mapping=[0, 0, 2, 3, 4, 5] from source indices Fin(6)={0,…,5} into Fin(9)={0,…,8}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(6, 9, No, No, size_condition_failure_does_not_imply_given_map_is_injective)","explanation":"Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (6, 9, No, No, size_condition_failure_does_not_imply_given_map_is_injective)."},"skills":["A Smaller Domain Does Not Generally Decide the Given Map Is Injective"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0090","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79e9e8f8b6d2b4ecc019bd341cf50f0936956d9ca97787db51fef0d7f2305c34","family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075090,"semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":10,\"mapping\":[0,0,2,3,4,5,6]}"},"fields":{"title":"A Smaller Domain Does Not Generally Decide the Given Map Is Injective","stem":"Use the complete total mapping=[0, 0, 2, 3, 4, 5, 6] from source indices Fin(7)={0,…,6} into Fin(10)={0,…,9}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(7, 10, No, No, size_condition_failure_does_not_imply_given_map_is_injective)","explanation":"Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (7, 10, No, No, size_condition_failure_does_not_imply_given_map_is_injective)."},"skills":["A Smaller Domain Does Not Generally Decide the Given Map Is Injective"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0091","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d21b6d6b6dec8db70379e6a386c1a49cdc820292e6ad9fdf15aa6c1c3738389","family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075091,"semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":7,\"mapping\":[0,0,2]}"},"fields":{"title":"A Smaller Domain Does Not Generally Decide the Given Map Is Injective","stem":"Use the complete total mapping=[0, 0, 2] from source indices Fin(3)={0,…,2} into Fin(7)={0,…,6}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(3, 7, No, No, size_condition_failure_does_not_imply_given_map_is_injective)","explanation":"Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (3, 7, No, No, size_condition_failure_does_not_imply_given_map_is_injective)."},"skills":["A Smaller Domain Does Not Generally Decide the Given Map Is Injective"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0092","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b19eed8b026cc6cde62d49b9feb6859cd840f1474351ab82747255e8098737c8","family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075092,"semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":8,\"mapping\":[0,0,2,3]}"},"fields":{"title":"A Smaller Domain Does Not Generally Decide the Given Map Is Injective","stem":"Use the complete total mapping=[0, 0, 2, 3] from source indices Fin(4)={0,…,3} into Fin(8)={0,…,7}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(4, 8, No, No, size_condition_failure_does_not_imply_given_map_is_injective)","explanation":"Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (4, 8, No, No, size_condition_failure_does_not_imply_given_map_is_injective)."},"skills":["A Smaller Domain Does Not Generally Decide the Given Map Is Injective"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0093","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5763c042f5b99145a0f663ef29b70af8e0fa6b0f685ca219c0bf699f2da4dd5f","family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075093,"semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":9,\"mapping\":[0,0,2,3,4]}"},"fields":{"title":"A Smaller Domain Does Not Generally Decide the Given Map Is Injective","stem":"Use the complete total mapping=[0, 0, 2, 3, 4] from source indices Fin(5)={0,…,4} into Fin(9)={0,…,8}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(5, 9, No, No, size_condition_failure_does_not_imply_given_map_is_injective)","explanation":"Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (5, 9, No, No, size_condition_failure_does_not_imply_given_map_is_injective)."},"skills":["A Smaller Domain Does Not Generally Decide the Given Map Is Injective"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0094","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6bc93eb0e9deed9636c3e143f446c208c62fa24740828bad4485b29fff9467e6","family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075094,"semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":10,\"mapping\":[0,0,2,3,4,5]}"},"fields":{"title":"A Smaller Domain Does Not Generally Decide the Given Map Is Injective","stem":"Use the complete total mapping=[0, 0, 2, 3, 4, 5] from source indices Fin(6)={0,…,5} into Fin(10)={0,…,9}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(6, 10, No, No, size_condition_failure_does_not_imply_given_map_is_injective)","explanation":"Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (6, 10, No, No, size_condition_failure_does_not_imply_given_map_is_injective)."},"skills":["A Smaller Domain Does Not Generally Decide the Given Map Is Injective"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0095","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd40c1837b783dd43f0d998fefe513db05aafb9b83092761d4174ca59554f0a8","family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075095,"semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":11,\"mapping\":[0,0,2,3,4,5,6]}"},"fields":{"title":"A Smaller Domain Does Not Generally Decide the Given Map Is Injective","stem":"Use the complete total mapping=[0, 0, 2, 3, 4, 5, 6] from source indices Fin(7)={0,…,6} into Fin(11)={0,…,10}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(7, 11, No, No, size_condition_failure_does_not_imply_given_map_is_injective)","explanation":"Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (7, 11, No, No, size_condition_failure_does_not_imply_given_map_is_injective)."},"skills":["A Smaller Domain Does Not Generally Decide the Given Map Is Injective"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0096","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68c0bb63818f42967db1c7f0329b8d4645740c12388e47e73ac4e1c74a94dab5","family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075096,"semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":8,\"mapping\":[0,0,2]}"},"fields":{"title":"A Smaller Domain Does Not Generally Decide the Given Map Is Injective","stem":"Use the complete total mapping=[0, 0, 2] from source indices Fin(3)={0,…,2} into Fin(8)={0,…,7}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(3, 8, No, No, size_condition_failure_does_not_imply_given_map_is_injective)","explanation":"Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (3, 8, No, No, size_condition_failure_does_not_imply_given_map_is_injective)."},"skills":["A Smaller Domain Does Not Generally Decide the Given Map Is Injective"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0097","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"870764eb1e0ce863869084f777dc8f75e6284355973e48e8957a5edcf85586a0","family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075097,"semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":9,\"mapping\":[0,0,2,3]}"},"fields":{"title":"A Smaller Domain Does Not Generally Decide the Given Map Is Injective","stem":"Use the complete total mapping=[0, 0, 2, 3] from source indices Fin(4)={0,…,3} into Fin(9)={0,…,8}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(4, 9, No, No, size_condition_failure_does_not_imply_given_map_is_injective)","explanation":"Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (4, 9, No, No, size_condition_failure_does_not_imply_given_map_is_injective)."},"skills":["A Smaller Domain Does Not Generally Decide the Given Map Is Injective"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0098","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"915e2caf401aa84b062156b5fb0bb25af86c994be229258d0ca371333460b543","family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075098,"semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":10,\"mapping\":[0,0,2,3,4]}"},"fields":{"title":"A Smaller Domain Does Not Generally Decide the Given Map Is Injective","stem":"Use the complete total mapping=[0, 0, 2, 3, 4] from source indices Fin(5)={0,…,4} into Fin(10)={0,…,9}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(5, 10, No, No, size_condition_failure_does_not_imply_given_map_is_injective)","explanation":"Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (5, 10, No, No, size_condition_failure_does_not_imply_given_map_is_injective)."},"skills":["A Smaller Domain Does Not Generally Decide the Given Map Is Injective"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0099","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2b5d26c0bf0500c46c4df28e2f097e8762226e4eb40c459de7a690e53826fab","family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075099,"semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":11,\"mapping\":[0,0,2,3,4,5]}"},"fields":{"title":"A Smaller Domain Does Not Generally Decide the Given Map Is Injective","stem":"Use the complete total mapping=[0, 0, 2, 3, 4, 5] from source indices Fin(6)={0,…,5} into Fin(11)={0,…,10}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(6, 11, No, No, size_condition_failure_does_not_imply_given_map_is_injective)","explanation":"Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (6, 11, No, No, size_condition_failure_does_not_imply_given_map_is_injective)."},"skills":["A Smaller Domain Does Not Generally Decide the Given Map Is Injective"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261107-r05-pigeonhole-principle-0100","batch_id":"batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f90a4f2b897bc02d4e9788547f0ef359be0748132f6e48049839e5183790240b","family_id":"qf-mod38-pigeon-converse-boundary","seed":202611075100,"semantic_template":"qf-mod38-pigeon-converse-boundary/{\"codomain_count\":12,\"mapping\":[0,0,2,3,4,5,6]}"},"fields":{"title":"A Smaller Domain Does Not Generally Decide the Given Map Is Injective","stem":"Use the complete total mapping=[0, 0, 2, 3, 4, 5, 6] from source indices Fin(7)={0,…,6} into Fin(12)={0,…,11}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).","answer":{"kind":"expression","value":"(7, 12, No, No, size_condition_failure_does_not_imply_given_map_is_injective)","explanation":"Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (7, 12, No, No, size_condition_failure_does_not_imply_given_map_is_injective)."},"skills":["A Smaller Domain Does Not Generally Decide the Given Map Is Injective"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}