{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0001","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"87b2933ba3a2d351ed36afaff8e6fa475d0ebee7fbb4e80ef7d9fb49d3bac891","family_id":"qf-density12-counting-disjoint-choice-sizes","seed":202610122001,"semantic_template":"qf-density12-counting-disjoint-choice-sizes/[]"},"fields":{"title":"Disjoint Tagged Choices and Prefix Counts","stem":"Mutually exclusive tagged categories have ordered sizes=[]. Give (total choices, each prefix’s total), retaining an empty list.","answer":{"kind":"expression","value":"(0, [])","explanation":"Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (0, [])."},"skills":["Disjoint Tagged Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0002","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"519dbd6b92defc1446fe79ce1b70fb863b8d171d80693373099bf9251f30ee43","family_id":"qf-density12-counting-disjoint-choice-sizes","seed":202610122002,"semantic_template":"qf-density12-counting-disjoint-choice-sizes/[0]"},"fields":{"title":"Disjoint Tagged Choices and Prefix Counts","stem":"Mutually exclusive tagged categories have ordered sizes=[0]. Give (total choices, each prefix’s total), retaining an empty list.","answer":{"kind":"expression","value":"(0, [0])","explanation":"Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (0, [0])."},"skills":["Disjoint Tagged Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0003","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"451642d64c668cfa5998722b1f1835ac3a6aad08cc46fca2f8750d25890fa245","family_id":"qf-density12-counting-disjoint-choice-sizes","seed":202610122003,"semantic_template":"qf-density12-counting-disjoint-choice-sizes/[1]"},"fields":{"title":"Disjoint Tagged Choices and Prefix Counts","stem":"Mutually exclusive tagged categories have ordered sizes=[1]. Give (total choices, each prefix’s total), retaining an empty list.","answer":{"kind":"expression","value":"(1, [1])","explanation":"Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (1, [1])."},"skills":["Disjoint Tagged Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0004","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0288db5d49c98e0d9f8fd8851b2d616a70a1c54841453313bf65b9497e067de","family_id":"qf-density12-counting-disjoint-choice-sizes","seed":202610122004,"semantic_template":"qf-density12-counting-disjoint-choice-sizes/[2,0,3,1]"},"fields":{"title":"Disjoint Tagged Choices and Prefix Counts","stem":"Mutually exclusive tagged categories have ordered sizes=[2, 0, 3, 1]. Give (total choices, each prefix’s total), retaining an empty list.","answer":{"kind":"expression","value":"(6, [2, 2, 5, 6])","explanation":"Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (6, [2, 2, 5, 6])."},"skills":["Disjoint Tagged Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0005","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"299495f0a5ac15037d1063b36659992831d686235816a9923774e5ec457f57cc","family_id":"qf-density12-counting-disjoint-choice-sizes","seed":202610122005,"semantic_template":"qf-density12-counting-disjoint-choice-sizes/[1,1]"},"fields":{"title":"Disjoint Tagged Choices and Prefix Counts","stem":"Mutually exclusive tagged categories have ordered sizes=[1, 1]. Give (total choices, each prefix’s total), retaining an empty list.","answer":{"kind":"expression","value":"(2, [1, 2])","explanation":"Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (2, [1, 2])."},"skills":["Disjoint Tagged Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0006","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5fd0d23c7733d2a11f6e569195e9e50da8b77477647bdc982dc3ba876b3a073e","family_id":"qf-density12-counting-disjoint-choice-sizes","seed":202610122006,"semantic_template":"qf-density12-counting-disjoint-choice-sizes/[1,2]"},"fields":{"title":"Disjoint Tagged Choices and Prefix Counts","stem":"Mutually exclusive tagged categories have ordered sizes=[1, 2]. Give (total choices, each prefix’s total), retaining an empty list.","answer":{"kind":"expression","value":"(3, [1, 3])","explanation":"Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (3, [1, 3])."},"skills":["Disjoint Tagged Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0007","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f48c640ce6d5b3d71775fed479debfde83e86a106bc07ea711837684ea8dee7c","family_id":"qf-density12-counting-disjoint-choice-sizes","seed":202610122007,"semantic_template":"qf-density12-counting-disjoint-choice-sizes/[1,3]"},"fields":{"title":"Disjoint Tagged Choices and Prefix Counts","stem":"Mutually exclusive tagged categories have ordered sizes=[1, 3]. Give (total choices, each prefix’s total), retaining an empty list.","answer":{"kind":"expression","value":"(4, [1, 4])","explanation":"Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (4, [1, 4])."},"skills":["Disjoint Tagged Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0008","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07656c6b3bce231a204f98ad4a0978ad122258a15307ed22a33ea0614795c9c0","family_id":"qf-density12-counting-disjoint-choice-sizes","seed":202610122008,"semantic_template":"qf-density12-counting-disjoint-choice-sizes/[1,4]"},"fields":{"title":"Disjoint Tagged Choices and Prefix Counts","stem":"Mutually exclusive tagged categories have ordered sizes=[1, 4]. Give (total choices, each prefix’s total), retaining an empty list.","answer":{"kind":"expression","value":"(5, [1, 5])","explanation":"Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (5, [1, 5])."},"skills":["Disjoint Tagged Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0009","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1afa73383321bb8b580c8edf07f7b0a0ee31f589e7adb8ec6ff4363a6068030","family_id":"qf-density12-counting-disjoint-choice-sizes","seed":202610122009,"semantic_template":"qf-density12-counting-disjoint-choice-sizes/[1,5]"},"fields":{"title":"Disjoint Tagged Choices and Prefix Counts","stem":"Mutually exclusive tagged categories have ordered sizes=[1, 5]. Give (total choices, each prefix’s total), retaining an empty list.","answer":{"kind":"expression","value":"(6, [1, 6])","explanation":"Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (6, [1, 6])."},"skills":["Disjoint Tagged Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0010","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61b2627241f5aca02549c48d618b7e9fae218bd8a007b9b74eb3c35ccf33fa7b","family_id":"qf-density12-counting-disjoint-choice-sizes","seed":202610122010,"semantic_template":"qf-density12-counting-disjoint-choice-sizes/[1,6]"},"fields":{"title":"Disjoint Tagged Choices and Prefix Counts","stem":"Mutually exclusive tagged categories have ordered sizes=[1, 6]. Give (total choices, each prefix’s total), retaining an empty list.","answer":{"kind":"expression","value":"(7, [1, 7])","explanation":"Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (7, [1, 7])."},"skills":["Disjoint Tagged Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0011","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a5d7d27e6e70616a0e2cca6d44be419d50298378b546ea5941bcc3f52e16ca3","family_id":"qf-density12-counting-disjoint-choice-sizes","seed":202610122011,"semantic_template":"qf-density12-counting-disjoint-choice-sizes/[1,7]"},"fields":{"title":"Disjoint Tagged Choices and Prefix Counts","stem":"Mutually exclusive tagged categories have ordered sizes=[1, 7]. Give (total choices, each prefix’s total), retaining an empty list.","answer":{"kind":"expression","value":"(8, [1, 8])","explanation":"Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (8, [1, 8])."},"skills":["Disjoint Tagged Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0012","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ec9e7b4701239a0ad92011c818c02e7d13fe96c4a648aec27c4ace93b82b654","family_id":"qf-density12-counting-disjoint-choice-sizes","seed":202610122012,"semantic_template":"qf-density12-counting-disjoint-choice-sizes/[1,8]"},"fields":{"title":"Disjoint Tagged Choices and Prefix Counts","stem":"Mutually exclusive tagged categories have ordered sizes=[1, 8]. Give (total choices, each prefix’s total), retaining an empty list.","answer":{"kind":"expression","value":"(9, [1, 9])","explanation":"Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (9, [1, 9])."},"skills":["Disjoint Tagged Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0013","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8006ae2da3dab69062057b5e4c9aaccf8ba6af4bb2f3d1b607af5378a22d8e31","family_id":"qf-density12-counting-disjoint-choice-sizes","seed":202610122013,"semantic_template":"qf-density12-counting-disjoint-choice-sizes/[2,1]"},"fields":{"title":"Disjoint Tagged Choices and Prefix Counts","stem":"Mutually exclusive tagged categories have ordered sizes=[2, 1]. Give (total choices, each prefix’s total), retaining an empty list.","answer":{"kind":"expression","value":"(3, [2, 3])","explanation":"Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (3, [2, 3])."},"skills":["Disjoint Tagged Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0014","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a99430b31b4207b940654442313ccb0e4f2ae1c9058fc9898eaf984290339dc2","family_id":"qf-density12-counting-disjoint-choice-sizes","seed":202610122014,"semantic_template":"qf-density12-counting-disjoint-choice-sizes/[2,2]"},"fields":{"title":"Disjoint Tagged Choices and Prefix Counts","stem":"Mutually exclusive tagged categories have ordered sizes=[2, 2]. Give (total choices, each prefix’s total), retaining an empty list.","answer":{"kind":"expression","value":"(4, [2, 4])","explanation":"Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (4, [2, 4])."},"skills":["Disjoint Tagged Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0015","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4c74e6a0beaf82ecb93e7228f17285793daa2e8790b347fbdd49bbee08dd0d6","family_id":"qf-density12-counting-disjoint-choice-sizes","seed":202610122015,"semantic_template":"qf-density12-counting-disjoint-choice-sizes/[2,3]"},"fields":{"title":"Disjoint Tagged Choices and Prefix Counts","stem":"Mutually exclusive tagged categories have ordered sizes=[2, 3]. Give (total choices, each prefix’s total), retaining an empty list.","answer":{"kind":"expression","value":"(5, [2, 5])","explanation":"Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (5, [2, 5])."},"skills":["Disjoint Tagged Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0016","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b101129a9da39b6a4d70070f7ef0eea529381081b3d4254c7c363b19ccf86098","family_id":"qf-density12-counting-disjoint-choice-sizes","seed":202610122016,"semantic_template":"qf-density12-counting-disjoint-choice-sizes/[2,4]"},"fields":{"title":"Disjoint Tagged Choices and Prefix Counts","stem":"Mutually exclusive tagged categories have ordered sizes=[2, 4]. Give (total choices, each prefix’s total), retaining an empty list.","answer":{"kind":"expression","value":"(6, [2, 6])","explanation":"Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (6, [2, 6])."},"skills":["Disjoint Tagged Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0017","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"816af48736f7b4f8091d8616d126db1ee4c840589d597c851f7ee636da3d8cea","family_id":"qf-density12-counting-disjoint-choice-sizes","seed":202610122017,"semantic_template":"qf-density12-counting-disjoint-choice-sizes/[2,5]"},"fields":{"title":"Disjoint Tagged Choices and Prefix Counts","stem":"Mutually exclusive tagged categories have ordered sizes=[2, 5]. Give (total choices, each prefix’s total), retaining an empty list.","answer":{"kind":"expression","value":"(7, [2, 7])","explanation":"Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (7, [2, 7])."},"skills":["Disjoint Tagged Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0018","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dfb4750712e75e61d6abb4b98c5c36b1830311d4024601eb944a641d00e77e37","family_id":"qf-density12-counting-disjoint-choice-sizes","seed":202610122018,"semantic_template":"qf-density12-counting-disjoint-choice-sizes/[2,6]"},"fields":{"title":"Disjoint Tagged Choices and Prefix Counts","stem":"Mutually exclusive tagged categories have ordered sizes=[2, 6]. Give (total choices, each prefix’s total), retaining an empty list.","answer":{"kind":"expression","value":"(8, [2, 8])","explanation":"Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (8, [2, 8])."},"skills":["Disjoint Tagged Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0019","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"352e18d3bf9ccb5016e03977c00db77a1fa141f999f0cd765949b09a644293a4","family_id":"qf-density12-counting-disjoint-choice-sizes","seed":202610122019,"semantic_template":"qf-density12-counting-disjoint-choice-sizes/[2,7]"},"fields":{"title":"Disjoint Tagged Choices and Prefix Counts","stem":"Mutually exclusive tagged categories have ordered sizes=[2, 7]. Give (total choices, each prefix’s total), retaining an empty list.","answer":{"kind":"expression","value":"(9, [2, 9])","explanation":"Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (9, [2, 9])."},"skills":["Disjoint Tagged Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0020","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0ff42473bba99ee471722e1511cb8d3a2aa030b57e47743bc652de9fca93a7b","family_id":"qf-density12-counting-disjoint-choice-sizes","seed":202610122020,"semantic_template":"qf-density12-counting-disjoint-choice-sizes/[2,8]"},"fields":{"title":"Disjoint Tagged Choices and Prefix Counts","stem":"Mutually exclusive tagged categories have ordered sizes=[2, 8]. Give (total choices, each prefix’s total), retaining an empty list.","answer":{"kind":"expression","value":"(10, [2, 10])","explanation":"Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (10, [2, 10])."},"skills":["Disjoint Tagged Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0021","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eac11666d4d8d1c8c0b9dd6d05d6af1f0c8abb27b82da07b1b1b8456b5547f92","family_id":"qf-density12-counting-disjoint-choice-sizes","seed":202610122021,"semantic_template":"qf-density12-counting-disjoint-choice-sizes/[3,1]"},"fields":{"title":"Disjoint Tagged Choices and Prefix Counts","stem":"Mutually exclusive tagged categories have ordered sizes=[3, 1]. Give (total choices, each prefix’s total), retaining an empty list.","answer":{"kind":"expression","value":"(4, [3, 4])","explanation":"Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (4, [3, 4])."},"skills":["Disjoint Tagged Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0022","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d0b4fe0f307528f8031304dc4dedb0a04a2d9fb443fe7d35e28568d51bb0475","family_id":"qf-density12-counting-disjoint-choice-sizes","seed":202610122022,"semantic_template":"qf-density12-counting-disjoint-choice-sizes/[3,2]"},"fields":{"title":"Disjoint Tagged Choices and Prefix Counts","stem":"Mutually exclusive tagged categories have ordered sizes=[3, 2]. Give (total choices, each prefix’s total), retaining an empty list.","answer":{"kind":"expression","value":"(5, [3, 5])","explanation":"Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (5, [3, 5])."},"skills":["Disjoint Tagged Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0023","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6feb62f676c2ff329988c24905a2ac5d8aaa51819dac6410d484d3f6934dfa8","family_id":"qf-density12-counting-disjoint-choice-sizes","seed":202610122023,"semantic_template":"qf-density12-counting-disjoint-choice-sizes/[3,3]"},"fields":{"title":"Disjoint Tagged Choices and Prefix Counts","stem":"Mutually exclusive tagged categories have ordered sizes=[3, 3]. Give (total choices, each prefix’s total), retaining an empty list.","answer":{"kind":"expression","value":"(6, [3, 6])","explanation":"Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (6, [3, 6])."},"skills":["Disjoint Tagged Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0024","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d08e4dfd9558e76ae89af4c2a6b00ee85f86bae931492463717892fbebd2e655","family_id":"qf-density12-counting-disjoint-choice-sizes","seed":202610122024,"semantic_template":"qf-density12-counting-disjoint-choice-sizes/[3,4]"},"fields":{"title":"Disjoint Tagged Choices and Prefix Counts","stem":"Mutually exclusive tagged categories have ordered sizes=[3, 4]. Give (total choices, each prefix’s total), retaining an empty list.","answer":{"kind":"expression","value":"(7, [3, 7])","explanation":"Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (7, [3, 7])."},"skills":["Disjoint Tagged Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0025","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab6a3eca2fc4f244666491af1f32c0bc0b885c444f1bf8aca680d47a5a64ea35","family_id":"qf-density12-counting-disjoint-choice-sizes","seed":202610122025,"semantic_template":"qf-density12-counting-disjoint-choice-sizes/[3,5]"},"fields":{"title":"Disjoint Tagged Choices and Prefix Counts","stem":"Mutually exclusive tagged categories have ordered sizes=[3, 5]. Give (total choices, each prefix’s total), retaining an empty list.","answer":{"kind":"expression","value":"(8, [3, 8])","explanation":"Add category sizes in order; no categories give total zero. This is a choice protocol, not a probability-independence assumption. The exact result, retaining the requested order, is (8, [3, 8])."},"skills":["Disjoint Tagged Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0026","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ab0715ff7a08a4da4311d9ca54ff3ae9adcf7cb1df43c42808651e3a5f40f31","family_id":"qf-density12-counting-full-product-prefix","seed":202610122026,"semantic_template":"qf-density12-counting-full-product-prefix/[]"},"fields":{"title":"Full Product Choices and Prefix Counts","stem":"Ordered stage sizes=[]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.","answer":{"kind":"expression","value":"(1, [])","explanation":"Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (1, [])."},"skills":["Full Product Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0027","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fba998730facce0a70bf44d5351a4a47c1349e9bd1217860d4163e4ffa97133b","family_id":"qf-density12-counting-full-product-prefix","seed":202610122027,"semantic_template":"qf-density12-counting-full-product-prefix/[0]"},"fields":{"title":"Full Product Choices and Prefix Counts","stem":"Ordered stage sizes=[0]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.","answer":{"kind":"expression","value":"(0, [0])","explanation":"Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (0, [0])."},"skills":["Full Product Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0028","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"98a3b9d41c1add4a19517cdb38ce9b2e48709d5bff0ca848b65eb28176c275e1","family_id":"qf-density12-counting-full-product-prefix","seed":202610122028,"semantic_template":"qf-density12-counting-full-product-prefix/[2,0,0]"},"fields":{"title":"Full Product Choices and Prefix Counts","stem":"Ordered stage sizes=[2, 0, 3]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.","answer":{"kind":"expression","value":"(0, [2, 0, 0])","explanation":"Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (0, [2, 0, 0])."},"skills":["Full Product Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0029","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7b611cbb2b8509fd1fe883038ffa2e7e7da195a4089929386e3fd8f3aa8ee12","family_id":"qf-density12-counting-full-product-prefix","seed":202610122029,"semantic_template":"qf-density12-counting-full-product-prefix/[0,0,0]"},"fields":{"title":"Full Product Choices and Prefix Counts","stem":"Ordered stage sizes=[0, 4, 5]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.","answer":{"kind":"expression","value":"(0, [0, 0, 0])","explanation":"Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (0, [0, 0, 0])."},"skills":["Full Product Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0030","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7c18a4cec92bdca429c912a9cd7bd215809a02cdbac214ef6524a293b96fef6","family_id":"qf-density12-counting-full-product-prefix","seed":202610122030,"semantic_template":"qf-density12-counting-full-product-prefix/[2,3,0,0]"},"fields":{"title":"Full Product Choices and Prefix Counts","stem":"Ordered stage sizes=[2, 3, 0, 8]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.","answer":{"kind":"expression","value":"(0, [2, 6, 0, 0])","explanation":"Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (0, [2, 6, 0, 0])."},"skills":["Full Product Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0031","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8638f1d0b35dd399fa834141cc08b1c66210eb73fb0fc9ae59f300e199fc3566","family_id":"qf-density12-counting-full-product-prefix","seed":202610122031,"semantic_template":"qf-density12-counting-full-product-prefix/[1,1]"},"fields":{"title":"Full Product Choices and Prefix Counts","stem":"Ordered stage sizes=[1, 1]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.","answer":{"kind":"expression","value":"(1, [1, 1])","explanation":"Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (1, [1, 1])."},"skills":["Full Product Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0032","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac0d54600d282f3379770e4e3ebf2160fe26ee6506e6a5c0c633dd206f942972","family_id":"qf-density12-counting-full-product-prefix","seed":202610122032,"semantic_template":"qf-density12-counting-full-product-prefix/[1,2]"},"fields":{"title":"Full Product Choices and Prefix Counts","stem":"Ordered stage sizes=[1, 2]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.","answer":{"kind":"expression","value":"(2, [1, 2])","explanation":"Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (2, [1, 2])."},"skills":["Full Product Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0033","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96c6579960d8d76e03adad45a81ab3c998a7b827e62c9d7a795f5f2eecf9fe40","family_id":"qf-density12-counting-full-product-prefix","seed":202610122033,"semantic_template":"qf-density12-counting-full-product-prefix/[1,3]"},"fields":{"title":"Full Product Choices and Prefix Counts","stem":"Ordered stage sizes=[1, 3]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.","answer":{"kind":"expression","value":"(3, [1, 3])","explanation":"Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (3, [1, 3])."},"skills":["Full Product Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0034","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cbb913c235083f4178c881db45cae2f8b6d15065f64a7a139e051a01aae5913e","family_id":"qf-density12-counting-full-product-prefix","seed":202610122034,"semantic_template":"qf-density12-counting-full-product-prefix/[1,4]"},"fields":{"title":"Full Product Choices and Prefix Counts","stem":"Ordered stage sizes=[1, 4]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.","answer":{"kind":"expression","value":"(4, [1, 4])","explanation":"Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (4, [1, 4])."},"skills":["Full Product Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0035","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4676de699f351f9e75b9851bfcb1b01ed9f5d209baea039a372cac43397de3fa","family_id":"qf-density12-counting-full-product-prefix","seed":202610122035,"semantic_template":"qf-density12-counting-full-product-prefix/[1,5]"},"fields":{"title":"Full Product Choices and Prefix Counts","stem":"Ordered stage sizes=[1, 5]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.","answer":{"kind":"expression","value":"(5, [1, 5])","explanation":"Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (5, [1, 5])."},"skills":["Full Product Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0036","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3764f75774746ef2c97f64ffe596f9f6ae0c3cedae0fae34838cdd050805c56","family_id":"qf-density12-counting-full-product-prefix","seed":202610122036,"semantic_template":"qf-density12-counting-full-product-prefix/[1,6]"},"fields":{"title":"Full Product Choices and Prefix Counts","stem":"Ordered stage sizes=[1, 6]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.","answer":{"kind":"expression","value":"(6, [1, 6])","explanation":"Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (6, [1, 6])."},"skills":["Full Product Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0037","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f5751968368f37b265cc84bfb6151cea79189d2b6c5e7316986950739af347a","family_id":"qf-density12-counting-full-product-prefix","seed":202610122037,"semantic_template":"qf-density12-counting-full-product-prefix/[1,7]"},"fields":{"title":"Full Product Choices and Prefix Counts","stem":"Ordered stage sizes=[1, 7]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.","answer":{"kind":"expression","value":"(7, [1, 7])","explanation":"Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (7, [1, 7])."},"skills":["Full Product Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0038","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04447fb13a0085f8eafb52d58e021ca41a3f6ff910902c7f9b110511d89af86a","family_id":"qf-density12-counting-full-product-prefix","seed":202610122038,"semantic_template":"qf-density12-counting-full-product-prefix/[1,8]"},"fields":{"title":"Full Product Choices and Prefix Counts","stem":"Ordered stage sizes=[1, 8]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.","answer":{"kind":"expression","value":"(8, [1, 8])","explanation":"Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (8, [1, 8])."},"skills":["Full Product Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0039","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a0052087a081e7935b752cf81d5fbc06ad810c93e04543a006d78dd0b01eb5f","family_id":"qf-density12-counting-full-product-prefix","seed":202610122039,"semantic_template":"qf-density12-counting-full-product-prefix/[2,1]"},"fields":{"title":"Full Product Choices and Prefix Counts","stem":"Ordered stage sizes=[2, 1]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.","answer":{"kind":"expression","value":"(2, [2, 2])","explanation":"Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (2, [2, 2])."},"skills":["Full Product Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0040","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a04d723dfff2cbfc12056f6be3906157f8e3fa965be29b7138aa80c8bdb7422","family_id":"qf-density12-counting-full-product-prefix","seed":202610122040,"semantic_template":"qf-density12-counting-full-product-prefix/[2,2]"},"fields":{"title":"Full Product Choices and Prefix Counts","stem":"Ordered stage sizes=[2, 2]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.","answer":{"kind":"expression","value":"(4, [2, 4])","explanation":"Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (4, [2, 4])."},"skills":["Full Product Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0041","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea14b8c77208876338a2c17d6e3fea474fb7842e4eacd61e740644df07e6887f","family_id":"qf-density12-counting-full-product-prefix","seed":202610122041,"semantic_template":"qf-density12-counting-full-product-prefix/[2,3]"},"fields":{"title":"Full Product Choices and Prefix Counts","stem":"Ordered stage sizes=[2, 3]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.","answer":{"kind":"expression","value":"(6, [2, 6])","explanation":"Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (6, [2, 6])."},"skills":["Full Product Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0042","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8ca55d2c608bfff0ca9d6493dfc4875dd518b74fc5c5ebd69a778199fe90271","family_id":"qf-density12-counting-full-product-prefix","seed":202610122042,"semantic_template":"qf-density12-counting-full-product-prefix/[2,4]"},"fields":{"title":"Full Product Choices and Prefix Counts","stem":"Ordered stage sizes=[2, 4]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.","answer":{"kind":"expression","value":"(8, [2, 8])","explanation":"Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (8, [2, 8])."},"skills":["Full Product Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0043","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37b3ddf433dd70701f7a64918543f76fd48f83aa85af6c0059f9110ab2cffccb","family_id":"qf-density12-counting-full-product-prefix","seed":202610122043,"semantic_template":"qf-density12-counting-full-product-prefix/[2,5]"},"fields":{"title":"Full Product Choices and Prefix Counts","stem":"Ordered stage sizes=[2, 5]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.","answer":{"kind":"expression","value":"(10, [2, 10])","explanation":"Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (10, [2, 10])."},"skills":["Full Product Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0044","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"25e6fb5f040abbd3bad38e6af237a86c0f87146781c6f7416bd3a7d78dee6aa6","family_id":"qf-density12-counting-full-product-prefix","seed":202610122044,"semantic_template":"qf-density12-counting-full-product-prefix/[2,6]"},"fields":{"title":"Full Product Choices and Prefix Counts","stem":"Ordered stage sizes=[2, 6]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.","answer":{"kind":"expression","value":"(12, [2, 12])","explanation":"Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (12, [2, 12])."},"skills":["Full Product Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0045","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"945987c50459a23ebbeedec048bf3b0a0e0796dc54fdf0016b0d5b7fa8e2d290","family_id":"qf-density12-counting-full-product-prefix","seed":202610122045,"semantic_template":"qf-density12-counting-full-product-prefix/[2,7]"},"fields":{"title":"Full Product Choices and Prefix Counts","stem":"Ordered stage sizes=[2, 7]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.","answer":{"kind":"expression","value":"(14, [2, 14])","explanation":"Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (14, [2, 14])."},"skills":["Full Product Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0046","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0196cfe0a86b27b28acd27dca8db84822786b807c85cdfdadff7b79d5a71395","family_id":"qf-density12-counting-full-product-prefix","seed":202610122046,"semantic_template":"qf-density12-counting-full-product-prefix/[2,8]"},"fields":{"title":"Full Product Choices and Prefix Counts","stem":"Ordered stage sizes=[2, 8]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.","answer":{"kind":"expression","value":"(16, [2, 16])","explanation":"Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (16, [2, 16])."},"skills":["Full Product Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0047","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3bfc5e8d907a2b56a5c257fe666cf1675e3a7ba7799e05d15a091a1e980c97bc","family_id":"qf-density12-counting-full-product-prefix","seed":202610122047,"semantic_template":"qf-density12-counting-full-product-prefix/[3,1]"},"fields":{"title":"Full Product Choices and Prefix Counts","stem":"Ordered stage sizes=[3, 1]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.","answer":{"kind":"expression","value":"(3, [3, 3])","explanation":"Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (3, [3, 3])."},"skills":["Full Product Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0048","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac2c6250085d4123663c2befab9b05387d3c866c5fc1bfa2ef70eb287887f5ea","family_id":"qf-density12-counting-full-product-prefix","seed":202610122048,"semantic_template":"qf-density12-counting-full-product-prefix/[3,2]"},"fields":{"title":"Full Product Choices and Prefix Counts","stem":"Ordered stage sizes=[3, 2]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.","answer":{"kind":"expression","value":"(6, [3, 6])","explanation":"Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (6, [3, 6])."},"skills":["Full Product Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0049","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"adb085a361e1d53cb06ab717d1759c59670e425c2afcd4702f494332dcbb1700","family_id":"qf-density12-counting-full-product-prefix","seed":202610122049,"semantic_template":"qf-density12-counting-full-product-prefix/[3,3]"},"fields":{"title":"Full Product Choices and Prefix Counts","stem":"Ordered stage sizes=[3, 3]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.","answer":{"kind":"expression","value":"(9, [3, 9])","explanation":"Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (9, [3, 9])."},"skills":["Full Product Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0050","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"564e26b3bb1db7c2d0474e0ac4568ff69705bfaa12e1e9aeb2536cda9e241547","family_id":"qf-density12-counting-full-product-prefix","seed":202610122050,"semantic_template":"qf-density12-counting-full-product-prefix/[3,4]"},"fields":{"title":"Full Product Choices and Prefix Counts","stem":"Ordered stage sizes=[3, 4]; every cross-stage combination is allowed. Give (full product count, each prefix product). Zero stages have one empty selection sequence.","answer":{"kind":"expression","value":"(12, [3, 12])","explanation":"Multiply successive choice counts; an empty product is one and a zero stage makes later products zero. No stochastic independence is asserted. The exact result, retaining the requested order, is (12, [3, 12])."},"skills":["Full Product Choices and Prefix Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0051","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08082f46f79403c0fa1ddf0c851d1f9eb367e04223967a058e1b6cea6a8b92fd","family_id":"qf-density12-counting-conditional-candidate-defect","seed":202610122051,"semantic_template":"qf-density12-counting-conditional-candidate-defect/[\"empty_first_stage\"]"},"fields":{"title":"Actual Branch Widths versus a Candidate Width","stem":"Ordered second-stage branch counts=[], n=0; candidate common width h=7. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).","answer":{"kind":"expression","value":"(0, 0, Yes, 0)","explanation":"Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (0, 0, Yes, 0)."},"skills":["Actual Branch Widths versus a Candidate Width"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0052","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c3d437f3a427491f67cc82e9ac79282b0b9ccc0188c760e96d8808ab1e85f1d","family_id":"qf-density12-counting-conditional-candidate-defect","seed":202610122052,"semantic_template":"qf-density12-counting-conditional-candidate-defect/[2,4,2,false]"},"fields":{"title":"Actual Branch Widths versus a Candidate Width","stem":"Ordered second-stage branch counts=[1, 3], n=2; candidate common width h=2. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).","answer":{"kind":"expression","value":"(4, 4, No, 0)","explanation":"Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (4, 4, No, 0)."},"skills":["Actual Branch Widths versus a Candidate Width"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0053","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8920dee9c800154c736639727753434e6fa1284a8bb759fa49c40cc786e4407","family_id":"qf-density12-counting-conditional-candidate-defect","seed":202610122053,"semantic_template":"qf-density12-counting-conditional-candidate-defect/[2,4,2,true]"},"fields":{"title":"Actual Branch Widths versus a Candidate Width","stem":"Ordered second-stage branch counts=[2, 2], n=2; candidate common width h=2. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).","answer":{"kind":"expression","value":"(4, 4, Yes, 0)","explanation":"Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (4, 4, Yes, 0)."},"skills":["Actual Branch Widths versus a Candidate Width"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0054","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46eeae75df531632542b9166c4a99fc8a28da281ce0354ca00631b4060c0531e","family_id":"qf-density12-counting-conditional-candidate-defect","seed":202610122054,"semantic_template":"qf-density12-counting-conditional-candidate-defect/[1,0,0,true]"},"fields":{"title":"Actual Branch Widths versus a Candidate Width","stem":"Ordered second-stage branch counts=[0], n=1; candidate common width h=0. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).","answer":{"kind":"expression","value":"(0, 0, Yes, 0)","explanation":"Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (0, 0, Yes, 0)."},"skills":["Actual Branch Widths versus a Candidate Width"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0055","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"788b50c392f39e972ff71f52121ae52504e82f92982835726bb5eaa0ee81cd44","family_id":"qf-density12-counting-conditional-candidate-defect","seed":202610122055,"semantic_template":"qf-density12-counting-conditional-candidate-defect/[1,3,2,false]"},"fields":{"title":"Actual Branch Widths versus a Candidate Width","stem":"Ordered second-stage branch counts=[3], n=1; candidate common width h=2. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).","answer":{"kind":"expression","value":"(3, 2, No, -1)","explanation":"Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (3, 2, No, -1)."},"skills":["Actual Branch Widths versus a Candidate Width"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0056","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c9cee3adb8355a8f44f8c882d3bc311092c3bda5d50c40827a6c135ad565ef5","family_id":"qf-density12-counting-conditional-candidate-defect","seed":202610122056,"semantic_template":"qf-density12-counting-conditional-candidate-defect/[2,11,0,false]"},"fields":{"title":"Actual Branch Widths versus a Candidate Width","stem":"Ordered second-stage branch counts=[5, 6], n=2; candidate common width h=0. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).","answer":{"kind":"expression","value":"(11, 0, No, -11)","explanation":"Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (11, 0, No, -11)."},"skills":["Actual Branch Widths versus a Candidate Width"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0057","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6204694b900605c5b5b0e44b6695584460cac4e55a3a065b0d505f89e9d5804e","family_id":"qf-density12-counting-conditional-candidate-defect","seed":202610122057,"semantic_template":"qf-density12-counting-conditional-candidate-defect/[2,13,0,false]"},"fields":{"title":"Actual Branch Widths versus a Candidate Width","stem":"Ordered second-stage branch counts=[6, 7], n=2; candidate common width h=0. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).","answer":{"kind":"expression","value":"(13, 0, No, -13)","explanation":"Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (13, 0, No, -13)."},"skills":["Actual Branch Widths versus a Candidate Width"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0058","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc5b6a06844dca061671a81fadeea607499e41c087b761a477950b9b92a758bb","family_id":"qf-density12-counting-conditional-candidate-defect","seed":202610122058,"semantic_template":"qf-density12-counting-conditional-candidate-defect/[2,15,0,false]"},"fields":{"title":"Actual Branch Widths versus a Candidate Width","stem":"Ordered second-stage branch counts=[7, 8], n=2; candidate common width h=0. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).","answer":{"kind":"expression","value":"(15, 0, No, -15)","explanation":"Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (15, 0, No, -15)."},"skills":["Actual Branch Widths versus a Candidate Width"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0059","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5aef9afb9f9410d3dc7e34dd474c7b9b125e8f201379b2e2dd53604aef4434b3","family_id":"qf-density12-counting-conditional-candidate-defect","seed":202610122059,"semantic_template":"qf-density12-counting-conditional-candidate-defect/[2,8,0,false]"},"fields":{"title":"Actual Branch Widths versus a Candidate Width","stem":"Ordered second-stage branch counts=[8, 0], n=2; candidate common width h=0. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).","answer":{"kind":"expression","value":"(8, 0, No, -8)","explanation":"Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (8, 0, No, -8)."},"skills":["Actual Branch Widths versus a Candidate Width"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0060","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"427dd8cfb809d915a892950982425e9d96883231d56c732f8f1bcd1f8089b444","family_id":"qf-density12-counting-conditional-candidate-defect","seed":202610122060,"semantic_template":"qf-density12-counting-conditional-candidate-defect/[2,1,1,false]"},"fields":{"title":"Actual Branch Widths versus a Candidate Width","stem":"Ordered second-stage branch counts=[0, 1], n=2; candidate common width h=1. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).","answer":{"kind":"expression","value":"(1, 2, No, 1)","explanation":"Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (1, 2, No, 1)."},"skills":["Actual Branch Widths versus a Candidate Width"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0061","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ca7745927796c87c171850c3b0c3ff3269707e500337f37dd5a4d835663216b","family_id":"qf-density12-counting-conditional-candidate-defect","seed":202610122061,"semantic_template":"qf-density12-counting-conditional-candidate-defect/[2,3,1,false]"},"fields":{"title":"Actual Branch Widths versus a Candidate Width","stem":"Ordered second-stage branch counts=[1, 2], n=2; candidate common width h=1. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).","answer":{"kind":"expression","value":"(3, 2, No, -1)","explanation":"Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (3, 2, No, -1)."},"skills":["Actual Branch Widths versus a Candidate Width"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0062","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58005a5ca467eec2a98a3d286146be81a3c7bd6e083cf590186e1a9b78652959","family_id":"qf-density12-counting-conditional-candidate-defect","seed":202610122062,"semantic_template":"qf-density12-counting-conditional-candidate-defect/[2,5,1,false]"},"fields":{"title":"Actual Branch Widths versus a Candidate Width","stem":"Ordered second-stage branch counts=[2, 3], n=2; candidate common width h=1. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).","answer":{"kind":"expression","value":"(5, 2, No, -3)","explanation":"Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (5, 2, No, -3)."},"skills":["Actual Branch Widths versus a Candidate Width"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0063","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a16761567e296fca3fa36bdbf2aefc54e0ce66a1b9f43d5da4ce4e468b733f8","family_id":"qf-density12-counting-conditional-candidate-defect","seed":202610122063,"semantic_template":"qf-density12-counting-conditional-candidate-defect/[2,7,1,false]"},"fields":{"title":"Actual Branch Widths versus a Candidate Width","stem":"Ordered second-stage branch counts=[3, 4], n=2; candidate common width h=1. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).","answer":{"kind":"expression","value":"(7, 2, No, -5)","explanation":"Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (7, 2, No, -5)."},"skills":["Actual Branch Widths versus a Candidate Width"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0064","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"80634799558f04662371c513abe72b8b9ac0f0de6e10df655824da9691d4b2e6","family_id":"qf-density12-counting-conditional-candidate-defect","seed":202610122064,"semantic_template":"qf-density12-counting-conditional-candidate-defect/[2,9,1,false]"},"fields":{"title":"Actual Branch Widths versus a Candidate Width","stem":"Ordered second-stage branch counts=[4, 5], n=2; candidate common width h=1. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).","answer":{"kind":"expression","value":"(9, 2, No, -7)","explanation":"Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (9, 2, No, -7)."},"skills":["Actual Branch Widths versus a Candidate Width"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0065","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52c2c4e4b109ed5d90accd55b46301ccce1e36fb4ff5a444b95d4041957e95a0","family_id":"qf-density12-counting-conditional-candidate-defect","seed":202610122065,"semantic_template":"qf-density12-counting-conditional-candidate-defect/[2,11,1,false]"},"fields":{"title":"Actual Branch Widths versus a Candidate Width","stem":"Ordered second-stage branch counts=[5, 6], n=2; candidate common width h=1. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).","answer":{"kind":"expression","value":"(11, 2, No, -9)","explanation":"Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (11, 2, No, -9)."},"skills":["Actual Branch Widths versus a Candidate Width"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0066","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"764026a0e6fb03d855fef58a56bc1b98643563c1a1342e3e9ac70a2213dec5b7","family_id":"qf-density12-counting-conditional-candidate-defect","seed":202610122066,"semantic_template":"qf-density12-counting-conditional-candidate-defect/[2,13,1,false]"},"fields":{"title":"Actual Branch Widths versus a Candidate Width","stem":"Ordered second-stage branch counts=[6, 7], n=2; candidate common width h=1. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).","answer":{"kind":"expression","value":"(13, 2, No, -11)","explanation":"Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (13, 2, No, -11)."},"skills":["Actual Branch Widths versus a Candidate Width"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0067","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69853ba4871e9f5b459bc2be3c4a3955f7bdaffadeae1b8948c4414d47786bc1","family_id":"qf-density12-counting-conditional-candidate-defect","seed":202610122067,"semantic_template":"qf-density12-counting-conditional-candidate-defect/[2,15,1,false]"},"fields":{"title":"Actual Branch Widths versus a Candidate Width","stem":"Ordered second-stage branch counts=[7, 8], n=2; candidate common width h=1. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).","answer":{"kind":"expression","value":"(15, 2, No, -13)","explanation":"Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (15, 2, No, -13)."},"skills":["Actual Branch Widths versus a Candidate Width"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0068","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"737a4d88e01940549c27bbc1f3264905533a5ce97ba201e677b22fdf37d63f01","family_id":"qf-density12-counting-conditional-candidate-defect","seed":202610122068,"semantic_template":"qf-density12-counting-conditional-candidate-defect/[2,8,1,false]"},"fields":{"title":"Actual Branch Widths versus a Candidate Width","stem":"Ordered second-stage branch counts=[8, 0], n=2; candidate common width h=1. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).","answer":{"kind":"expression","value":"(8, 2, No, -6)","explanation":"Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (8, 2, No, -6)."},"skills":["Actual Branch Widths versus a Candidate Width"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0069","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99baba5d85facf4ccd8b248efdc7cf855a28a3bd73e708cd4025d4c054b925bb","family_id":"qf-density12-counting-conditional-candidate-defect","seed":202610122069,"semantic_template":"qf-density12-counting-conditional-candidate-defect/[2,1,2,false]"},"fields":{"title":"Actual Branch Widths versus a Candidate Width","stem":"Ordered second-stage branch counts=[0, 1], n=2; candidate common width h=2. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).","answer":{"kind":"expression","value":"(1, 4, No, 3)","explanation":"Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (1, 4, No, 3)."},"skills":["Actual Branch Widths versus a Candidate Width"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0070","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a249007401fb450b836a0203c5dfdcfdd997ea3f900128f62e1d5ed8d95bb19","family_id":"qf-density12-counting-conditional-candidate-defect","seed":202610122070,"semantic_template":"qf-density12-counting-conditional-candidate-defect/[2,3,2,false]"},"fields":{"title":"Actual Branch Widths versus a Candidate Width","stem":"Ordered second-stage branch counts=[1, 2], n=2; candidate common width h=2. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).","answer":{"kind":"expression","value":"(3, 4, No, 1)","explanation":"Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (3, 4, No, 1)."},"skills":["Actual Branch Widths versus a Candidate Width"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0071","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9085d7807a055b8875eeb69cbddd9cf21a9c3ae6bf4820d4931d2e9ba96e755","family_id":"qf-density12-counting-conditional-candidate-defect","seed":202610122071,"semantic_template":"qf-density12-counting-conditional-candidate-defect/[2,5,2,false]"},"fields":{"title":"Actual Branch Widths versus a Candidate Width","stem":"Ordered second-stage branch counts=[2, 3], n=2; candidate common width h=2. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).","answer":{"kind":"expression","value":"(5, 4, No, -1)","explanation":"Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (5, 4, No, -1)."},"skills":["Actual Branch Widths versus a Candidate Width"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0072","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"415620bd22417b0e2eeb2553b1dbddd77cbd8ea5e8d8c3bc6d5cb3779ef49a4b","family_id":"qf-density12-counting-conditional-candidate-defect","seed":202610122072,"semantic_template":"qf-density12-counting-conditional-candidate-defect/[2,7,2,false]"},"fields":{"title":"Actual Branch Widths versus a Candidate Width","stem":"Ordered second-stage branch counts=[3, 4], n=2; candidate common width h=2. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).","answer":{"kind":"expression","value":"(7, 4, No, -3)","explanation":"Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (7, 4, No, -3)."},"skills":["Actual Branch Widths versus a Candidate Width"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0073","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d05d70eacfd0d0b236daa147c4c02c44140ad288ac09df737ac746649a500e9","family_id":"qf-density12-counting-conditional-candidate-defect","seed":202610122073,"semantic_template":"qf-density12-counting-conditional-candidate-defect/[2,9,2,false]"},"fields":{"title":"Actual Branch Widths versus a Candidate Width","stem":"Ordered second-stage branch counts=[4, 5], n=2; candidate common width h=2. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).","answer":{"kind":"expression","value":"(9, 4, No, -5)","explanation":"Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (9, 4, No, -5)."},"skills":["Actual Branch Widths versus a Candidate Width"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0074","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76daadb8fbcba666c880d459e832f8a0098f375238f33a42b0cbb72e27336cd0","family_id":"qf-density12-counting-conditional-candidate-defect","seed":202610122074,"semantic_template":"qf-density12-counting-conditional-candidate-defect/[2,11,2,false]"},"fields":{"title":"Actual Branch Widths versus a Candidate Width","stem":"Ordered second-stage branch counts=[5, 6], n=2; candidate common width h=2. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).","answer":{"kind":"expression","value":"(11, 4, No, -7)","explanation":"Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (11, 4, No, -7)."},"skills":["Actual Branch Widths versus a Candidate Width"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0075","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f90b2296f01ec042453bbd55e61411d25626f415883f0b9b2c7a60df166409d","family_id":"qf-density12-counting-conditional-candidate-defect","seed":202610122075,"semantic_template":"qf-density12-counting-conditional-candidate-defect/[2,13,2,false]"},"fields":{"title":"Actual Branch Widths versus a Candidate Width","stem":"Ordered second-stage branch counts=[6, 7], n=2; candidate common width h=2. Give (actual total, assumed n*h total, does every branch equal h?, n*h−actual total).","answer":{"kind":"expression","value":"(13, 4, No, -9)","explanation":"Sum actual branches and check every width separately; equal totals do not establish equal widths. For no first-stage branches the width condition is vacuously Yes. The exact result, retaining the requested order, is (13, 4, No, -9)."},"skills":["Actual Branch Widths versus a Candidate Width"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0076","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02b05ba618611d923a27d3524821523e797b0f5b7dd192ff616f8c0428148e5f","family_id":"qf-density12-counting-overlap-correction","seed":202610122076,"semantic_template":"qf-density12-counting-overlap-correction/[[0,0],0]"},"fields":{"title":"Overlap-Corrected Finite Choice Counts","stem":"Finite integer choice sets A=[], B=[]. Give (|A|+|B|, |A∪B|, |A∩B|).","answer":{"kind":"expression","value":"(0, 0, 0)","explanation":"Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (0, 0, 0)."},"skills":["Overlap-Corrected Finite Choice Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0077","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35092f73237fe926b3f0b082ab2ce7d01087dff6ece2340b4fd58e385c68d54b","family_id":"qf-density12-counting-overlap-correction","seed":202610122077,"semantic_template":"qf-density12-counting-overlap-correction/[[0,0],1]"},"fields":{"title":"Overlap-Corrected Finite Choice Counts","stem":"Finite integer choice sets A=[0], B=[0]. Give (|A|+|B|, |A∪B|, |A∩B|).","answer":{"kind":"expression","value":"(2, 1, 1)","explanation":"Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (2, 1, 1)."},"skills":["Overlap-Corrected Finite Choice Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0078","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6d74a76d0f8a73a673fd45f4840272fa93fe6fcef08038439439684f4d74043","family_id":"qf-density12-counting-overlap-correction","seed":202610122078,"semantic_template":"qf-density12-counting-overlap-correction/[[0,1],0]"},"fields":{"title":"Overlap-Corrected Finite Choice Counts","stem":"Finite integer choice sets A=[], B=[100]. Give (|A|+|B|, |A∪B|, |A∩B|).","answer":{"kind":"expression","value":"(1, 1, 0)","explanation":"Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (1, 1, 0)."},"skills":["Overlap-Corrected Finite Choice Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0079","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff4ec8d30589a54a204b40d3cc019c596f21217ec9a88f298979aea344f7b5c9","family_id":"qf-density12-counting-overlap-correction","seed":202610122079,"semantic_template":"qf-density12-counting-overlap-correction/[[0,0],2]"},"fields":{"title":"Overlap-Corrected Finite Choice Counts","stem":"Finite integer choice sets A=[0, 1], B=[0, 1]. Give (|A|+|B|, |A∪B|, |A∩B|).","answer":{"kind":"expression","value":"(4, 2, 2)","explanation":"Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (4, 2, 2)."},"skills":["Overlap-Corrected Finite Choice Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0080","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b92030979a9ef61a0f9455c066be41afe56f1e9e1b7b7399fee7b424a952e43","family_id":"qf-density12-counting-overlap-correction","seed":202610122080,"semantic_template":"qf-density12-counting-overlap-correction/[[0,1],1]"},"fields":{"title":"Overlap-Corrected Finite Choice Counts","stem":"Finite integer choice sets A=[0], B=[0, 100]. Give (|A|+|B|, |A∪B|, |A∩B|).","answer":{"kind":"expression","value":"(3, 2, 1)","explanation":"Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (3, 2, 1)."},"skills":["Overlap-Corrected Finite Choice Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0081","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12eb934d5d76ea35763d5919497f9b1b3b73b969850f2b2f7789a4ee38da5baf","family_id":"qf-density12-counting-overlap-correction","seed":202610122081,"semantic_template":"qf-density12-counting-overlap-correction/[[0,2],0]"},"fields":{"title":"Overlap-Corrected Finite Choice Counts","stem":"Finite integer choice sets A=[], B=[100, 101]. Give (|A|+|B|, |A∪B|, |A∩B|).","answer":{"kind":"expression","value":"(2, 2, 0)","explanation":"Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (2, 2, 0)."},"skills":["Overlap-Corrected Finite Choice Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0082","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cbca87f2a5a53c78c8abd7708d44847ef1463e97b49cc852ab0c39b2aa53d55b","family_id":"qf-density12-counting-overlap-correction","seed":202610122082,"semantic_template":"qf-density12-counting-overlap-correction/[[1,1],0]"},"fields":{"title":"Overlap-Corrected Finite Choice Counts","stem":"Finite integer choice sets A=[-1], B=[100]. Give (|A|+|B|, |A∪B|, |A∩B|).","answer":{"kind":"expression","value":"(2, 2, 0)","explanation":"Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (2, 2, 0)."},"skills":["Overlap-Corrected Finite Choice Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0083","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"127957daaf6b9dca18da21d68368faa6ce8e4a6402a9f4b775cc514f6a6ca144","family_id":"qf-density12-counting-overlap-correction","seed":202610122083,"semantic_template":"qf-density12-counting-overlap-correction/[[0,0],3]"},"fields":{"title":"Overlap-Corrected Finite Choice Counts","stem":"Finite integer choice sets A=[0, 1, 2], B=[0, 1, 2]. Give (|A|+|B|, |A∪B|, |A∩B|).","answer":{"kind":"expression","value":"(6, 3, 3)","explanation":"Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (6, 3, 3)."},"skills":["Overlap-Corrected Finite Choice Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0084","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16a0b93b15d697b07e3958b70c03db1fe69147bf32c7ea547a28158b439d1677","family_id":"qf-density12-counting-overlap-correction","seed":202610122084,"semantic_template":"qf-density12-counting-overlap-correction/[[0,1],2]"},"fields":{"title":"Overlap-Corrected Finite Choice Counts","stem":"Finite integer choice sets A=[0, 1], B=[0, 1, 100]. Give (|A|+|B|, |A∪B|, |A∩B|).","answer":{"kind":"expression","value":"(5, 3, 2)","explanation":"Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (5, 3, 2)."},"skills":["Overlap-Corrected Finite Choice Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0085","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7701cb7c6e213f612b47aae88dc85f1a3c5027b2428d4b12f928740e67c95d3d","family_id":"qf-density12-counting-overlap-correction","seed":202610122085,"semantic_template":"qf-density12-counting-overlap-correction/[[0,2],1]"},"fields":{"title":"Overlap-Corrected Finite Choice Counts","stem":"Finite integer choice sets A=[0], B=[0, 100, 101]. Give (|A|+|B|, |A∪B|, |A∩B|).","answer":{"kind":"expression","value":"(4, 3, 1)","explanation":"Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (4, 3, 1)."},"skills":["Overlap-Corrected Finite Choice Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0086","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a40168e0ea15a798300f03a7ec9fc7cfa5323b2106334698196d52aeaabbe81","family_id":"qf-density12-counting-overlap-correction","seed":202610122086,"semantic_template":"qf-density12-counting-overlap-correction/[[0,3],0]"},"fields":{"title":"Overlap-Corrected Finite Choice Counts","stem":"Finite integer choice sets A=[], B=[100, 101, 102]. Give (|A|+|B|, |A∪B|, |A∩B|).","answer":{"kind":"expression","value":"(3, 3, 0)","explanation":"Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (3, 3, 0)."},"skills":["Overlap-Corrected Finite Choice Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0087","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd5194309c35e16711fec45b47769f5860e749bb221036cf286ad469e1c2db08","family_id":"qf-density12-counting-overlap-correction","seed":202610122087,"semantic_template":"qf-density12-counting-overlap-correction/[[1,1],1]"},"fields":{"title":"Overlap-Corrected Finite Choice Counts","stem":"Finite integer choice sets A=[0, -1], B=[0, 100]. Give (|A|+|B|, |A∪B|, |A∩B|).","answer":{"kind":"expression","value":"(4, 3, 1)","explanation":"Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (4, 3, 1)."},"skills":["Overlap-Corrected Finite Choice Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0088","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd8ba2d8537e532f99cf91dee2480caba9f3a20d64e5ddd72da3c8c834c92ef8","family_id":"qf-density12-counting-overlap-correction","seed":202610122088,"semantic_template":"qf-density12-counting-overlap-correction/[[1,2],0]"},"fields":{"title":"Overlap-Corrected Finite Choice Counts","stem":"Finite integer choice sets A=[-1], B=[100, 101]. Give (|A|+|B|, |A∪B|, |A∩B|).","answer":{"kind":"expression","value":"(3, 3, 0)","explanation":"Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (3, 3, 0)."},"skills":["Overlap-Corrected Finite Choice Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0089","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2e09497de73111568441455b6bec91677fcae1d84da3f51f40dc0d5ee0baacc","family_id":"qf-density12-counting-overlap-correction","seed":202610122089,"semantic_template":"qf-density12-counting-overlap-correction/[[0,0],4]"},"fields":{"title":"Overlap-Corrected Finite Choice Counts","stem":"Finite integer choice sets A=[0, 1, 2, 3], B=[0, 1, 2, 3]. Give (|A|+|B|, |A∪B|, |A∩B|).","answer":{"kind":"expression","value":"(8, 4, 4)","explanation":"Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (8, 4, 4)."},"skills":["Overlap-Corrected Finite Choice Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0090","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3bf14611d3bf291c9889142673cd304ed87bf803772e798cb69fce8c21401cd9","family_id":"qf-density12-counting-overlap-correction","seed":202610122090,"semantic_template":"qf-density12-counting-overlap-correction/[[0,1],3]"},"fields":{"title":"Overlap-Corrected Finite Choice Counts","stem":"Finite integer choice sets A=[0, 1, 2], B=[0, 1, 2, 100]. Give (|A|+|B|, |A∪B|, |A∩B|).","answer":{"kind":"expression","value":"(7, 4, 3)","explanation":"Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (7, 4, 3)."},"skills":["Overlap-Corrected Finite Choice Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0091","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73400c1feae04f65a932c91b4b00483b3148c04405a70645153fdcf123e98ed2","family_id":"qf-density12-counting-overlap-correction","seed":202610122091,"semantic_template":"qf-density12-counting-overlap-correction/[[0,2],2]"},"fields":{"title":"Overlap-Corrected Finite Choice Counts","stem":"Finite integer choice sets A=[0, 1], B=[0, 1, 100, 101]. Give (|A|+|B|, |A∪B|, |A∩B|).","answer":{"kind":"expression","value":"(6, 4, 2)","explanation":"Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (6, 4, 2)."},"skills":["Overlap-Corrected Finite Choice Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0092","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f4e83481b6b6c94b39a1bc74ce3301eeac5748518fdbd56d6e975244bd6d1d5","family_id":"qf-density12-counting-overlap-correction","seed":202610122092,"semantic_template":"qf-density12-counting-overlap-correction/[[0,3],1]"},"fields":{"title":"Overlap-Corrected Finite Choice Counts","stem":"Finite integer choice sets A=[0], B=[0, 100, 101, 102]. Give (|A|+|B|, |A∪B|, |A∩B|).","answer":{"kind":"expression","value":"(5, 4, 1)","explanation":"Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (5, 4, 1)."},"skills":["Overlap-Corrected Finite Choice Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0093","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ccc21cc46d55bab8d5e7121574828c2449db94cc7f6f0b0bd4db28e5d75abf7b","family_id":"qf-density12-counting-overlap-correction","seed":202610122093,"semantic_template":"qf-density12-counting-overlap-correction/[[0,4],0]"},"fields":{"title":"Overlap-Corrected Finite Choice Counts","stem":"Finite integer choice sets A=[], B=[100, 101, 102, 103]. Give (|A|+|B|, |A∪B|, |A∩B|).","answer":{"kind":"expression","value":"(4, 4, 0)","explanation":"Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (4, 4, 0)."},"skills":["Overlap-Corrected Finite Choice Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0094","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d60189b6c58ed49122260559a47418442a9a6ca83ad47563aca4813e6fd6054","family_id":"qf-density12-counting-overlap-correction","seed":202610122094,"semantic_template":"qf-density12-counting-overlap-correction/[[1,1],2]"},"fields":{"title":"Overlap-Corrected Finite Choice Counts","stem":"Finite integer choice sets A=[0, 1, -1], B=[0, 1, 100]. Give (|A|+|B|, |A∪B|, |A∩B|).","answer":{"kind":"expression","value":"(6, 4, 2)","explanation":"Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (6, 4, 2)."},"skills":["Overlap-Corrected Finite Choice Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0095","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6fe251ba8a218322d24f202bdc84586fc0324affc60d84be2fc44f79363c05ff","family_id":"qf-density12-counting-overlap-correction","seed":202610122095,"semantic_template":"qf-density12-counting-overlap-correction/[[1,2],1]"},"fields":{"title":"Overlap-Corrected Finite Choice Counts","stem":"Finite integer choice sets A=[0, -1], B=[0, 100, 101]. Give (|A|+|B|, |A∪B|, |A∩B|).","answer":{"kind":"expression","value":"(5, 4, 1)","explanation":"Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (5, 4, 1)."},"skills":["Overlap-Corrected Finite Choice Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0096","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0fbe5f3f508f0a1603cda4ef8faca506e6cb37e8fed1e590a435c9da7df2623f","family_id":"qf-density12-counting-overlap-correction","seed":202610122096,"semantic_template":"qf-density12-counting-overlap-correction/[[1,3],0]"},"fields":{"title":"Overlap-Corrected Finite Choice Counts","stem":"Finite integer choice sets A=[-1], B=[100, 101, 102]. Give (|A|+|B|, |A∪B|, |A∩B|).","answer":{"kind":"expression","value":"(4, 4, 0)","explanation":"Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (4, 4, 0)."},"skills":["Overlap-Corrected Finite Choice Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0097","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"567adf54974b6243c82725e058d15fbe01babb9e1cd085841b71cb1393d5037e","family_id":"qf-density12-counting-overlap-correction","seed":202610122097,"semantic_template":"qf-density12-counting-overlap-correction/[[2,2],0]"},"fields":{"title":"Overlap-Corrected Finite Choice Counts","stem":"Finite integer choice sets A=[-2, -1], B=[100, 101]. Give (|A|+|B|, |A∪B|, |A∩B|).","answer":{"kind":"expression","value":"(4, 4, 0)","explanation":"Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (4, 4, 0)."},"skills":["Overlap-Corrected Finite Choice Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0098","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd0ba8c48ab1aea604897d7ebee21033f9ce2a150eb4d8eb2cfa380e006ee7ce","family_id":"qf-density12-counting-overlap-correction","seed":202610122098,"semantic_template":"qf-density12-counting-overlap-correction/[[0,0],5]"},"fields":{"title":"Overlap-Corrected Finite Choice Counts","stem":"Finite integer choice sets A=[0, 1, 2, 3, 4], B=[0, 1, 2, 3, 4]. Give (|A|+|B|, |A∪B|, |A∩B|).","answer":{"kind":"expression","value":"(10, 5, 5)","explanation":"Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (10, 5, 5)."},"skills":["Overlap-Corrected Finite Choice Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0099","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07c0bffa2d61e5f295195a6cf7f95602ef8eb96755e232a8832b9cc7981936f4","family_id":"qf-density12-counting-overlap-correction","seed":202610122099,"semantic_template":"qf-density12-counting-overlap-correction/[[0,1],4]"},"fields":{"title":"Overlap-Corrected Finite Choice Counts","stem":"Finite integer choice sets A=[0, 1, 2, 3], B=[0, 1, 2, 3, 100]. Give (|A|+|B|, |A∪B|, |A∩B|).","answer":{"kind":"expression","value":"(9, 5, 4)","explanation":"Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (9, 5, 4)."},"skills":["Overlap-Corrected Finite Choice Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261012-r02-counting-principle-0100","batch_id":"batch-20261008-advance-20261012-r02-counting-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70058c01d89985259fc13f2e516f5fd6f47386b0b7e0a1da5bfc14ec1f80e0b7","family_id":"qf-density12-counting-overlap-correction","seed":202610122100,"semantic_template":"qf-density12-counting-overlap-correction/[[0,2],3]"},"fields":{"title":"Overlap-Corrected Finite Choice Counts","stem":"Finite integer choice sets A=[0, 1, 2], B=[0, 1, 2, 100, 101]. Give (|A|+|B|, |A∪B|, |A∩B|).","answer":{"kind":"expression","value":"(8, 5, 3)","explanation":"Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (8, 5, 3)."},"skills":["Overlap-Corrected Finite Choice Counts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}