{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0001","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a42dc3a7a33f6659fc724ca6839b64724541b8df0f99402f53e4a71f4bd90187","family_id":"qf-mod40-mobius-canonical-value","seed":202611096001,"semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":2}"},"fields":{"title":"Complete Prime Multiplicities Determine the Mobius Value","stem":"Use positive integer n=2. Give (n, prime_multiset, distinct_primes, mu, certificate).","answer":{"kind":"expression","value":"(2, [2], [2], -1, squarefree_parity_sign)","explanation":"Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (2, [2], [2], -1, squarefree_parity_sign)."},"skills":["Complete Prime Multiplicities Determine the Mobius Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0002","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dfd3a1c523a737dc157e7421b158591261759a40aa317f757026bb0b565b5ec1","family_id":"qf-mod40-mobius-canonical-value","seed":202611096002,"semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":6}"},"fields":{"title":"Complete Prime Multiplicities Determine the Mobius Value","stem":"Use positive integer n=6. Give (n, prime_multiset, distinct_primes, mu, certificate).","answer":{"kind":"expression","value":"(6, [2, 3], [2, 3], 1, squarefree_parity_sign)","explanation":"Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (6, [2, 3], [2, 3], 1, squarefree_parity_sign)."},"skills":["Complete Prime Multiplicities Determine the Mobius Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0003","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58a01035434162ce11859207ef94585f2f05b9858ee3846e90932e38abac993d","family_id":"qf-mod40-mobius-canonical-value","seed":202611096003,"semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":12}"},"fields":{"title":"Complete Prime Multiplicities Determine the Mobius Value","stem":"Use positive integer n=12. Give (n, prime_multiset, distinct_primes, mu, certificate).","answer":{"kind":"expression","value":"(12, [2, 2, 3], [2, 3], 0, square_prime_factor_gives_zero)","explanation":"Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (12, [2, 2, 3], [2, 3], 0, square_prime_factor_gives_zero)."},"skills":["Complete Prime Multiplicities Determine the Mobius Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0004","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"33cb77cda82821fb56d995ad96b21d5559779ac28dc0b7a73d0f8b36735e0243","family_id":"qf-mod40-mobius-canonical-value","seed":202611096004,"semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":20}"},"fields":{"title":"Complete Prime Multiplicities Determine the Mobius Value","stem":"Use positive integer n=20. Give (n, prime_multiset, distinct_primes, mu, certificate).","answer":{"kind":"expression","value":"(20, [2, 2, 5], [2, 5], 0, square_prime_factor_gives_zero)","explanation":"Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (20, [2, 2, 5], [2, 5], 0, square_prime_factor_gives_zero)."},"skills":["Complete Prime Multiplicities Determine the Mobius Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0005","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a2bc7bb2fddf15767c1161e9665a3892daede65445e292bb8c9e2b46d108734","family_id":"qf-mod40-mobius-canonical-value","seed":202611096005,"semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":30}"},"fields":{"title":"Complete Prime Multiplicities Determine the Mobius Value","stem":"Use positive integer n=30. Give (n, prime_multiset, distinct_primes, mu, certificate).","answer":{"kind":"expression","value":"(30, [2, 3, 5], [2, 3, 5], -1, squarefree_parity_sign)","explanation":"Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (30, [2, 3, 5], [2, 3, 5], -1, squarefree_parity_sign)."},"skills":["Complete Prime Multiplicities Determine the Mobius Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0006","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"454e71aa44704715ae2f76cf37b262d890ec8025773214938f4b54915beeffc2","family_id":"qf-mod40-mobius-canonical-value","seed":202611096006,"semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":42}"},"fields":{"title":"Complete Prime Multiplicities Determine the Mobius Value","stem":"Use positive integer n=42. Give (n, prime_multiset, distinct_primes, mu, certificate).","answer":{"kind":"expression","value":"(42, [2, 3, 7], [2, 3, 7], -1, squarefree_parity_sign)","explanation":"Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (42, [2, 3, 7], [2, 3, 7], -1, squarefree_parity_sign)."},"skills":["Complete Prime Multiplicities Determine the Mobius Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0007","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f68832486f55eb8c692e7b1117ba586e2bee6dead316628f882838f79a297045","family_id":"qf-mod40-mobius-canonical-value","seed":202611096007,"semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":56}"},"fields":{"title":"Complete Prime Multiplicities Determine the Mobius Value","stem":"Use positive integer n=56. Give (n, prime_multiset, distinct_primes, mu, certificate).","answer":{"kind":"expression","value":"(56, [2, 2, 2, 7], [2, 7], 0, square_prime_factor_gives_zero)","explanation":"Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (56, [2, 2, 2, 7], [2, 7], 0, square_prime_factor_gives_zero)."},"skills":["Complete Prime Multiplicities Determine the Mobius Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0008","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f9171c17cc3a71063bdfbba03de5baab1ae3d55329e232b84dd49b30d10dd02","family_id":"qf-mod40-mobius-canonical-value","seed":202611096008,"semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":72}"},"fields":{"title":"Complete Prime Multiplicities Determine the Mobius Value","stem":"Use positive integer n=72. Give (n, prime_multiset, distinct_primes, mu, certificate).","answer":{"kind":"expression","value":"(72, [2, 2, 2, 3, 3], [2, 3], 0, square_prime_factor_gives_zero)","explanation":"Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (72, [2, 2, 2, 3, 3], [2, 3], 0, square_prime_factor_gives_zero)."},"skills":["Complete Prime Multiplicities Determine the Mobius Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0009","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b255ecdc2a7814e4fbcc007a46eaf8360051c4b32d28ce56a1dded7729b5119","family_id":"qf-mod40-mobius-canonical-value","seed":202611096009,"semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":90}"},"fields":{"title":"Complete Prime Multiplicities Determine the Mobius Value","stem":"Use positive integer n=90. Give (n, prime_multiset, distinct_primes, mu, certificate).","answer":{"kind":"expression","value":"(90, [2, 3, 3, 5], [2, 3, 5], 0, square_prime_factor_gives_zero)","explanation":"Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (90, [2, 3, 3, 5], [2, 3, 5], 0, square_prime_factor_gives_zero)."},"skills":["Complete Prime Multiplicities Determine the Mobius Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0010","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4b65b1ae93d95843fb2cfc957ed8f48205744355dec4fcb0a6af2a4ad333a7a","family_id":"qf-mod40-mobius-canonical-value","seed":202611096010,"semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":110}"},"fields":{"title":"Complete Prime Multiplicities Determine the Mobius Value","stem":"Use positive integer n=110. Give (n, prime_multiset, distinct_primes, mu, certificate).","answer":{"kind":"expression","value":"(110, [2, 5, 11], [2, 5, 11], -1, squarefree_parity_sign)","explanation":"Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (110, [2, 5, 11], [2, 5, 11], -1, squarefree_parity_sign)."},"skills":["Complete Prime Multiplicities Determine the Mobius Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0011","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b300a806cf73ba40febd6099f7ffe5f3c755bc7020d93e9c908c332d59115942","family_id":"qf-mod40-mobius-canonical-value","seed":202611096011,"semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":132}"},"fields":{"title":"Complete Prime Multiplicities Determine the Mobius Value","stem":"Use positive integer n=132. Give (n, prime_multiset, distinct_primes, mu, certificate).","answer":{"kind":"expression","value":"(132, [2, 2, 3, 11], [2, 3, 11], 0, square_prime_factor_gives_zero)","explanation":"Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (132, [2, 2, 3, 11], [2, 3, 11], 0, square_prime_factor_gives_zero)."},"skills":["Complete Prime Multiplicities Determine the Mobius Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0012","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab57880d01db57a384a911be3fcf32de65e80785addc0a2eefac239cd13a91c6","family_id":"qf-mod40-mobius-canonical-value","seed":202611096012,"semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":156}"},"fields":{"title":"Complete Prime Multiplicities Determine the Mobius Value","stem":"Use positive integer n=156. Give (n, prime_multiset, distinct_primes, mu, certificate).","answer":{"kind":"expression","value":"(156, [2, 2, 3, 13], [2, 3, 13], 0, square_prime_factor_gives_zero)","explanation":"Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (156, [2, 2, 3, 13], [2, 3, 13], 0, square_prime_factor_gives_zero)."},"skills":["Complete Prime Multiplicities Determine the Mobius Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0013","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9dfb56376f4301a17715aa6ee7cbe751632f1670e140693e1f7bbdd684ba90a6","family_id":"qf-mod40-mobius-canonical-value","seed":202611096013,"semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":182}"},"fields":{"title":"Complete Prime Multiplicities Determine the Mobius Value","stem":"Use positive integer n=182. Give (n, prime_multiset, distinct_primes, mu, certificate).","answer":{"kind":"expression","value":"(182, [2, 7, 13], [2, 7, 13], -1, squarefree_parity_sign)","explanation":"Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (182, [2, 7, 13], [2, 7, 13], -1, squarefree_parity_sign)."},"skills":["Complete Prime Multiplicities Determine the Mobius Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0014","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3fa8b59dd4b1c18cc9290325879f9ceb39e5ac86715b88de921205c2570628a","family_id":"qf-mod40-mobius-canonical-value","seed":202611096014,"semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":210}"},"fields":{"title":"Complete Prime Multiplicities Determine the Mobius Value","stem":"Use positive integer n=210. Give (n, prime_multiset, distinct_primes, mu, certificate).","answer":{"kind":"expression","value":"(210, [2, 3, 5, 7], [2, 3, 5, 7], 1, squarefree_parity_sign)","explanation":"Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (210, [2, 3, 5, 7], [2, 3, 5, 7], 1, squarefree_parity_sign)."},"skills":["Complete Prime Multiplicities Determine the Mobius Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0015","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b0cdc438b50ec57039ac9de8e5ee31fe6516bd73a42b8e28e088bfc5c5eacf3","family_id":"qf-mod40-mobius-canonical-value","seed":202611096015,"semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":240}"},"fields":{"title":"Complete Prime Multiplicities Determine the Mobius Value","stem":"Use positive integer n=240. Give (n, prime_multiset, distinct_primes, mu, certificate).","answer":{"kind":"expression","value":"(240, [2, 2, 2, 2, 3, 5], [2, 3, 5], 0, square_prime_factor_gives_zero)","explanation":"Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (240, [2, 2, 2, 2, 3, 5], [2, 3, 5], 0, square_prime_factor_gives_zero)."},"skills":["Complete Prime Multiplicities Determine the Mobius Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0016","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6078920244010e947d3333ed8dd4d05b5557e685731c728c2f19a8098fc829d5","family_id":"qf-mod40-mobius-canonical-value","seed":202611096016,"semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":272}"},"fields":{"title":"Complete Prime Multiplicities Determine the Mobius Value","stem":"Use positive integer n=272. Give (n, prime_multiset, distinct_primes, mu, certificate).","answer":{"kind":"expression","value":"(272, [2, 2, 2, 2, 17], [2, 17], 0, square_prime_factor_gives_zero)","explanation":"Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (272, [2, 2, 2, 2, 17], [2, 17], 0, square_prime_factor_gives_zero)."},"skills":["Complete Prime Multiplicities Determine the Mobius Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0017","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d4d4373473a62072029b894af97e3002cc945ef433e1c87793240aaf6c418b7d","family_id":"qf-mod40-mobius-canonical-value","seed":202611096017,"semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":306}"},"fields":{"title":"Complete Prime Multiplicities Determine the Mobius Value","stem":"Use positive integer n=306. Give (n, prime_multiset, distinct_primes, mu, certificate).","answer":{"kind":"expression","value":"(306, [2, 3, 3, 17], [2, 3, 17], 0, square_prime_factor_gives_zero)","explanation":"Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (306, [2, 3, 3, 17], [2, 3, 17], 0, square_prime_factor_gives_zero)."},"skills":["Complete Prime Multiplicities Determine the Mobius Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0018","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4bc67060e4bbf1ee80169cdc17268aa19ffe91e941adb24adf4c23257a79c572","family_id":"qf-mod40-mobius-canonical-value","seed":202611096018,"semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":342}"},"fields":{"title":"Complete Prime Multiplicities Determine the Mobius Value","stem":"Use positive integer n=342. Give (n, prime_multiset, distinct_primes, mu, certificate).","answer":{"kind":"expression","value":"(342, [2, 3, 3, 19], [2, 3, 19], 0, square_prime_factor_gives_zero)","explanation":"Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (342, [2, 3, 3, 19], [2, 3, 19], 0, square_prime_factor_gives_zero)."},"skills":["Complete Prime Multiplicities Determine the Mobius Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0019","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d27fa11cebfdfdc0d460e2446aa1e8e26c154c31d576e35be311784a51ee33a3","family_id":"qf-mod40-mobius-canonical-value","seed":202611096019,"semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":380}"},"fields":{"title":"Complete Prime Multiplicities Determine the Mobius Value","stem":"Use positive integer n=380. Give (n, prime_multiset, distinct_primes, mu, certificate).","answer":{"kind":"expression","value":"(380, [2, 2, 5, 19], [2, 5, 19], 0, square_prime_factor_gives_zero)","explanation":"Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (380, [2, 2, 5, 19], [2, 5, 19], 0, square_prime_factor_gives_zero)."},"skills":["Complete Prime Multiplicities Determine the Mobius Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0020","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"827b6d74f2528f863824e533ec96d3e4ad9127dfe170bfaa3c212dba055a2647","family_id":"qf-mod40-mobius-canonical-value","seed":202611096020,"semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":420}"},"fields":{"title":"Complete Prime Multiplicities Determine the Mobius Value","stem":"Use positive integer n=420. Give (n, prime_multiset, distinct_primes, mu, certificate).","answer":{"kind":"expression","value":"(420, [2, 2, 3, 5, 7], [2, 3, 5, 7], 0, square_prime_factor_gives_zero)","explanation":"Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (420, [2, 2, 3, 5, 7], [2, 3, 5, 7], 0, square_prime_factor_gives_zero)."},"skills":["Complete Prime Multiplicities Determine the Mobius Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0021","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e85684657906c91c7c94f8e20c5e09b220fc7e11b3e7b12a7d13a704b84f9fd4","family_id":"qf-mod40-mobius-canonical-value","seed":202611096021,"semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":462}"},"fields":{"title":"Complete Prime Multiplicities Determine the Mobius Value","stem":"Use positive integer n=462. Give (n, prime_multiset, distinct_primes, mu, certificate).","answer":{"kind":"expression","value":"(462, [2, 3, 7, 11], [2, 3, 7, 11], 1, squarefree_parity_sign)","explanation":"Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (462, [2, 3, 7, 11], [2, 3, 7, 11], 1, squarefree_parity_sign)."},"skills":["Complete Prime Multiplicities Determine the Mobius Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0022","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"efda13d569ba43d40c0b26d23e70d275113b574fd0feed698f78be0ab8086981","family_id":"qf-mod40-mobius-canonical-value","seed":202611096022,"semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":506}"},"fields":{"title":"Complete Prime Multiplicities Determine the Mobius Value","stem":"Use positive integer n=506. Give (n, prime_multiset, distinct_primes, mu, certificate).","answer":{"kind":"expression","value":"(506, [2, 11, 23], [2, 11, 23], -1, squarefree_parity_sign)","explanation":"Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (506, [2, 11, 23], [2, 11, 23], -1, squarefree_parity_sign)."},"skills":["Complete Prime Multiplicities Determine the Mobius Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0023","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6622eb40c5f69351f8e45f276957d10f1758f90fcef4b5dba31f088312a8104","family_id":"qf-mod40-mobius-canonical-value","seed":202611096023,"semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":552}"},"fields":{"title":"Complete Prime Multiplicities Determine the Mobius Value","stem":"Use positive integer n=552. Give (n, prime_multiset, distinct_primes, mu, certificate).","answer":{"kind":"expression","value":"(552, [2, 2, 2, 3, 23], [2, 3, 23], 0, square_prime_factor_gives_zero)","explanation":"Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (552, [2, 2, 2, 3, 23], [2, 3, 23], 0, square_prime_factor_gives_zero)."},"skills":["Complete Prime Multiplicities Determine the Mobius Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0024","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f5608382c2e89e35ec30693907eadd561c31799769f086979bfb6e63bd7a1f6","family_id":"qf-mod40-mobius-canonical-value","seed":202611096024,"semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":600}"},"fields":{"title":"Complete Prime Multiplicities Determine the Mobius Value","stem":"Use positive integer n=600. Give (n, prime_multiset, distinct_primes, mu, certificate).","answer":{"kind":"expression","value":"(600, [2, 2, 2, 3, 5, 5], [2, 3, 5], 0, square_prime_factor_gives_zero)","explanation":"Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (600, [2, 2, 2, 3, 5, 5], [2, 3, 5], 0, square_prime_factor_gives_zero)."},"skills":["Complete Prime Multiplicities Determine the Mobius Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0025","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1ecc0b03917fec9083b866fd1cba924d12b320093d638789ceac8daa9f5e05e","family_id":"qf-mod40-mobius-canonical-value","seed":202611096025,"semantic_template":"qf-mod40-mobius-canonical-value/{\"n\":650}"},"fields":{"title":"Complete Prime Multiplicities Determine the Mobius Value","stem":"Use positive integer n=650. Give (n, prime_multiset, distinct_primes, mu, certificate).","answer":{"kind":"expression","value":"(650, [2, 5, 5, 13], [2, 5, 13], 0, square_prime_factor_gives_zero)","explanation":"Complete trial division gives the increasing prime multiset and distinct-prime list. Set mu(1)=1 for the empty product, use0 when any prime repeats, and otherwise use (−1) to the distinct-prime count. A partial factor search is insufficient. The exact result, retaining the requested order, is (650, [2, 5, 5, 13], [2, 5, 13], 0, square_prime_factor_gives_zero)."},"skills":["Complete Prime Multiplicities Determine the Mobius Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0026","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"577f109af28bb2fb1de00bcf870a6db7d60f625981a3142d9a7a3c45b29c31f4","family_id":"qf-mod40-mobius-factorization-audit","seed":202611096026,"semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,2],\"n\":4}"},"fields":{"title":"A Matching Composite Product Does Not Qualify as a Prime Factorization","stem":"Use positive integer n=4, supplied factors=[2, 2] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).","answer":{"kind":"expression","value":"(4, [2, 2], Yes, 0, Yes, complete_UPF_multiplicity_and_distinct_prime_count)","explanation":"Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (4, [2, 2], Yes, 0, Yes, complete_UPF_multiplicity_and_distinct_prime_count)."},"skills":["A Matching Composite Product Does Not Qualify as a Prime Factorization"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0027","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d5faf28135994fb30c1a2470adddafd81cdcc8aeba8a694a7024b8ca9c44ec7","family_id":"qf-mod40-mobius-factorization-audit","seed":202611096027,"semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,3],\"n\":6}"},"fields":{"title":"A Matching Composite Product Does Not Qualify as a Prime Factorization","stem":"Use positive integer n=6, supplied factors=[2, 3] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).","answer":{"kind":"expression","value":"(6, [2, 3], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)","explanation":"Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (6, [2, 3], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)."},"skills":["A Matching Composite Product Does Not Qualify as a Prime Factorization"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0028","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"927ffe4aa103f7115ac40b6fcf85931172e190f2d9f8ce2f5e6c2a3fdddff5a8","family_id":"qf-mod40-mobius-factorization-audit","seed":202611096028,"semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,5],\"n\":10}"},"fields":{"title":"A Matching Composite Product Does Not Qualify as a Prime Factorization","stem":"Use positive integer n=10, supplied factors=[2, 5] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).","answer":{"kind":"expression","value":"(10, [2, 5], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)","explanation":"Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (10, [2, 5], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)."},"skills":["A Matching Composite Product Does Not Qualify as a Prime Factorization"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0029","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5aa059abcf043f70bef30ede9248eb5b7df95be85194b6939bb7ce61a14bdaf8","family_id":"qf-mod40-mobius-factorization-audit","seed":202611096029,"semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,7],\"n\":14}"},"fields":{"title":"A Matching Composite Product Does Not Qualify as a Prime Factorization","stem":"Use positive integer n=14, supplied factors=[2, 7] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).","answer":{"kind":"expression","value":"(14, [2, 7], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)","explanation":"Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (14, [2, 7], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)."},"skills":["A Matching Composite Product Does Not Qualify as a Prime Factorization"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0030","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b76baa6f30e79536b2505cbbc7723806d447cd7b4a75f42298ccf53f0ef55ec7","family_id":"qf-mod40-mobius-factorization-audit","seed":202611096030,"semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,11],\"n\":22}"},"fields":{"title":"A Matching Composite Product Does Not Qualify as a Prime Factorization","stem":"Use positive integer n=22, supplied factors=[2, 11] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).","answer":{"kind":"expression","value":"(22, [2, 11], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)","explanation":"Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (22, [2, 11], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)."},"skills":["A Matching Composite Product Does Not Qualify as a Prime Factorization"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0031","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"407b84f93eb5196ec963a81015bd419f597d546b713627b4d3cd64802d1234d3","family_id":"qf-mod40-mobius-factorization-audit","seed":202611096031,"semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,13],\"n\":26}"},"fields":{"title":"A Matching Composite Product Does Not Qualify as a Prime Factorization","stem":"Use positive integer n=26, supplied factors=[2, 13] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).","answer":{"kind":"expression","value":"(26, [2, 13], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)","explanation":"Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (26, [2, 13], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)."},"skills":["A Matching Composite Product Does Not Qualify as a Prime Factorization"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0032","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"010f6b3b260a3b403d7b374d77728507c13652f9d6fb20170e0070b0c0611a07","family_id":"qf-mod40-mobius-factorization-audit","seed":202611096032,"semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,17],\"n\":34}"},"fields":{"title":"A Matching Composite Product Does Not Qualify as a Prime Factorization","stem":"Use positive integer n=34, supplied factors=[2, 17] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).","answer":{"kind":"expression","value":"(34, [2, 17], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)","explanation":"Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (34, [2, 17], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)."},"skills":["A Matching Composite Product Does Not Qualify as a Prime Factorization"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0033","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"94cf7b7e23040a47c9be350c1405e9210e2857ae6c6f7c1a6638b135c4efb50d","family_id":"qf-mod40-mobius-factorization-audit","seed":202611096033,"semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,19],\"n\":38}"},"fields":{"title":"A Matching Composite Product Does Not Qualify as a Prime Factorization","stem":"Use positive integer n=38, supplied factors=[2, 19] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).","answer":{"kind":"expression","value":"(38, [2, 19], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)","explanation":"Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (38, [2, 19], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)."},"skills":["A Matching Composite Product Does Not Qualify as a Prime Factorization"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0034","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81b3e59a6fc5aa5f1af0855a02b6395352837287883bf4c76d66d2cc6f0c3ad3","family_id":"qf-mod40-mobius-factorization-audit","seed":202611096034,"semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,23],\"n\":46}"},"fields":{"title":"A Matching Composite Product Does Not Qualify as a Prime Factorization","stem":"Use positive integer n=46, supplied factors=[2, 23] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).","answer":{"kind":"expression","value":"(46, [2, 23], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)","explanation":"Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (46, [2, 23], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)."},"skills":["A Matching Composite Product Does Not Qualify as a Prime Factorization"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0035","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e617cc58b69cd2d05a3e234add23d21a64c3448e7cb49ee801948cb4017ad211","family_id":"qf-mod40-mobius-factorization-audit","seed":202611096035,"semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,29],\"n\":58}"},"fields":{"title":"A Matching Composite Product Does Not Qualify as a Prime Factorization","stem":"Use positive integer n=58, supplied factors=[2, 29] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).","answer":{"kind":"expression","value":"(58, [2, 29], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)","explanation":"Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (58, [2, 29], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)."},"skills":["A Matching Composite Product Does Not Qualify as a Prime Factorization"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0036","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5689be140b95de7ef4c61dff7d74d1458f77ec2c82d2f1e411142f68bb669f47","family_id":"qf-mod40-mobius-factorization-audit","seed":202611096036,"semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,31],\"n\":62}"},"fields":{"title":"A Matching Composite Product Does Not Qualify as a Prime Factorization","stem":"Use positive integer n=62, supplied factors=[2, 31] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).","answer":{"kind":"expression","value":"(62, [2, 31], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)","explanation":"Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (62, [2, 31], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)."},"skills":["A Matching Composite Product Does Not Qualify as a Prime Factorization"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0037","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd1a44b22519fe5dd1d5b8ef1338e2fa0d6c6bb5c7ef5909a7a666e5d07a0488","family_id":"qf-mod40-mobius-factorization-audit","seed":202611096037,"semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,37],\"n\":74}"},"fields":{"title":"A Matching Composite Product Does Not Qualify as a Prime Factorization","stem":"Use positive integer n=74, supplied factors=[2, 37] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).","answer":{"kind":"expression","value":"(74, [2, 37], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)","explanation":"Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (74, [2, 37], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)."},"skills":["A Matching Composite Product Does Not Qualify as a Prime Factorization"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0038","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e89e0ed9d9fa060d0ededd6e2d5d5d3b90c20ccaefc6b46b48375e912442a6b","family_id":"qf-mod40-mobius-factorization-audit","seed":202611096038,"semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,41],\"n\":82}"},"fields":{"title":"A Matching Composite Product Does Not Qualify as a Prime Factorization","stem":"Use positive integer n=82, supplied factors=[2, 41] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).","answer":{"kind":"expression","value":"(82, [2, 41], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)","explanation":"Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (82, [2, 41], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)."},"skills":["A Matching Composite Product Does Not Qualify as a Prime Factorization"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0039","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"930e73890e9e147ed7d51918d9a5c3710d8171eaad8efc482e4e60c00c613f68","family_id":"qf-mod40-mobius-factorization-audit","seed":202611096039,"semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,43],\"n\":86}"},"fields":{"title":"A Matching Composite Product Does Not Qualify as a Prime Factorization","stem":"Use positive integer n=86, supplied factors=[2, 43] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).","answer":{"kind":"expression","value":"(86, [2, 43], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)","explanation":"Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (86, [2, 43], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)."},"skills":["A Matching Composite Product Does Not Qualify as a Prime Factorization"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0040","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e7bbf454c479ce9e6a541b36dfcfaac71bbf3107527d3a673279dae8a49aa60","family_id":"qf-mod40-mobius-factorization-audit","seed":202611096040,"semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,47],\"n\":94}"},"fields":{"title":"A Matching Composite Product Does Not Qualify as a Prime Factorization","stem":"Use positive integer n=94, supplied factors=[2, 47] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).","answer":{"kind":"expression","value":"(94, [2, 47], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)","explanation":"Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (94, [2, 47], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)."},"skills":["A Matching Composite Product Does Not Qualify as a Prime Factorization"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0041","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48e8fb7a028978b0a258c170de6f0bc384ba83bb9611486bfdbaf2fa9340ab29","family_id":"qf-mod40-mobius-factorization-audit","seed":202611096041,"semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,53],\"n\":106}"},"fields":{"title":"A Matching Composite Product Does Not Qualify as a Prime Factorization","stem":"Use positive integer n=106, supplied factors=[2, 53] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).","answer":{"kind":"expression","value":"(106, [2, 53], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)","explanation":"Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (106, [2, 53], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)."},"skills":["A Matching Composite Product Does Not Qualify as a Prime Factorization"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0042","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c662a251bd8bd08a14ebda92417ecaa3ccf6a9899c72afbdfe22a27caf42f53","family_id":"qf-mod40-mobius-factorization-audit","seed":202611096042,"semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,59],\"n\":118}"},"fields":{"title":"A Matching Composite Product Does Not Qualify as a Prime Factorization","stem":"Use positive integer n=118, supplied factors=[2, 59] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).","answer":{"kind":"expression","value":"(118, [2, 59], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)","explanation":"Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (118, [2, 59], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)."},"skills":["A Matching Composite Product Does Not Qualify as a Prime Factorization"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0043","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0dbf70e7c8bc056c74ee0ff040e7ee6c6a2851eb0f0e0b8e31efaef65b24331a","family_id":"qf-mod40-mobius-factorization-audit","seed":202611096043,"semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,61],\"n\":122}"},"fields":{"title":"A Matching Composite Product Does Not Qualify as a Prime Factorization","stem":"Use positive integer n=122, supplied factors=[2, 61] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).","answer":{"kind":"expression","value":"(122, [2, 61], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)","explanation":"Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (122, [2, 61], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)."},"skills":["A Matching Composite Product Does Not Qualify as a Prime Factorization"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0044","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e75777b0248b3ed679fce888cd05fc2417198af1b7b0cca7836aeeb08c95d622","family_id":"qf-mod40-mobius-factorization-audit","seed":202611096044,"semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,67],\"n\":134}"},"fields":{"title":"A Matching Composite Product Does Not Qualify as a Prime Factorization","stem":"Use positive integer n=134, supplied factors=[2, 67] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).","answer":{"kind":"expression","value":"(134, [2, 67], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)","explanation":"Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (134, [2, 67], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)."},"skills":["A Matching Composite Product Does Not Qualify as a Prime Factorization"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0045","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86e793556d33279c6a2ab1f7137d78d5f4d4ec9802f375ddf8e7082423471fe2","family_id":"qf-mod40-mobius-factorization-audit","seed":202611096045,"semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,71],\"n\":142}"},"fields":{"title":"A Matching Composite Product Does Not Qualify as a Prime Factorization","stem":"Use positive integer n=142, supplied factors=[2, 71] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).","answer":{"kind":"expression","value":"(142, [2, 71], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)","explanation":"Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (142, [2, 71], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)."},"skills":["A Matching Composite Product Does Not Qualify as a Prime Factorization"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0046","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"785c21c11a5e35d1af15f654bf436eca1dbc56f9c05d4ec3228ebf235e900680","family_id":"qf-mod40-mobius-factorization-audit","seed":202611096046,"semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,73],\"n\":146}"},"fields":{"title":"A Matching Composite Product Does Not Qualify as a Prime Factorization","stem":"Use positive integer n=146, supplied factors=[2, 73] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).","answer":{"kind":"expression","value":"(146, [2, 73], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)","explanation":"Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (146, [2, 73], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)."},"skills":["A Matching Composite Product Does Not Qualify as a Prime Factorization"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0047","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec5f335ad6c97bbe7e2355de237d1bf9d5f17a6d11724cb8c4ffb2bf58c71f69","family_id":"qf-mod40-mobius-factorization-audit","seed":202611096047,"semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,79],\"n\":158}"},"fields":{"title":"A Matching Composite Product Does Not Qualify as a Prime Factorization","stem":"Use positive integer n=158, supplied factors=[2, 79] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).","answer":{"kind":"expression","value":"(158, [2, 79], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)","explanation":"Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (158, [2, 79], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)."},"skills":["A Matching Composite Product Does Not Qualify as a Prime Factorization"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0048","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d7c7ee403622319872cabb803f7deefca962bab64c7dc7d099812482e80b67a","family_id":"qf-mod40-mobius-factorization-audit","seed":202611096048,"semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,83],\"n\":166}"},"fields":{"title":"A Matching Composite Product Does Not Qualify as a Prime Factorization","stem":"Use positive integer n=166, supplied factors=[2, 83] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).","answer":{"kind":"expression","value":"(166, [2, 83], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)","explanation":"Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (166, [2, 83], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)."},"skills":["A Matching Composite Product Does Not Qualify as a Prime Factorization"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0049","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20268cbdcb7b85b97c11195a655e8cc24562f4a37a270ef4b91aa4afd7c92239","family_id":"qf-mod40-mobius-factorization-audit","seed":202611096049,"semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,89],\"n\":178}"},"fields":{"title":"A Matching Composite Product Does Not Qualify as a Prime Factorization","stem":"Use positive integer n=178, supplied factors=[2, 89] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).","answer":{"kind":"expression","value":"(178, [2, 89], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)","explanation":"Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (178, [2, 89], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)."},"skills":["A Matching Composite Product Does Not Qualify as a Prime Factorization"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0050","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"259787f31de4895bda30413ecbe0095aed9c0c42ff73457165fc41b1ce0bc2cb","family_id":"qf-mod40-mobius-factorization-audit","seed":202611096050,"semantic_template":"qf-mod40-mobius-factorization-audit/{\"factors\":[2,97],\"n\":194}"},"fields":{"title":"A Matching Composite Product Does Not Qualify as a Prime Factorization","stem":"Use positive integer n=194, supplied factors=[2, 97] in original order. Give (n, factors, valid_prime_product, mu_or_null, agrees_with_canonical, certificate).","answer":{"kind":"expression","value":"(194, [2, 97], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)","explanation":"Require every factor to be prime and their complete product to equal n before computing a value from multiplicities. An invalid candidate gives null and No canonical agreement, even if its composite product numerically equals n. For n=1 the empty prime product is valid and mu=1. The exact result, retaining the requested order, is (194, [2, 97], Yes, 1, Yes, complete_UPF_multiplicity_and_distinct_prime_count)."},"skills":["A Matching Composite Product Does Not Qualify as a Prime Factorization"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0051","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8bf747907440a8cc71ed6d7672d8f3bb61345447e2e0277594e301a289a4e2df","family_id":"qf-mod40-mobius-divisor-sum","seed":202611096051,"semantic_template":"qf-mod40-mobius-divisor-sum/{\"n\":1}"},"fields":{"title":"All Positive Divisors Supply the Exact Mobius Sum Identity","stem":"Use positive integer n=1. Give (n, divisor_mu_rows, sum, target, equal).","answer":{"kind":"expression","value":"(1, [[1, 1]], 1, 1, Yes)","explanation":"Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (1, [[1, 1]], 1, 1, Yes)."},"skills":["All Positive Divisors Supply the Exact Mobius Sum Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0052","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba7c6735920a04b0c6a44aef5aac1462b65917e491ae1c3022835b244682bc5d","family_id":"qf-mod40-mobius-divisor-sum","seed":202611096052,"semantic_template":"qf-mod40-mobius-divisor-sum/{\"n\":2}"},"fields":{"title":"All Positive Divisors Supply the Exact Mobius Sum Identity","stem":"Use positive integer n=2. Give (n, divisor_mu_rows, sum, target, equal).","answer":{"kind":"expression","value":"(2, [[1, 1], [2, -1]], 0, 0, Yes)","explanation":"Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (2, [[1, 1], [2, -1]], 0, 0, Yes)."},"skills":["All Positive Divisors Supply the Exact Mobius Sum Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0053","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84aac71c72429f4be62f7101b0d45d70da1008168c3d028c4a30dc4a73b3e317","family_id":"qf-mod40-mobius-divisor-sum","seed":202611096053,"semantic_template":"qf-mod40-mobius-divisor-sum/{\"n\":3}"},"fields":{"title":"All Positive Divisors Supply the Exact Mobius Sum Identity","stem":"Use positive integer n=3. Give (n, divisor_mu_rows, sum, target, equal).","answer":{"kind":"expression","value":"(3, [[1, 1], [3, -1]], 0, 0, Yes)","explanation":"Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (3, [[1, 1], [3, -1]], 0, 0, Yes)."},"skills":["All Positive Divisors Supply the Exact Mobius Sum Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0054","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"235f00462b9cdfd6feb858f227c46073b98703199c7a60aa0fb322844205ed61","family_id":"qf-mod40-mobius-divisor-sum","seed":202611096054,"semantic_template":"qf-mod40-mobius-divisor-sum/{\"n\":4}"},"fields":{"title":"All Positive Divisors Supply the Exact Mobius Sum Identity","stem":"Use positive integer n=4. Give (n, divisor_mu_rows, sum, target, equal).","answer":{"kind":"expression","value":"(4, [[1, 1], [2, -1], [4, 0]], 0, 0, Yes)","explanation":"Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (4, [[1, 1], [2, -1], [4, 0]], 0, 0, Yes)."},"skills":["All Positive Divisors Supply the Exact Mobius Sum Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0055","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"82833224675f56075a8dd8504b0a10a4d4e45383888e55eed95105e5036fc190","family_id":"qf-mod40-mobius-divisor-sum","seed":202611096055,"semantic_template":"qf-mod40-mobius-divisor-sum/{\"n\":5}"},"fields":{"title":"All Positive Divisors Supply the Exact Mobius Sum Identity","stem":"Use positive integer n=5. Give (n, divisor_mu_rows, sum, target, equal).","answer":{"kind":"expression","value":"(5, [[1, 1], [5, -1]], 0, 0, Yes)","explanation":"Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (5, [[1, 1], [5, -1]], 0, 0, Yes)."},"skills":["All Positive Divisors Supply the Exact Mobius Sum Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0056","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"836ac6ebf5abc108901031ff41609099f27ce5c3c3a986e716284ab1561f2054","family_id":"qf-mod40-mobius-divisor-sum","seed":202611096056,"semantic_template":"qf-mod40-mobius-divisor-sum/{\"n\":6}"},"fields":{"title":"All Positive Divisors Supply the Exact Mobius Sum Identity","stem":"Use positive integer n=6. Give (n, divisor_mu_rows, sum, target, equal).","answer":{"kind":"expression","value":"(6, [[1, 1], [2, -1], [3, -1], [6, 1]], 0, 0, Yes)","explanation":"Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (6, [[1, 1], [2, -1], [3, -1], [6, 1]], 0, 0, Yes)."},"skills":["All Positive Divisors Supply the Exact Mobius Sum Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0057","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a33985997ac92d76d1621c6b2f1ef957f52b918d97090ecfbabfb40f1007cbf","family_id":"qf-mod40-mobius-divisor-sum","seed":202611096057,"semantic_template":"qf-mod40-mobius-divisor-sum/{\"n\":7}"},"fields":{"title":"All Positive Divisors Supply the Exact Mobius Sum Identity","stem":"Use positive integer n=7. Give (n, divisor_mu_rows, sum, target, equal).","answer":{"kind":"expression","value":"(7, [[1, 1], [7, -1]], 0, 0, Yes)","explanation":"Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (7, [[1, 1], [7, -1]], 0, 0, Yes)."},"skills":["All Positive Divisors Supply the Exact Mobius Sum Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0058","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"715638f408024c5923d9eab2f4eff0de973815d3f6fececa8a943837fbdbc2fb","family_id":"qf-mod40-mobius-divisor-sum","seed":202611096058,"semantic_template":"qf-mod40-mobius-divisor-sum/{\"n\":8}"},"fields":{"title":"All Positive Divisors Supply the Exact Mobius Sum Identity","stem":"Use positive integer n=8. Give (n, divisor_mu_rows, sum, target, equal).","answer":{"kind":"expression","value":"(8, [[1, 1], [2, -1], [4, 0], [8, 0]], 0, 0, Yes)","explanation":"Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (8, [[1, 1], [2, -1], [4, 0], [8, 0]], 0, 0, Yes)."},"skills":["All Positive Divisors Supply the Exact Mobius Sum Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0059","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8da0b8f77141243470541b4c57119a153ebe602265a97962a0ff925eb20887c5","family_id":"qf-mod40-mobius-divisor-sum","seed":202611096059,"semantic_template":"qf-mod40-mobius-divisor-sum/{\"n\":9}"},"fields":{"title":"All Positive Divisors Supply the Exact Mobius Sum Identity","stem":"Use positive integer n=9. Give (n, divisor_mu_rows, sum, target, equal).","answer":{"kind":"expression","value":"(9, [[1, 1], [3, -1], [9, 0]], 0, 0, Yes)","explanation":"Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (9, [[1, 1], [3, -1], [9, 0]], 0, 0, Yes)."},"skills":["All Positive Divisors Supply the Exact Mobius Sum Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0060","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b6842730cc6a3001fc01688e5025c7b48dfbfb5efa9a1f46733a9550f67b89a","family_id":"qf-mod40-mobius-divisor-sum","seed":202611096060,"semantic_template":"qf-mod40-mobius-divisor-sum/{\"n\":10}"},"fields":{"title":"All Positive Divisors Supply the Exact Mobius Sum Identity","stem":"Use positive integer n=10. Give (n, divisor_mu_rows, sum, target, equal).","answer":{"kind":"expression","value":"(10, [[1, 1], [2, -1], [5, -1], [10, 1]], 0, 0, Yes)","explanation":"Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (10, [[1, 1], [2, -1], [5, -1], [10, 1]], 0, 0, Yes)."},"skills":["All Positive Divisors Supply the Exact Mobius Sum Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0061","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf5d34012b1cfa8a390bb2a130827f07d98b38f1801b56233f7c05a650a201fb","family_id":"qf-mod40-mobius-divisor-sum","seed":202611096061,"semantic_template":"qf-mod40-mobius-divisor-sum/{\"n\":11}"},"fields":{"title":"All Positive Divisors Supply the Exact Mobius Sum Identity","stem":"Use positive integer n=11. Give (n, divisor_mu_rows, sum, target, equal).","answer":{"kind":"expression","value":"(11, [[1, 1], [11, -1]], 0, 0, Yes)","explanation":"Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (11, [[1, 1], [11, -1]], 0, 0, Yes)."},"skills":["All Positive Divisors Supply the Exact Mobius Sum Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0062","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"15b238c8df568c72fe8b1487ca1b4bc9652ecd9ed37634f017814cb973f39c2b","family_id":"qf-mod40-mobius-divisor-sum","seed":202611096062,"semantic_template":"qf-mod40-mobius-divisor-sum/{\"n\":12}"},"fields":{"title":"All Positive Divisors Supply the Exact Mobius Sum Identity","stem":"Use positive integer n=12. Give (n, divisor_mu_rows, sum, target, equal).","answer":{"kind":"expression","value":"(12, [[1, 1], [2, -1], [3, -1], [4, 0], [6, 1], [12, 0]], 0, 0, Yes)","explanation":"Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (12, [[1, 1], [2, -1], [3, -1], [4, 0], [6, 1], [12, 0]], 0, 0, Yes)."},"skills":["All Positive Divisors Supply the Exact Mobius Sum Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0063","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb94a2a07b8b60ae49051107bfb897dc4c25d2fb44269e0301dde04c2678ca3c","family_id":"qf-mod40-mobius-divisor-sum","seed":202611096063,"semantic_template":"qf-mod40-mobius-divisor-sum/{\"n\":13}"},"fields":{"title":"All Positive Divisors Supply the Exact Mobius Sum Identity","stem":"Use positive integer n=13. Give (n, divisor_mu_rows, sum, target, equal).","answer":{"kind":"expression","value":"(13, [[1, 1], [13, -1]], 0, 0, Yes)","explanation":"Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (13, [[1, 1], [13, -1]], 0, 0, Yes)."},"skills":["All Positive Divisors Supply the Exact Mobius Sum Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0064","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b720cd5afc4c1a3a8ccb41db7e80c440ceb92b06ed4a0dc1d149fcf772c7144f","family_id":"qf-mod40-mobius-divisor-sum","seed":202611096064,"semantic_template":"qf-mod40-mobius-divisor-sum/{\"n\":14}"},"fields":{"title":"All Positive Divisors Supply the Exact Mobius Sum Identity","stem":"Use positive integer n=14. Give (n, divisor_mu_rows, sum, target, equal).","answer":{"kind":"expression","value":"(14, [[1, 1], [2, -1], [7, -1], [14, 1]], 0, 0, Yes)","explanation":"Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (14, [[1, 1], [2, -1], [7, -1], [14, 1]], 0, 0, Yes)."},"skills":["All Positive Divisors Supply the Exact Mobius Sum Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0065","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e057f70af734795deab94b9b06dc8a310aaa8d8bb452425f39c46660ff84906","family_id":"qf-mod40-mobius-divisor-sum","seed":202611096065,"semantic_template":"qf-mod40-mobius-divisor-sum/{\"n\":15}"},"fields":{"title":"All Positive Divisors Supply the Exact Mobius Sum Identity","stem":"Use positive integer n=15. Give (n, divisor_mu_rows, sum, target, equal).","answer":{"kind":"expression","value":"(15, [[1, 1], [3, -1], [5, -1], [15, 1]], 0, 0, Yes)","explanation":"Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (15, [[1, 1], [3, -1], [5, -1], [15, 1]], 0, 0, Yes)."},"skills":["All Positive Divisors Supply the Exact Mobius Sum Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0066","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2074da1a2eebb6f3389484a5af34e6108bf0e97ccdd5d0e3e7108ed639f0e1e2","family_id":"qf-mod40-mobius-divisor-sum","seed":202611096066,"semantic_template":"qf-mod40-mobius-divisor-sum/{\"n\":16}"},"fields":{"title":"All Positive Divisors Supply the Exact Mobius Sum Identity","stem":"Use positive integer n=16. Give (n, divisor_mu_rows, sum, target, equal).","answer":{"kind":"expression","value":"(16, [[1, 1], [2, -1], [4, 0], [8, 0], [16, 0]], 0, 0, Yes)","explanation":"Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (16, [[1, 1], [2, -1], [4, 0], [8, 0], [16, 0]], 0, 0, Yes)."},"skills":["All Positive Divisors Supply the Exact Mobius Sum Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0067","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52567672d37be668b9565fabd5fa5991ab5d9f7f0682239182c1d8424aa8bde4","family_id":"qf-mod40-mobius-divisor-sum","seed":202611096067,"semantic_template":"qf-mod40-mobius-divisor-sum/{\"n\":17}"},"fields":{"title":"All Positive Divisors Supply the Exact Mobius Sum Identity","stem":"Use positive integer n=17. Give (n, divisor_mu_rows, sum, target, equal).","answer":{"kind":"expression","value":"(17, [[1, 1], [17, -1]], 0, 0, Yes)","explanation":"Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (17, [[1, 1], [17, -1]], 0, 0, Yes)."},"skills":["All Positive Divisors Supply the Exact Mobius Sum Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0068","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ac084452379e93003b808847d7d5cccae31e399305e7dfc80c9f253b5058883","family_id":"qf-mod40-mobius-divisor-sum","seed":202611096068,"semantic_template":"qf-mod40-mobius-divisor-sum/{\"n\":18}"},"fields":{"title":"All Positive Divisors Supply the Exact Mobius Sum Identity","stem":"Use positive integer n=18. Give (n, divisor_mu_rows, sum, target, equal).","answer":{"kind":"expression","value":"(18, [[1, 1], [2, -1], [3, -1], [6, 1], [9, 0], [18, 0]], 0, 0, Yes)","explanation":"Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (18, [[1, 1], [2, -1], [3, -1], [6, 1], [9, 0], [18, 0]], 0, 0, Yes)."},"skills":["All Positive Divisors Supply the Exact Mobius Sum Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0069","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b5827477460d15dad093953c78c7bdba11b1d1f194fac4e228fee206915424b","family_id":"qf-mod40-mobius-divisor-sum","seed":202611096069,"semantic_template":"qf-mod40-mobius-divisor-sum/{\"n\":19}"},"fields":{"title":"All Positive Divisors Supply the Exact Mobius Sum Identity","stem":"Use positive integer n=19. Give (n, divisor_mu_rows, sum, target, equal).","answer":{"kind":"expression","value":"(19, [[1, 1], [19, -1]], 0, 0, Yes)","explanation":"Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (19, [[1, 1], [19, -1]], 0, 0, Yes)."},"skills":["All Positive Divisors Supply the Exact Mobius Sum Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0070","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"87ffa40275581c080875022f239328a10210a9251db8e4e2f6265ecc3bc73356","family_id":"qf-mod40-mobius-divisor-sum","seed":202611096070,"semantic_template":"qf-mod40-mobius-divisor-sum/{\"n\":20}"},"fields":{"title":"All Positive Divisors Supply the Exact Mobius Sum Identity","stem":"Use positive integer n=20. Give (n, divisor_mu_rows, sum, target, equal).","answer":{"kind":"expression","value":"(20, [[1, 1], [2, -1], [4, 0], [5, -1], [10, 1], [20, 0]], 0, 0, Yes)","explanation":"Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (20, [[1, 1], [2, -1], [4, 0], [5, -1], [10, 1], [20, 0]], 0, 0, Yes)."},"skills":["All Positive Divisors Supply the Exact Mobius Sum Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0071","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a29ed660cd7bb342f7df878e63071cfa63bcefa07e70c9bcfd450a7c100c5d0c","family_id":"qf-mod40-mobius-divisor-sum","seed":202611096071,"semantic_template":"qf-mod40-mobius-divisor-sum/{\"n\":21}"},"fields":{"title":"All Positive Divisors Supply the Exact Mobius Sum Identity","stem":"Use positive integer n=21. Give (n, divisor_mu_rows, sum, target, equal).","answer":{"kind":"expression","value":"(21, [[1, 1], [3, -1], [7, -1], [21, 1]], 0, 0, Yes)","explanation":"Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (21, [[1, 1], [3, -1], [7, -1], [21, 1]], 0, 0, Yes)."},"skills":["All Positive Divisors Supply the Exact Mobius Sum Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0072","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b97851ab1b5da689d53e2e2ba0820c4e6676b336f4742f5b5e16454c686e7921","family_id":"qf-mod40-mobius-divisor-sum","seed":202611096072,"semantic_template":"qf-mod40-mobius-divisor-sum/{\"n\":22}"},"fields":{"title":"All Positive Divisors Supply the Exact Mobius Sum Identity","stem":"Use positive integer n=22. Give (n, divisor_mu_rows, sum, target, equal).","answer":{"kind":"expression","value":"(22, [[1, 1], [2, -1], [11, -1], [22, 1]], 0, 0, Yes)","explanation":"Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (22, [[1, 1], [2, -1], [11, -1], [22, 1]], 0, 0, Yes)."},"skills":["All Positive Divisors Supply the Exact Mobius Sum Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0073","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6770f587f34104e4d5a10ba7c3ae3cbaad321d231f22b21251011d0f8880628d","family_id":"qf-mod40-mobius-divisor-sum","seed":202611096073,"semantic_template":"qf-mod40-mobius-divisor-sum/{\"n\":23}"},"fields":{"title":"All Positive Divisors Supply the Exact Mobius Sum Identity","stem":"Use positive integer n=23. Give (n, divisor_mu_rows, sum, target, equal).","answer":{"kind":"expression","value":"(23, [[1, 1], [23, -1]], 0, 0, Yes)","explanation":"Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (23, [[1, 1], [23, -1]], 0, 0, Yes)."},"skills":["All Positive Divisors Supply the Exact Mobius Sum Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0074","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c2db8af2a4755dff119ec6fe3252cd2ccf6c4ffeb518fb3031316ad172d7dec","family_id":"qf-mod40-mobius-divisor-sum","seed":202611096074,"semantic_template":"qf-mod40-mobius-divisor-sum/{\"n\":24}"},"fields":{"title":"All Positive Divisors Supply the Exact Mobius Sum Identity","stem":"Use positive integer n=24. Give (n, divisor_mu_rows, sum, target, equal).","answer":{"kind":"expression","value":"(24, [[1, 1], [2, -1], [3, -1], [4, 0], [6, 1], [8, 0], [12, 0], [24, 0]], 0, 0, Yes)","explanation":"Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (24, [[1, 1], [2, -1], [3, -1], [4, 0], [6, 1], [8, 0], [12, 0], [24, 0]], 0, 0, Yes)."},"skills":["All Positive Divisors Supply the Exact Mobius Sum Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0075","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1ab6197ac3d9d9f3b6c0a720e0b5caee54bbe091f96a281ff8ad44263baa653","family_id":"qf-mod40-mobius-divisor-sum","seed":202611096075,"semantic_template":"qf-mod40-mobius-divisor-sum/{\"n\":25}"},"fields":{"title":"All Positive Divisors Supply the Exact Mobius Sum Identity","stem":"Use positive integer n=25. Give (n, divisor_mu_rows, sum, target, equal).","answer":{"kind":"expression","value":"(25, [[1, 1], [5, -1], [25, 0]], 0, 0, Yes)","explanation":"Enumerate every positive divisor d increasingly and its complete mu(d). Their sum is1 when n=1 and0 otherwise. This exact instance check does not justify classifying an incompletely factored input. The exact result, retaining the requested order, is (25, [[1, 1], [5, -1], [25, 0]], 0, 0, Yes)."},"skills":["All Positive Divisors Supply the Exact Mobius Sum Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0076","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75b21b0af7bb4f782ada47dd088293ec9c42e772982dbc14960107165d33476e","family_id":"qf-mod40-mobius-domain-boundary","seed":202611096076,"semantic_template":"qf-mod40-mobius-domain-boundary/{\"n\":0}"},"fields":{"title":"Zero and Negative Integers Are Outside This Mobius Domain","stem":"Use nonpositive bounded integer n=0. Give (n, in_positive_domain, mu_or_null, boundary).","answer":{"kind":"expression","value":"(0, No, null, Mobius_domain_positive_integers_only)","explanation":"Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (0, No, null, Mobius_domain_positive_integers_only)."},"skills":["Zero and Negative Integers Are Outside This Mobius Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0077","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd9579a2fff312e67a681519dd510caad126e58100c5c7bfa2de00e5b66f2289","family_id":"qf-mod40-mobius-domain-boundary","seed":202611096077,"semantic_template":"qf-mod40-mobius-domain-boundary/{\"n\":-1}"},"fields":{"title":"Zero and Negative Integers Are Outside This Mobius Domain","stem":"Use nonpositive bounded integer n=-1. Give (n, in_positive_domain, mu_or_null, boundary).","answer":{"kind":"expression","value":"(-1, No, null, Mobius_domain_positive_integers_only)","explanation":"Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-1, No, null, Mobius_domain_positive_integers_only)."},"skills":["Zero and Negative Integers Are Outside This Mobius Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0078","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53744dd1568fe88254a06d6064cc75547958c1aeef89a3226a3657b684063774","family_id":"qf-mod40-mobius-domain-boundary","seed":202611096078,"semantic_template":"qf-mod40-mobius-domain-boundary/{\"n\":-2}"},"fields":{"title":"Zero and Negative Integers Are Outside This Mobius Domain","stem":"Use nonpositive bounded integer n=-2. Give (n, in_positive_domain, mu_or_null, boundary).","answer":{"kind":"expression","value":"(-2, No, null, Mobius_domain_positive_integers_only)","explanation":"Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-2, No, null, Mobius_domain_positive_integers_only)."},"skills":["Zero and Negative Integers Are Outside This Mobius Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0079","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"052ff33b54698740f63305ea31610cd40d00903fed2bceeaa7508135869d8687","family_id":"qf-mod40-mobius-domain-boundary","seed":202611096079,"semantic_template":"qf-mod40-mobius-domain-boundary/{\"n\":-3}"},"fields":{"title":"Zero and Negative Integers Are Outside This Mobius Domain","stem":"Use nonpositive bounded integer n=-3. Give (n, in_positive_domain, mu_or_null, boundary).","answer":{"kind":"expression","value":"(-3, No, null, Mobius_domain_positive_integers_only)","explanation":"Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-3, No, null, Mobius_domain_positive_integers_only)."},"skills":["Zero and Negative Integers Are Outside This Mobius Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0080","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c45e3640ea1286b1963aa5bbaf42a4c273c7aaeba12ed8b5109251f4cc88929","family_id":"qf-mod40-mobius-domain-boundary","seed":202611096080,"semantic_template":"qf-mod40-mobius-domain-boundary/{\"n\":-4}"},"fields":{"title":"Zero and Negative Integers Are Outside This Mobius Domain","stem":"Use nonpositive bounded integer n=-4. Give (n, in_positive_domain, mu_or_null, boundary).","answer":{"kind":"expression","value":"(-4, No, null, Mobius_domain_positive_integers_only)","explanation":"Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-4, No, null, Mobius_domain_positive_integers_only)."},"skills":["Zero and Negative Integers Are Outside This Mobius Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0081","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f2f4a76532ef5c032d4187b8c532a404652cf670e8b60b2db932869bbf8da0c","family_id":"qf-mod40-mobius-domain-boundary","seed":202611096081,"semantic_template":"qf-mod40-mobius-domain-boundary/{\"n\":-5}"},"fields":{"title":"Zero and Negative Integers Are Outside This Mobius Domain","stem":"Use nonpositive bounded integer n=-5. Give (n, in_positive_domain, mu_or_null, boundary).","answer":{"kind":"expression","value":"(-5, No, null, Mobius_domain_positive_integers_only)","explanation":"Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-5, No, null, Mobius_domain_positive_integers_only)."},"skills":["Zero and Negative Integers Are Outside This Mobius Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0082","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16d0f62edfeaa9e189d2b0cdbd10258f7b96c5adb43d589e5f7343c5866956da","family_id":"qf-mod40-mobius-domain-boundary","seed":202611096082,"semantic_template":"qf-mod40-mobius-domain-boundary/{\"n\":-6}"},"fields":{"title":"Zero and Negative Integers Are Outside This Mobius Domain","stem":"Use nonpositive bounded integer n=-6. Give (n, in_positive_domain, mu_or_null, boundary).","answer":{"kind":"expression","value":"(-6, No, null, Mobius_domain_positive_integers_only)","explanation":"Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-6, No, null, Mobius_domain_positive_integers_only)."},"skills":["Zero and Negative Integers Are Outside This Mobius Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0083","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"853ed627e0fd4a695e99660564c303ebda7e1c31361ef38a30e3babb7f359a69","family_id":"qf-mod40-mobius-domain-boundary","seed":202611096083,"semantic_template":"qf-mod40-mobius-domain-boundary/{\"n\":-7}"},"fields":{"title":"Zero and Negative Integers Are Outside This Mobius Domain","stem":"Use nonpositive bounded integer n=-7. Give (n, in_positive_domain, mu_or_null, boundary).","answer":{"kind":"expression","value":"(-7, No, null, Mobius_domain_positive_integers_only)","explanation":"Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-7, No, null, Mobius_domain_positive_integers_only)."},"skills":["Zero and Negative Integers Are Outside This Mobius Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0084","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29003feea632f499c2eebd4b74e52ebb971040494c2948c2836a22355bbbf808","family_id":"qf-mod40-mobius-domain-boundary","seed":202611096084,"semantic_template":"qf-mod40-mobius-domain-boundary/{\"n\":-8}"},"fields":{"title":"Zero and Negative Integers Are Outside This Mobius Domain","stem":"Use nonpositive bounded integer n=-8. Give (n, in_positive_domain, mu_or_null, boundary).","answer":{"kind":"expression","value":"(-8, No, null, Mobius_domain_positive_integers_only)","explanation":"Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-8, No, null, Mobius_domain_positive_integers_only)."},"skills":["Zero and Negative Integers Are Outside This Mobius Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0085","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3d97488364c7dbe4c556b6aa661698f022eb0306781d7bb7cb15ff8735a0fd0","family_id":"qf-mod40-mobius-domain-boundary","seed":202611096085,"semantic_template":"qf-mod40-mobius-domain-boundary/{\"n\":-9}"},"fields":{"title":"Zero and Negative Integers Are Outside This Mobius Domain","stem":"Use nonpositive bounded integer n=-9. Give (n, in_positive_domain, mu_or_null, boundary).","answer":{"kind":"expression","value":"(-9, No, null, Mobius_domain_positive_integers_only)","explanation":"Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-9, No, null, Mobius_domain_positive_integers_only)."},"skills":["Zero and Negative Integers Are Outside This Mobius Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0086","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6d975d824154750ebc4ff38c8cf32e760050549e57ffe8c4f1763e2f80b5c6d","family_id":"qf-mod40-mobius-domain-boundary","seed":202611096086,"semantic_template":"qf-mod40-mobius-domain-boundary/{\"n\":-10}"},"fields":{"title":"Zero and Negative Integers Are Outside This Mobius Domain","stem":"Use nonpositive bounded integer n=-10. Give (n, in_positive_domain, mu_or_null, boundary).","answer":{"kind":"expression","value":"(-10, No, null, Mobius_domain_positive_integers_only)","explanation":"Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-10, No, null, Mobius_domain_positive_integers_only)."},"skills":["Zero and Negative Integers Are Outside This Mobius Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0087","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc898c1077065f365698a1650130ccccd9f6a9c51297f02d3ccd69763731fa32","family_id":"qf-mod40-mobius-domain-boundary","seed":202611096087,"semantic_template":"qf-mod40-mobius-domain-boundary/{\"n\":-11}"},"fields":{"title":"Zero and Negative Integers Are Outside This Mobius Domain","stem":"Use nonpositive bounded integer n=-11. Give (n, in_positive_domain, mu_or_null, boundary).","answer":{"kind":"expression","value":"(-11, No, null, Mobius_domain_positive_integers_only)","explanation":"Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-11, No, null, Mobius_domain_positive_integers_only)."},"skills":["Zero and Negative Integers Are Outside This Mobius Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0088","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b9b84bc0e1485b34b432542a084612155d33b980f3a8b71b51da8b385d49803","family_id":"qf-mod40-mobius-domain-boundary","seed":202611096088,"semantic_template":"qf-mod40-mobius-domain-boundary/{\"n\":-12}"},"fields":{"title":"Zero and Negative Integers Are Outside This Mobius Domain","stem":"Use nonpositive bounded integer n=-12. Give (n, in_positive_domain, mu_or_null, boundary).","answer":{"kind":"expression","value":"(-12, No, null, Mobius_domain_positive_integers_only)","explanation":"Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-12, No, null, Mobius_domain_positive_integers_only)."},"skills":["Zero and Negative Integers Are Outside This Mobius Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0089","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6852e45c97ec11eb88422be763eaa4854f999f73ab9a0d49ff9694930a2cc40a","family_id":"qf-mod40-mobius-domain-boundary","seed":202611096089,"semantic_template":"qf-mod40-mobius-domain-boundary/{\"n\":-13}"},"fields":{"title":"Zero and Negative Integers Are Outside This Mobius Domain","stem":"Use nonpositive bounded integer n=-13. Give (n, in_positive_domain, mu_or_null, boundary).","answer":{"kind":"expression","value":"(-13, No, null, Mobius_domain_positive_integers_only)","explanation":"Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-13, No, null, Mobius_domain_positive_integers_only)."},"skills":["Zero and Negative Integers Are Outside This Mobius Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0090","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a7884b69c27ba7a82b6c23d64ad796dc7ce6282e133525baef73be6ebe4f9f7","family_id":"qf-mod40-mobius-domain-boundary","seed":202611096090,"semantic_template":"qf-mod40-mobius-domain-boundary/{\"n\":-14}"},"fields":{"title":"Zero and Negative Integers Are Outside This Mobius Domain","stem":"Use nonpositive bounded integer n=-14. Give (n, in_positive_domain, mu_or_null, boundary).","answer":{"kind":"expression","value":"(-14, No, null, Mobius_domain_positive_integers_only)","explanation":"Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-14, No, null, Mobius_domain_positive_integers_only)."},"skills":["Zero and Negative Integers Are Outside This Mobius Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0091","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cebdb9d910eac1041810c4d1c0daf90336f27a5716c6a63176c2fe2daa0ce8fe","family_id":"qf-mod40-mobius-domain-boundary","seed":202611096091,"semantic_template":"qf-mod40-mobius-domain-boundary/{\"n\":-15}"},"fields":{"title":"Zero and Negative Integers Are Outside This Mobius Domain","stem":"Use nonpositive bounded integer n=-15. Give (n, in_positive_domain, mu_or_null, boundary).","answer":{"kind":"expression","value":"(-15, No, null, Mobius_domain_positive_integers_only)","explanation":"Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-15, No, null, Mobius_domain_positive_integers_only)."},"skills":["Zero and Negative Integers Are Outside This Mobius Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0092","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6476a0d6b213de8f275848272594fc3fe397c60477b871c9b6104359a2664891","family_id":"qf-mod40-mobius-domain-boundary","seed":202611096092,"semantic_template":"qf-mod40-mobius-domain-boundary/{\"n\":-16}"},"fields":{"title":"Zero and Negative Integers Are Outside This Mobius Domain","stem":"Use nonpositive bounded integer n=-16. Give (n, in_positive_domain, mu_or_null, boundary).","answer":{"kind":"expression","value":"(-16, No, null, Mobius_domain_positive_integers_only)","explanation":"Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-16, No, null, Mobius_domain_positive_integers_only)."},"skills":["Zero and Negative Integers Are Outside This Mobius Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0093","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3fbc09a45352f9067eb846476f37487d84d6c8c7a49f1f60194e224c27df93de","family_id":"qf-mod40-mobius-domain-boundary","seed":202611096093,"semantic_template":"qf-mod40-mobius-domain-boundary/{\"n\":-17}"},"fields":{"title":"Zero and Negative Integers Are Outside This Mobius Domain","stem":"Use nonpositive bounded integer n=-17. Give (n, in_positive_domain, mu_or_null, boundary).","answer":{"kind":"expression","value":"(-17, No, null, Mobius_domain_positive_integers_only)","explanation":"Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-17, No, null, Mobius_domain_positive_integers_only)."},"skills":["Zero and Negative Integers Are Outside This Mobius Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0094","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5e0d329aa1bcb6e759a62e7b74a2f05ed766f0a1d22edeb38ea672ca6a043a8","family_id":"qf-mod40-mobius-domain-boundary","seed":202611096094,"semantic_template":"qf-mod40-mobius-domain-boundary/{\"n\":-18}"},"fields":{"title":"Zero and Negative Integers Are Outside This Mobius Domain","stem":"Use nonpositive bounded integer n=-18. Give (n, in_positive_domain, mu_or_null, boundary).","answer":{"kind":"expression","value":"(-18, No, null, Mobius_domain_positive_integers_only)","explanation":"Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-18, No, null, Mobius_domain_positive_integers_only)."},"skills":["Zero and Negative Integers Are Outside This Mobius Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0095","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b99203adf591db520fbe83b16c9fda44c7900f818894950f89f022ce4c58de9","family_id":"qf-mod40-mobius-domain-boundary","seed":202611096095,"semantic_template":"qf-mod40-mobius-domain-boundary/{\"n\":-19}"},"fields":{"title":"Zero and Negative Integers Are Outside This Mobius Domain","stem":"Use nonpositive bounded integer n=-19. Give (n, in_positive_domain, mu_or_null, boundary).","answer":{"kind":"expression","value":"(-19, No, null, Mobius_domain_positive_integers_only)","explanation":"Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-19, No, null, Mobius_domain_positive_integers_only)."},"skills":["Zero and Negative Integers Are Outside This Mobius Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0096","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"186c1d3b6002a57f3dda53e7624e70122584bb1d7b46d268cabbffa6fe98a49d","family_id":"qf-mod40-mobius-domain-boundary","seed":202611096096,"semantic_template":"qf-mod40-mobius-domain-boundary/{\"n\":-20}"},"fields":{"title":"Zero and Negative Integers Are Outside This Mobius Domain","stem":"Use nonpositive bounded integer n=-20. Give (n, in_positive_domain, mu_or_null, boundary).","answer":{"kind":"expression","value":"(-20, No, null, Mobius_domain_positive_integers_only)","explanation":"Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-20, No, null, Mobius_domain_positive_integers_only)."},"skills":["Zero and Negative Integers Are Outside This Mobius Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0097","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29cd586537fca0ae1ff1c00f99574181673693f8912d9ec05db69f8e2c8860f7","family_id":"qf-mod40-mobius-domain-boundary","seed":202611096097,"semantic_template":"qf-mod40-mobius-domain-boundary/{\"n\":-21}"},"fields":{"title":"Zero and Negative Integers Are Outside This Mobius Domain","stem":"Use nonpositive bounded integer n=-21. Give (n, in_positive_domain, mu_or_null, boundary).","answer":{"kind":"expression","value":"(-21, No, null, Mobius_domain_positive_integers_only)","explanation":"Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-21, No, null, Mobius_domain_positive_integers_only)."},"skills":["Zero and Negative Integers Are Outside This Mobius Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0098","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8efddeb334e2d80fdebe8540c019259f72ad2ad258ee6c84f9f6871b2068d35e","family_id":"qf-mod40-mobius-domain-boundary","seed":202611096098,"semantic_template":"qf-mod40-mobius-domain-boundary/{\"n\":-22}"},"fields":{"title":"Zero and Negative Integers Are Outside This Mobius Domain","stem":"Use nonpositive bounded integer n=-22. Give (n, in_positive_domain, mu_or_null, boundary).","answer":{"kind":"expression","value":"(-22, No, null, Mobius_domain_positive_integers_only)","explanation":"Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-22, No, null, Mobius_domain_positive_integers_only)."},"skills":["Zero and Negative Integers Are Outside This Mobius Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0099","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6927d90062c087b77fd37f85a5d71d1e99756bae3fd33c07ea705ae121ffcff","family_id":"qf-mod40-mobius-domain-boundary","seed":202611096099,"semantic_template":"qf-mod40-mobius-domain-boundary/{\"n\":-23}"},"fields":{"title":"Zero and Negative Integers Are Outside This Mobius Domain","stem":"Use nonpositive bounded integer n=-23. Give (n, in_positive_domain, mu_or_null, boundary).","answer":{"kind":"expression","value":"(-23, No, null, Mobius_domain_positive_integers_only)","explanation":"Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-23, No, null, Mobius_domain_positive_integers_only)."},"skills":["Zero and Negative Integers Are Outside This Mobius Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r06-mobius-function-0100","batch_id":"batch-20261014-advance-20261109-r06-mobius-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d92fb420c34dcf35c64252da266cd87760cdf0119586dc608317ccd98f28d1bd","family_id":"qf-mod40-mobius-domain-boundary","seed":202611096100,"semantic_template":"qf-mod40-mobius-domain-boundary/{\"n\":-24}"},"fields":{"title":"Zero and Negative Integers Are Outside This Mobius Domain","stem":"Use nonpositive bounded integer n=-24. Give (n, in_positive_domain, mu_or_null, boundary).","answer":{"kind":"expression","value":"(-24, No, null, Mobius_domain_positive_integers_only)","explanation":"Report No positive-domain membership and null value. Do not assign the squarefree parity rule to zero or a negative input in this declared arithmetic-function model. The exact result, retaining the requested order, is (-24, No, null, Mobius_domain_positive_integers_only)."},"skills":["Zero and Negative Integers Are Outside This Mobius Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}