{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0001","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d041102a5a39d2b75db0f1553dca640c5e39f9fb636f6eae736245878fdb6ef","family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106001,"semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":1,\"b\":-12,\"n\":1}"},"fields":{"title":"The Forward Transform Sums over All Positive Divisors","stem":"Use positive integer n=1, exact integer arithmetic function f(k)=a*k+b with a=1, b=-12. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).","answer":{"kind":"expression","value":"(1, [[1, -11]], -11, sum_over_all_positive_divisors)","explanation":"List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (1, [[1, -11]], -11, sum_over_all_positive_divisors)."},"skills":["The Forward Transform Sums over All Positive Divisors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0002","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17bfd0c83ca0aee5a0a6d6d0ed5066cd462db410fad332e81f5864ee10bbf88b","family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106002,"semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":2,\"b\":-11,\"n\":2}"},"fields":{"title":"The Forward Transform Sums over All Positive Divisors","stem":"Use positive integer n=2, exact integer arithmetic function f(k)=a*k+b with a=2, b=-11. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).","answer":{"kind":"expression","value":"(2, [[1, -9], [2, -7]], -16, sum_over_all_positive_divisors)","explanation":"List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (2, [[1, -9], [2, -7]], -16, sum_over_all_positive_divisors)."},"skills":["The Forward Transform Sums over All Positive Divisors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0003","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f45e659161ab645f2b8988ef59edd911252a0dfdc6db955b76899a0eca244dce","family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106003,"semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":3,\"b\":-10,\"n\":3}"},"fields":{"title":"The Forward Transform Sums over All Positive Divisors","stem":"Use positive integer n=3, exact integer arithmetic function f(k)=a*k+b with a=3, b=-10. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).","answer":{"kind":"expression","value":"(3, [[1, -7], [3, -1]], -8, sum_over_all_positive_divisors)","explanation":"List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (3, [[1, -7], [3, -1]], -8, sum_over_all_positive_divisors)."},"skills":["The Forward Transform Sums over All Positive Divisors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0004","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43fc6a79898936874c679a2992eb09e94a1ca15be2ed46a36c678ffabdf9185c","family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106004,"semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":4,\"b\":-9,\"n\":4}"},"fields":{"title":"The Forward Transform Sums over All Positive Divisors","stem":"Use positive integer n=4, exact integer arithmetic function f(k)=a*k+b with a=4, b=-9. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).","answer":{"kind":"expression","value":"(4, [[1, -5], [2, -1], [4, 7]], 1, sum_over_all_positive_divisors)","explanation":"List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (4, [[1, -5], [2, -1], [4, 7]], 1, sum_over_all_positive_divisors)."},"skills":["The Forward Transform Sums over All Positive Divisors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0005","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85a5ed7b91dea3d782585f1526302bd1e36929294d3d07c7eea2e2ff4dcb9d13","family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106005,"semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":5,\"b\":-8,\"n\":5}"},"fields":{"title":"The Forward Transform Sums over All Positive Divisors","stem":"Use positive integer n=5, exact integer arithmetic function f(k)=a*k+b with a=5, b=-8. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).","answer":{"kind":"expression","value":"(5, [[1, -3], [5, 17]], 14, sum_over_all_positive_divisors)","explanation":"List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (5, [[1, -3], [5, 17]], 14, sum_over_all_positive_divisors)."},"skills":["The Forward Transform Sums over All Positive Divisors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0006","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f73cd9b2c7364e3f7435a4031bf73500729da6b5e0760e5ab9c736a2ccf85ca9","family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106006,"semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":1,\"b\":-7,\"n\":6}"},"fields":{"title":"The Forward Transform Sums over All Positive Divisors","stem":"Use positive integer n=6, exact integer arithmetic function f(k)=a*k+b with a=1, b=-7. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).","answer":{"kind":"expression","value":"(6, [[1, -6], [2, -5], [3, -4], [6, -1]], -16, sum_over_all_positive_divisors)","explanation":"List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (6, [[1, -6], [2, -5], [3, -4], [6, -1]], -16, sum_over_all_positive_divisors)."},"skills":["The Forward Transform Sums over All Positive Divisors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0007","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"abb7e6298461a8509a4768a8245620da99c5ee053d51b8826dc192d24544143e","family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106007,"semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":2,\"b\":-6,\"n\":7}"},"fields":{"title":"The Forward Transform Sums over All Positive Divisors","stem":"Use positive integer n=7, exact integer arithmetic function f(k)=a*k+b with a=2, b=-6. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).","answer":{"kind":"expression","value":"(7, [[1, -4], [7, 8]], 4, sum_over_all_positive_divisors)","explanation":"List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (7, [[1, -4], [7, 8]], 4, sum_over_all_positive_divisors)."},"skills":["The Forward Transform Sums over All Positive Divisors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0008","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"257893eec03e585bfcbde06e1eb48cf93eefa7206d0b416abc13d304986c3c6f","family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106008,"semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":3,\"b\":-5,\"n\":8}"},"fields":{"title":"The Forward Transform Sums over All Positive Divisors","stem":"Use positive integer n=8, exact integer arithmetic function f(k)=a*k+b with a=3, b=-5. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).","answer":{"kind":"expression","value":"(8, [[1, -2], [2, 1], [4, 7], [8, 19]], 25, sum_over_all_positive_divisors)","explanation":"List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (8, [[1, -2], [2, 1], [4, 7], [8, 19]], 25, sum_over_all_positive_divisors)."},"skills":["The Forward Transform Sums over All Positive Divisors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0009","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67f016fec1882e809f0ffcbe4a1f599334edd4c80b5380b22d2534ad1c21b258","family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106009,"semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":4,\"b\":-4,\"n\":9}"},"fields":{"title":"The Forward Transform Sums over All Positive Divisors","stem":"Use positive integer n=9, exact integer arithmetic function f(k)=a*k+b with a=4, b=-4. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).","answer":{"kind":"expression","value":"(9, [[1, 0], [3, 8], [9, 32]], 40, sum_over_all_positive_divisors)","explanation":"List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (9, [[1, 0], [3, 8], [9, 32]], 40, sum_over_all_positive_divisors)."},"skills":["The Forward Transform Sums over All Positive Divisors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0010","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"916984ff6a3fb1c0d7a5b0beb6f4affdd5d7e88e955b81f6e0c987d4ea70cf99","family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106010,"semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":5,\"b\":-3,\"n\":10}"},"fields":{"title":"The Forward Transform Sums over All Positive Divisors","stem":"Use positive integer n=10, exact integer arithmetic function f(k)=a*k+b with a=5, b=-3. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).","answer":{"kind":"expression","value":"(10, [[1, 2], [2, 7], [5, 22], [10, 47]], 78, sum_over_all_positive_divisors)","explanation":"List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (10, [[1, 2], [2, 7], [5, 22], [10, 47]], 78, sum_over_all_positive_divisors)."},"skills":["The Forward Transform Sums over All Positive Divisors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0011","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e4dbfd74aedebed73faf60a3f73b0fd58c2a5ec389f6bb22a05a166a68e2c02","family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106011,"semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":1,\"b\":-2,\"n\":11}"},"fields":{"title":"The Forward Transform Sums over All Positive Divisors","stem":"Use positive integer n=11, exact integer arithmetic function f(k)=a*k+b with a=1, b=-2. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).","answer":{"kind":"expression","value":"(11, [[1, -1], [11, 9]], 8, sum_over_all_positive_divisors)","explanation":"List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (11, [[1, -1], [11, 9]], 8, sum_over_all_positive_divisors)."},"skills":["The Forward Transform Sums over All Positive Divisors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0012","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e213be5ce79e3a175ecaeeac6c5efbf766741533eb96b95aaa852a7f8ce9abbc","family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106012,"semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":2,\"b\":-1,\"n\":12}"},"fields":{"title":"The Forward Transform Sums over All Positive Divisors","stem":"Use positive integer n=12, exact integer arithmetic function f(k)=a*k+b with a=2, b=-1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).","answer":{"kind":"expression","value":"(12, [[1, 1], [2, 3], [3, 5], [4, 7], [6, 11], [12, 23]], 50, sum_over_all_positive_divisors)","explanation":"List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (12, [[1, 1], [2, 3], [3, 5], [4, 7], [6, 11], [12, 23]], 50, sum_over_all_positive_divisors)."},"skills":["The Forward Transform Sums over All Positive Divisors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0013","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0b6c791a68dedd64d8176e31da4d8649d29eb52cbbebf0f42fc37d2cd18e77a","family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106013,"semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":3,\"b\":0,\"n\":13}"},"fields":{"title":"The Forward Transform Sums over All Positive Divisors","stem":"Use positive integer n=13, exact integer arithmetic function f(k)=a*k+b with a=3, b=0. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).","answer":{"kind":"expression","value":"(13, [[1, 3], [13, 39]], 42, sum_over_all_positive_divisors)","explanation":"List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (13, [[1, 3], [13, 39]], 42, sum_over_all_positive_divisors)."},"skills":["The Forward Transform Sums over All Positive Divisors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0014","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22ef33c09ad2fecd17782f868c9652732f51a28a8ad32a8ab5841cd0e1a318c9","family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106014,"semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":4,\"b\":1,\"n\":14}"},"fields":{"title":"The Forward Transform Sums over All Positive Divisors","stem":"Use positive integer n=14, exact integer arithmetic function f(k)=a*k+b with a=4, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).","answer":{"kind":"expression","value":"(14, [[1, 5], [2, 9], [7, 29], [14, 57]], 100, sum_over_all_positive_divisors)","explanation":"List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (14, [[1, 5], [2, 9], [7, 29], [14, 57]], 100, sum_over_all_positive_divisors)."},"skills":["The Forward Transform Sums over All Positive Divisors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0015","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20a76d4a36a62a15f0f906dfa199ebdb88584a245f56db9ef684cc70dd7cc093","family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106015,"semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":5,\"b\":2,\"n\":15}"},"fields":{"title":"The Forward Transform Sums over All Positive Divisors","stem":"Use positive integer n=15, exact integer arithmetic function f(k)=a*k+b with a=5, b=2. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).","answer":{"kind":"expression","value":"(15, [[1, 7], [3, 17], [5, 27], [15, 77]], 128, sum_over_all_positive_divisors)","explanation":"List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (15, [[1, 7], [3, 17], [5, 27], [15, 77]], 128, sum_over_all_positive_divisors)."},"skills":["The Forward Transform Sums over All Positive Divisors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0016","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b40d0d67d5dba2a6e436d24de43695815e55b781e74b9010a562ee10564e06a6","family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106016,"semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":1,\"b\":3,\"n\":16}"},"fields":{"title":"The Forward Transform Sums over All Positive Divisors","stem":"Use positive integer n=16, exact integer arithmetic function f(k)=a*k+b with a=1, b=3. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).","answer":{"kind":"expression","value":"(16, [[1, 4], [2, 5], [4, 7], [8, 11], [16, 19]], 46, sum_over_all_positive_divisors)","explanation":"List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (16, [[1, 4], [2, 5], [4, 7], [8, 11], [16, 19]], 46, sum_over_all_positive_divisors)."},"skills":["The Forward Transform Sums over All Positive Divisors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0017","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e52b926bea52f6edf836bac07f07c7f166fae1b8b94ef91f93b614c1f0d2b449","family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106017,"semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":2,\"b\":4,\"n\":17}"},"fields":{"title":"The Forward Transform Sums over All Positive Divisors","stem":"Use positive integer n=17, exact integer arithmetic function f(k)=a*k+b with a=2, b=4. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).","answer":{"kind":"expression","value":"(17, [[1, 6], [17, 38]], 44, sum_over_all_positive_divisors)","explanation":"List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (17, [[1, 6], [17, 38]], 44, sum_over_all_positive_divisors)."},"skills":["The Forward Transform Sums over All Positive Divisors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0018","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c6d0571aec2b967e9aa231d196364c62ffd1c043400e424851fa2b3dd5d64c4","family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106018,"semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":3,\"b\":5,\"n\":18}"},"fields":{"title":"The Forward Transform Sums over All Positive Divisors","stem":"Use positive integer n=18, exact integer arithmetic function f(k)=a*k+b with a=3, b=5. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).","answer":{"kind":"expression","value":"(18, [[1, 8], [2, 11], [3, 14], [6, 23], [9, 32], [18, 59]], 147, sum_over_all_positive_divisors)","explanation":"List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (18, [[1, 8], [2, 11], [3, 14], [6, 23], [9, 32], [18, 59]], 147, sum_over_all_positive_divisors)."},"skills":["The Forward Transform Sums over All Positive Divisors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0019","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71f8d7921c29d055465ca4fc8249d69ccc55ddb598f4c99ae0ec2205b36c2c70","family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106019,"semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":4,\"b\":6,\"n\":19}"},"fields":{"title":"The Forward Transform Sums over All Positive Divisors","stem":"Use positive integer n=19, exact integer arithmetic function f(k)=a*k+b with a=4, b=6. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).","answer":{"kind":"expression","value":"(19, [[1, 10], [19, 82]], 92, sum_over_all_positive_divisors)","explanation":"List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (19, [[1, 10], [19, 82]], 92, sum_over_all_positive_divisors)."},"skills":["The Forward Transform Sums over All Positive Divisors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0020","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a78a4b48cf261c680c5d5ef5883dc5288644aaaa3d774544750b7fe7d80b5ffa","family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106020,"semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":5,\"b\":7,\"n\":20}"},"fields":{"title":"The Forward Transform Sums over All Positive Divisors","stem":"Use positive integer n=20, exact integer arithmetic function f(k)=a*k+b with a=5, b=7. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).","answer":{"kind":"expression","value":"(20, [[1, 12], [2, 17], [4, 27], [5, 32], [10, 57], [20, 107]], 252, sum_over_all_positive_divisors)","explanation":"List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (20, [[1, 12], [2, 17], [4, 27], [5, 32], [10, 57], [20, 107]], 252, sum_over_all_positive_divisors)."},"skills":["The Forward Transform Sums over All Positive Divisors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0021","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d92c82d38725fe24f097aa236773f94d448f489bf6ddd3d3d5fea31abe8dabf","family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106021,"semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":1,\"b\":8,\"n\":21}"},"fields":{"title":"The Forward Transform Sums over All Positive Divisors","stem":"Use positive integer n=21, exact integer arithmetic function f(k)=a*k+b with a=1, b=8. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).","answer":{"kind":"expression","value":"(21, [[1, 9], [3, 11], [7, 15], [21, 29]], 64, sum_over_all_positive_divisors)","explanation":"List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (21, [[1, 9], [3, 11], [7, 15], [21, 29]], 64, sum_over_all_positive_divisors)."},"skills":["The Forward Transform Sums over All Positive Divisors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0022","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e109c1d19fa92d66c5629c4c3a01e430993720fa16808a970942ed99c16e9db","family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106022,"semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":2,\"b\":9,\"n\":22}"},"fields":{"title":"The Forward Transform Sums over All Positive Divisors","stem":"Use positive integer n=22, exact integer arithmetic function f(k)=a*k+b with a=2, b=9. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).","answer":{"kind":"expression","value":"(22, [[1, 11], [2, 13], [11, 31], [22, 53]], 108, sum_over_all_positive_divisors)","explanation":"List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (22, [[1, 11], [2, 13], [11, 31], [22, 53]], 108, sum_over_all_positive_divisors)."},"skills":["The Forward Transform Sums over All Positive Divisors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0023","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68f73ef6ce5e3a5ac4749313e57c07ffbd4db86c74391fd71a27934cf6827c86","family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106023,"semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":3,\"b\":10,\"n\":23}"},"fields":{"title":"The Forward Transform Sums over All Positive Divisors","stem":"Use positive integer n=23, exact integer arithmetic function f(k)=a*k+b with a=3, b=10. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).","answer":{"kind":"expression","value":"(23, [[1, 13], [23, 79]], 92, sum_over_all_positive_divisors)","explanation":"List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (23, [[1, 13], [23, 79]], 92, sum_over_all_positive_divisors)."},"skills":["The Forward Transform Sums over All Positive Divisors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0024","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fecfa2d905b81bf58634043ba32444f4c648e1aa94220c092f8c114a89e5267c","family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106024,"semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":4,\"b\":11,\"n\":24}"},"fields":{"title":"The Forward Transform Sums over All Positive Divisors","stem":"Use positive integer n=24, exact integer arithmetic function f(k)=a*k+b with a=4, b=11. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).","answer":{"kind":"expression","value":"(24, [[1, 15], [2, 19], [3, 23], [4, 27], [6, 35], [8, 43], [12, 59], [24, 107]], 328, sum_over_all_positive_divisors)","explanation":"List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (24, [[1, 15], [2, 19], [3, 23], [4, 27], [6, 35], [8, 43], [12, 59], [24, 107]], 328, sum_over_all_positive_divisors)."},"skills":["The Forward Transform Sums over All Positive Divisors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0025","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"260a12db1a056fc579f7a0e53d7bd99bb31786b84ded934b62b1ef764239181f","family_id":"qf-mod41-mobinv-forward-divisor-transform","seed":202611106025,"semantic_template":"qf-mod41-mobinv-forward-divisor-transform/{\"a\":5,\"b\":12,\"n\":25}"},"fields":{"title":"The Forward Transform Sums over All Positive Divisors","stem":"Use positive integer n=25, exact integer arithmetic function f(k)=a*k+b with a=5, b=12. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, divisor_value_rows, F_n, certificate).","answer":{"kind":"expression","value":"(25, [[1, 17], [5, 37], [25, 137]], 191, sum_over_all_positive_divisors)","explanation":"List [d,f(d)] for every divisor increasingly and sum them to F(n), including d=1 and d=n. The exact result, retaining the requested order, is (25, [[1, 17], [5, 37], [25, 137]], 191, sum_over_all_positive_divisors)."},"skills":["The Forward Transform Sums over All Positive Divisors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0026","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae303839ae82ad5d830b0d87c33f02525ddd9827a6ee944f716633a8a8fe4f99","family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106026,"semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":1,\"b\":-12,\"n\":1}"},"fields":{"title":"Keep mu(d) Paired with F(n Divided by d)","stem":"Use positive integer n=1, exact integer arithmetic function f(k)=a*k+b with a=1, b=-12. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).","answer":{"kind":"expression","value":"(1, [[1, 1, 1, -11, -11]], -11, -11, Yes)","explanation":"Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (1, [[1, 1, 1, -11, -11]], -11, -11, Yes)."},"skills":["Keep mu(d) Paired with F(n Divided by d)"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0027","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d526066ad62c23a0f12cefdfc196692bfca0ce7c15e014289f19cc407fca67f","family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106027,"semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":2,\"b\":-11,\"n\":2}"},"fields":{"title":"Keep mu(d) Paired with F(n Divided by d)","stem":"Use positive integer n=2, exact integer arithmetic function f(k)=a*k+b with a=2, b=-11. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).","answer":{"kind":"expression","value":"(2, [[1, 1, 2, -16, -16], [2, -1, 1, -9, 9]], -7, -7, Yes)","explanation":"Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (2, [[1, 1, 2, -16, -16], [2, -1, 1, -9, 9]], -7, -7, Yes)."},"skills":["Keep mu(d) Paired with F(n Divided by d)"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0028","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"719a5af73a89553ab693c17f4bae5da60726608e7051dfa5faddc569149d720f","family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106028,"semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":3,\"b\":-10,\"n\":3}"},"fields":{"title":"Keep mu(d) Paired with F(n Divided by d)","stem":"Use positive integer n=3, exact integer arithmetic function f(k)=a*k+b with a=3, b=-10. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).","answer":{"kind":"expression","value":"(3, [[1, 1, 3, -8, -8], [3, -1, 1, -7, 7]], -1, -1, Yes)","explanation":"Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (3, [[1, 1, 3, -8, -8], [3, -1, 1, -7, 7]], -1, -1, Yes)."},"skills":["Keep mu(d) Paired with F(n Divided by d)"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0029","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc61cb83d425496938714703273f6d0e394d52b01bef7f98746b3b580ca69255","family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106029,"semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":4,\"b\":-9,\"n\":4}"},"fields":{"title":"Keep mu(d) Paired with F(n Divided by d)","stem":"Use positive integer n=4, exact integer arithmetic function f(k)=a*k+b with a=4, b=-9. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).","answer":{"kind":"expression","value":"(4, [[1, 1, 4, 1, 1], [2, -1, 2, -6, 6], [4, 0, 1, -5, 0]], 7, 7, Yes)","explanation":"Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (4, [[1, 1, 4, 1, 1], [2, -1, 2, -6, 6], [4, 0, 1, -5, 0]], 7, 7, Yes)."},"skills":["Keep mu(d) Paired with F(n Divided by d)"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0030","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9d2a921745ac07221ff536d3a38c319a86e70a40ee4000ff40a2c35af82ba29","family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106030,"semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":5,\"b\":-8,\"n\":5}"},"fields":{"title":"Keep mu(d) Paired with F(n Divided by d)","stem":"Use positive integer n=5, exact integer arithmetic function f(k)=a*k+b with a=5, b=-8. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).","answer":{"kind":"expression","value":"(5, [[1, 1, 5, 14, 14], [5, -1, 1, -3, 3]], 17, 17, Yes)","explanation":"Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (5, [[1, 1, 5, 14, 14], [5, -1, 1, -3, 3]], 17, 17, Yes)."},"skills":["Keep mu(d) Paired with F(n Divided by d)"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0031","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69ef1d25ce7ca1299b2b1e7663bdae4de992219226144a2c9b9360c459af090e","family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106031,"semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":1,\"b\":-7,\"n\":6}"},"fields":{"title":"Keep mu(d) Paired with F(n Divided by d)","stem":"Use positive integer n=6, exact integer arithmetic function f(k)=a*k+b with a=1, b=-7. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).","answer":{"kind":"expression","value":"(6, [[1, 1, 6, -16, -16], [2, -1, 3, -10, 10], [3, -1, 2, -11, 11], [6, 1, 1, -6, -6]], -1, -1, Yes)","explanation":"Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (6, [[1, 1, 6, -16, -16], [2, -1, 3, -10, 10], [3, -1, 2, -11, 11], [6, 1, 1, -6, -6]], -1, -1, Yes)."},"skills":["Keep mu(d) Paired with F(n Divided by d)"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0032","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2e6f154c507b2ba29c6b6d8162873577de8a1f60373c14380103d0e53d584ef","family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106032,"semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":2,\"b\":-6,\"n\":7}"},"fields":{"title":"Keep mu(d) Paired with F(n Divided by d)","stem":"Use positive integer n=7, exact integer arithmetic function f(k)=a*k+b with a=2, b=-6. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).","answer":{"kind":"expression","value":"(7, [[1, 1, 7, 4, 4], [7, -1, 1, -4, 4]], 8, 8, Yes)","explanation":"Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (7, [[1, 1, 7, 4, 4], [7, -1, 1, -4, 4]], 8, 8, Yes)."},"skills":["Keep mu(d) Paired with F(n Divided by d)"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0033","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a461d75417144d2838f82561eb3848c789de5870e6050fc2a87db7866feff777","family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106033,"semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":3,\"b\":-5,\"n\":8}"},"fields":{"title":"Keep mu(d) Paired with F(n Divided by d)","stem":"Use positive integer n=8, exact integer arithmetic function f(k)=a*k+b with a=3, b=-5. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).","answer":{"kind":"expression","value":"(8, [[1, 1, 8, 25, 25], [2, -1, 4, 6, -6], [4, 0, 2, -1, 0], [8, 0, 1, -2, 0]], 19, 19, Yes)","explanation":"Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (8, [[1, 1, 8, 25, 25], [2, -1, 4, 6, -6], [4, 0, 2, -1, 0], [8, 0, 1, -2, 0]], 19, 19, Yes)."},"skills":["Keep mu(d) Paired with F(n Divided by d)"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0034","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2ba635856bbf655c60413d3f997e09fa769159f1dd1dd3937244d04eef493e5","family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106034,"semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":4,\"b\":-4,\"n\":9}"},"fields":{"title":"Keep mu(d) Paired with F(n Divided by d)","stem":"Use positive integer n=9, exact integer arithmetic function f(k)=a*k+b with a=4, b=-4. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).","answer":{"kind":"expression","value":"(9, [[1, 1, 9, 40, 40], [3, -1, 3, 8, -8], [9, 0, 1, 0, 0]], 32, 32, Yes)","explanation":"Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (9, [[1, 1, 9, 40, 40], [3, -1, 3, 8, -8], [9, 0, 1, 0, 0]], 32, 32, Yes)."},"skills":["Keep mu(d) Paired with F(n Divided by d)"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0035","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75ef467a7618426da613c2bf1c88ecdaf486de26368ab1f36a803af8896b6da4","family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106035,"semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":5,\"b\":-3,\"n\":10}"},"fields":{"title":"Keep mu(d) Paired with F(n Divided by d)","stem":"Use positive integer n=10, exact integer arithmetic function f(k)=a*k+b with a=5, b=-3. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).","answer":{"kind":"expression","value":"(10, [[1, 1, 10, 78, 78], [2, -1, 5, 24, -24], [5, -1, 2, 9, -9], [10, 1, 1, 2, 2]], 47, 47, Yes)","explanation":"Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (10, [[1, 1, 10, 78, 78], [2, -1, 5, 24, -24], [5, -1, 2, 9, -9], [10, 1, 1, 2, 2]], 47, 47, Yes)."},"skills":["Keep mu(d) Paired with F(n Divided by d)"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0036","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b94cac94e9aa709c5c12a24593c6a6348f56c1252456e7a14ab81d0eef7b99b3","family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106036,"semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":1,\"b\":-2,\"n\":11}"},"fields":{"title":"Keep mu(d) Paired with F(n Divided by d)","stem":"Use positive integer n=11, exact integer arithmetic function f(k)=a*k+b with a=1, b=-2. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).","answer":{"kind":"expression","value":"(11, [[1, 1, 11, 8, 8], [11, -1, 1, -1, 1]], 9, 9, Yes)","explanation":"Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (11, [[1, 1, 11, 8, 8], [11, -1, 1, -1, 1]], 9, 9, Yes)."},"skills":["Keep mu(d) Paired with F(n Divided by d)"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0037","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ded7249386493f0e0bd4269b530e69cac4e01d6309154f1b2aee85b43d5c8ca1","family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106037,"semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":2,\"b\":-1,\"n\":12}"},"fields":{"title":"Keep mu(d) Paired with F(n Divided by d)","stem":"Use positive integer n=12, exact integer arithmetic function f(k)=a*k+b with a=2, b=-1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).","answer":{"kind":"expression","value":"(12, [[1, 1, 12, 50, 50], [2, -1, 6, 20, -20], [3, -1, 4, 11, -11], [4, 0, 3, 6, 0], [6, 1, 2, 4, 4], [12, 0, 1, 1, 0]], 23, 23, Yes)","explanation":"Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (12, [[1, 1, 12, 50, 50], [2, -1, 6, 20, -20], [3, -1, 4, 11, -11], [4, 0, 3, 6, 0], [6, 1, 2, 4, 4], [12, 0, 1, 1, 0]], 23, 23, Yes)."},"skills":["Keep mu(d) Paired with F(n Divided by d)"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0038","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f21c3cf6842b5ed6622674dd7008f4a5a0e8a048144482afb7d6663ebcb9ca8c","family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106038,"semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":3,\"b\":0,\"n\":13}"},"fields":{"title":"Keep mu(d) Paired with F(n Divided by d)","stem":"Use positive integer n=13, exact integer arithmetic function f(k)=a*k+b with a=3, b=0. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).","answer":{"kind":"expression","value":"(13, [[1, 1, 13, 42, 42], [13, -1, 1, 3, -3]], 39, 39, Yes)","explanation":"Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (13, [[1, 1, 13, 42, 42], [13, -1, 1, 3, -3]], 39, 39, Yes)."},"skills":["Keep mu(d) Paired with F(n Divided by d)"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0039","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69e2f4e9789d15b3e0ac731a9ae23ef882d89390fe7cc98a18fc1cdeb998123d","family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106039,"semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":4,\"b\":1,\"n\":14}"},"fields":{"title":"Keep mu(d) Paired with F(n Divided by d)","stem":"Use positive integer n=14, exact integer arithmetic function f(k)=a*k+b with a=4, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).","answer":{"kind":"expression","value":"(14, [[1, 1, 14, 100, 100], [2, -1, 7, 34, -34], [7, -1, 2, 14, -14], [14, 1, 1, 5, 5]], 57, 57, Yes)","explanation":"Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (14, [[1, 1, 14, 100, 100], [2, -1, 7, 34, -34], [7, -1, 2, 14, -14], [14, 1, 1, 5, 5]], 57, 57, Yes)."},"skills":["Keep mu(d) Paired with F(n Divided by d)"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0040","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1629e69824fdb663ca05258ba4de40bba8bdf3b50dbc4e00631fbca27f7a738","family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106040,"semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":5,\"b\":2,\"n\":15}"},"fields":{"title":"Keep mu(d) Paired with F(n Divided by d)","stem":"Use positive integer n=15, exact integer arithmetic function f(k)=a*k+b with a=5, b=2. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).","answer":{"kind":"expression","value":"(15, [[1, 1, 15, 128, 128], [3, -1, 5, 34, -34], [5, -1, 3, 24, -24], [15, 1, 1, 7, 7]], 77, 77, Yes)","explanation":"Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (15, [[1, 1, 15, 128, 128], [3, -1, 5, 34, -34], [5, -1, 3, 24, -24], [15, 1, 1, 7, 7]], 77, 77, Yes)."},"skills":["Keep mu(d) Paired with F(n Divided by d)"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0041","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ae06a7522ec95cec6c2e362901e22877baa9045580a76f62a1276d26cc7be90","family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106041,"semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":1,\"b\":3,\"n\":16}"},"fields":{"title":"Keep mu(d) Paired with F(n Divided by d)","stem":"Use positive integer n=16, exact integer arithmetic function f(k)=a*k+b with a=1, b=3. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).","answer":{"kind":"expression","value":"(16, [[1, 1, 16, 46, 46], [2, -1, 8, 27, -27], [4, 0, 4, 16, 0], [8, 0, 2, 9, 0], [16, 0, 1, 4, 0]], 19, 19, Yes)","explanation":"Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (16, [[1, 1, 16, 46, 46], [2, -1, 8, 27, -27], [4, 0, 4, 16, 0], [8, 0, 2, 9, 0], [16, 0, 1, 4, 0]], 19, 19, Yes)."},"skills":["Keep mu(d) Paired with F(n Divided by d)"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0042","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac3064f727ac75fb8a7e07a56c0455061b2c430a8ecab086e1cf39902a283c11","family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106042,"semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":2,\"b\":4,\"n\":17}"},"fields":{"title":"Keep mu(d) Paired with F(n Divided by d)","stem":"Use positive integer n=17, exact integer arithmetic function f(k)=a*k+b with a=2, b=4. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).","answer":{"kind":"expression","value":"(17, [[1, 1, 17, 44, 44], [17, -1, 1, 6, -6]], 38, 38, Yes)","explanation":"Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (17, [[1, 1, 17, 44, 44], [17, -1, 1, 6, -6]], 38, 38, Yes)."},"skills":["Keep mu(d) Paired with F(n Divided by d)"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0043","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"207a325547af7357b7b189092cb7e895602bba481426884899fbcd2d17d612cf","family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106043,"semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":3,\"b\":5,\"n\":18}"},"fields":{"title":"Keep mu(d) Paired with F(n Divided by d)","stem":"Use positive integer n=18, exact integer arithmetic function f(k)=a*k+b with a=3, b=5. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).","answer":{"kind":"expression","value":"(18, [[1, 1, 18, 147, 147], [2, -1, 9, 54, -54], [3, -1, 6, 56, -56], [6, 1, 3, 22, 22], [9, 0, 2, 19, 0], [18, 0, 1, 8, 0]], 59, 59, Yes)","explanation":"Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (18, [[1, 1, 18, 147, 147], [2, -1, 9, 54, -54], [3, -1, 6, 56, -56], [6, 1, 3, 22, 22], [9, 0, 2, 19, 0], [18, 0, 1, 8, 0]], 59, 59, Yes)."},"skills":["Keep mu(d) Paired with F(n Divided by d)"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0044","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7bb1032776d70d5822e6b0b1fb9e329d723f520b4a3fefd62fc1ad11b408af69","family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106044,"semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":4,\"b\":6,\"n\":19}"},"fields":{"title":"Keep mu(d) Paired with F(n Divided by d)","stem":"Use positive integer n=19, exact integer arithmetic function f(k)=a*k+b with a=4, b=6. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).","answer":{"kind":"expression","value":"(19, [[1, 1, 19, 92, 92], [19, -1, 1, 10, -10]], 82, 82, Yes)","explanation":"Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (19, [[1, 1, 19, 92, 92], [19, -1, 1, 10, -10]], 82, 82, Yes)."},"skills":["Keep mu(d) Paired with F(n Divided by d)"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0045","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34be98f6fd4b8088085494369cb9f4182e392a5c7e878e695c81d1dd1951ea00","family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106045,"semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":5,\"b\":7,\"n\":20}"},"fields":{"title":"Keep mu(d) Paired with F(n Divided by d)","stem":"Use positive integer n=20, exact integer arithmetic function f(k)=a*k+b with a=5, b=7. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).","answer":{"kind":"expression","value":"(20, [[1, 1, 20, 252, 252], [2, -1, 10, 118, -118], [4, 0, 5, 44, 0], [5, -1, 4, 56, -56], [10, 1, 2, 29, 29], [20, 0, 1, 12, 0]], 107, 107, Yes)","explanation":"Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (20, [[1, 1, 20, 252, 252], [2, -1, 10, 118, -118], [4, 0, 5, 44, 0], [5, -1, 4, 56, -56], [10, 1, 2, 29, 29], [20, 0, 1, 12, 0]], 107, 107, Yes)."},"skills":["Keep mu(d) Paired with F(n Divided by d)"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0046","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ba68e2f89b9ad44e7b217aaff5b09bdd6e7a624cb3fe098b1909fbcc1510820","family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106046,"semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":1,\"b\":8,\"n\":21}"},"fields":{"title":"Keep mu(d) Paired with F(n Divided by d)","stem":"Use positive integer n=21, exact integer arithmetic function f(k)=a*k+b with a=1, b=8. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).","answer":{"kind":"expression","value":"(21, [[1, 1, 21, 64, 64], [3, -1, 7, 24, -24], [7, -1, 3, 20, -20], [21, 1, 1, 9, 9]], 29, 29, Yes)","explanation":"Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (21, [[1, 1, 21, 64, 64], [3, -1, 7, 24, -24], [7, -1, 3, 20, -20], [21, 1, 1, 9, 9]], 29, 29, Yes)."},"skills":["Keep mu(d) Paired with F(n Divided by d)"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0047","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2965686a08536fefaa29e97ec599f8098480e62d1a46f8229c1ec0662cfe6944","family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106047,"semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":2,\"b\":9,\"n\":22}"},"fields":{"title":"Keep mu(d) Paired with F(n Divided by d)","stem":"Use positive integer n=22, exact integer arithmetic function f(k)=a*k+b with a=2, b=9. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).","answer":{"kind":"expression","value":"(22, [[1, 1, 22, 108, 108], [2, -1, 11, 42, -42], [11, -1, 2, 24, -24], [22, 1, 1, 11, 11]], 53, 53, Yes)","explanation":"Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (22, [[1, 1, 22, 108, 108], [2, -1, 11, 42, -42], [11, -1, 2, 24, -24], [22, 1, 1, 11, 11]], 53, 53, Yes)."},"skills":["Keep mu(d) Paired with F(n Divided by d)"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0048","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36a10cec56c0331f023b8bed319aa15c5eac68aefa77809378b3384d296bfb88","family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106048,"semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":3,\"b\":10,\"n\":23}"},"fields":{"title":"Keep mu(d) Paired with F(n Divided by d)","stem":"Use positive integer n=23, exact integer arithmetic function f(k)=a*k+b with a=3, b=10. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).","answer":{"kind":"expression","value":"(23, [[1, 1, 23, 92, 92], [23, -1, 1, 13, -13]], 79, 79, Yes)","explanation":"Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (23, [[1, 1, 23, 92, 92], [23, -1, 1, 13, -13]], 79, 79, Yes)."},"skills":["Keep mu(d) Paired with F(n Divided by d)"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0049","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7795facd10aad52ce5c5dbe1cca7ceb09bb15f2296865a0cfde66d5ef3bc16d2","family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106049,"semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":4,\"b\":11,\"n\":24}"},"fields":{"title":"Keep mu(d) Paired with F(n Divided by d)","stem":"Use positive integer n=24, exact integer arithmetic function f(k)=a*k+b with a=4, b=11. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).","answer":{"kind":"expression","value":"(24, [[1, 1, 24, 328, 328], [2, -1, 12, 178, -178], [3, -1, 8, 104, -104], [4, 0, 6, 92, 0], [6, 1, 4, 61, 61], [8, 0, 3, 38, 0], [12, 0, 2, 34, 0], [24, 0, 1, 15, 0]], 107, 107, Yes)","explanation":"Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (24, [[1, 1, 24, 328, 328], [2, -1, 12, 178, -178], [3, -1, 8, 104, -104], [4, 0, 6, 92, 0], [6, 1, 4, 61, 61], [8, 0, 3, 38, 0], [12, 0, 2, 34, 0], [24, 0, 1, 15, 0]], 107, 107, Yes)."},"skills":["Keep mu(d) Paired with F(n Divided by d)"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0050","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc670c57ed928e57048ea660fd64cae2b2d91d7178f4df02880b0231d39a63a9","family_id":"qf-mod41-mobinv-inverse-reconstruction","seed":202611106050,"semantic_template":"qf-mod41-mobinv-inverse-reconstruction/{\"a\":5,\"b\":12,\"n\":25}"},"fields":{"title":"Keep mu(d) Paired with F(n Divided by d)","stem":"Use positive integer n=25, exact integer arithmetic function f(k)=a*k+b with a=5, b=12. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).","answer":{"kind":"expression","value":"(25, [[1, 1, 25, 191, 191], [5, -1, 5, 54, -54], [25, 0, 1, 17, 0]], 137, 137, Yes)","explanation":"Each inverse row is [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)]. Sum every term and compare with f(n). The equivalent mu(n/d)*F(d) convention must not be mixed into these rows. The exact result, retaining the requested order, is (25, [[1, 1, 25, 191, 191], [5, -1, 5, 54, -54], [25, 0, 1, 17, 0]], 137, 137, Yes)."},"skills":["Keep mu(d) Paired with F(n Divided by d)"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0051","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8acaffb6a85653d72785f3fb0aaf56cb9314eadd2e84186179c6f5239ab3347b","family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106051,"semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":1,\"b\":-12,\"n\":1}"},"fields":{"title":"Every Divisor Row Recovers Its Original Arithmetic Value","stem":"Use positive integer n=1, exact integer arithmetic function f(k)=a*k+b with a=1, b=-12. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).","answer":{"kind":"expression","value":"(1, [[1, -11, -11, -11]], Yes, mu_one_boundary_preserved)","explanation":"For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (1, [[1, -11, -11, -11]], Yes, mu_one_boundary_preserved)."},"skills":["Every Divisor Row Recovers Its Original Arithmetic 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-20261110-r06-mobius-inversion-finite-divisors-0052","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6ef829501ca370b9167301e16183aee852ab76623628455aee924aac4d1fa2e","family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106052,"semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":2,\"b\":-11,\"n\":2}"},"fields":{"title":"Every Divisor Row Recovers Its Original Arithmetic Value","stem":"Use positive integer n=2, exact integer arithmetic function f(k)=a*k+b with a=2, b=-11. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).","answer":{"kind":"expression","value":"(2, [[1, -9, -9, -9], [2, -16, -7, -7]], Yes, finite_divisor_poset_roundtrip)","explanation":"For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (2, [[1, -9, -9, -9], [2, -16, -7, -7]], Yes, finite_divisor_poset_roundtrip)."},"skills":["Every Divisor Row Recovers Its Original Arithmetic 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-20261110-r06-mobius-inversion-finite-divisors-0053","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43beeb516e22b68f3911c76234875fc67ba31ae4563e6715c453b6b62e6d6767","family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106053,"semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":3,\"b\":-10,\"n\":3}"},"fields":{"title":"Every Divisor Row Recovers Its Original Arithmetic Value","stem":"Use positive integer n=3, exact integer arithmetic function f(k)=a*k+b with a=3, b=-10. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).","answer":{"kind":"expression","value":"(3, [[1, -7, -7, -7], [3, -8, -1, -1]], Yes, finite_divisor_poset_roundtrip)","explanation":"For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (3, [[1, -7, -7, -7], [3, -8, -1, -1]], Yes, finite_divisor_poset_roundtrip)."},"skills":["Every Divisor Row Recovers Its Original Arithmetic 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-20261110-r06-mobius-inversion-finite-divisors-0054","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"456b1291bd828016cf54e41ee69e754845c98f43e0b429894935b23762dbdc45","family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106054,"semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":4,\"b\":-9,\"n\":4}"},"fields":{"title":"Every Divisor Row Recovers Its Original Arithmetic Value","stem":"Use positive integer n=4, exact integer arithmetic function f(k)=a*k+b with a=4, b=-9. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).","answer":{"kind":"expression","value":"(4, [[1, -5, -5, -5], [2, -6, -1, -1], [4, 1, 7, 7]], Yes, finite_divisor_poset_roundtrip)","explanation":"For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (4, [[1, -5, -5, -5], [2, -6, -1, -1], [4, 1, 7, 7]], Yes, finite_divisor_poset_roundtrip)."},"skills":["Every Divisor Row Recovers Its Original Arithmetic 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-20261110-r06-mobius-inversion-finite-divisors-0055","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"400617d320a2313861ec675dfd87cf1ef0404b615a281a463a0c4acfbb990a8a","family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106055,"semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":5,\"b\":-8,\"n\":5}"},"fields":{"title":"Every Divisor Row Recovers Its Original Arithmetic Value","stem":"Use positive integer n=5, exact integer arithmetic function f(k)=a*k+b with a=5, b=-8. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).","answer":{"kind":"expression","value":"(5, [[1, -3, -3, -3], [5, 14, 17, 17]], Yes, finite_divisor_poset_roundtrip)","explanation":"For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (5, [[1, -3, -3, -3], [5, 14, 17, 17]], Yes, finite_divisor_poset_roundtrip)."},"skills":["Every Divisor Row Recovers Its Original Arithmetic 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-20261110-r06-mobius-inversion-finite-divisors-0056","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93b8d6e677a942b603e2c350784848cdbf00440023a3f8e0f73e8ccd6ac5ea7a","family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106056,"semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":1,\"b\":-7,\"n\":6}"},"fields":{"title":"Every Divisor Row Recovers Its Original Arithmetic Value","stem":"Use positive integer n=6, exact integer arithmetic function f(k)=a*k+b with a=1, b=-7. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).","answer":{"kind":"expression","value":"(6, [[1, -6, -6, -6], [2, -11, -5, -5], [3, -10, -4, -4], [6, -16, -1, -1]], Yes, finite_divisor_poset_roundtrip)","explanation":"For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (6, [[1, -6, -6, -6], [2, -11, -5, -5], [3, -10, -4, -4], [6, -16, -1, -1]], Yes, finite_divisor_poset_roundtrip)."},"skills":["Every Divisor Row Recovers Its Original Arithmetic 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-20261110-r06-mobius-inversion-finite-divisors-0057","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0fd1787a033901bfd315fa7325a6a3f373528d5cc1691f5305a7addc2412df5","family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106057,"semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":2,\"b\":-6,\"n\":7}"},"fields":{"title":"Every Divisor Row Recovers Its Original Arithmetic Value","stem":"Use positive integer n=7, exact integer arithmetic function f(k)=a*k+b with a=2, b=-6. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).","answer":{"kind":"expression","value":"(7, [[1, -4, -4, -4], [7, 4, 8, 8]], Yes, finite_divisor_poset_roundtrip)","explanation":"For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (7, [[1, -4, -4, -4], [7, 4, 8, 8]], Yes, finite_divisor_poset_roundtrip)."},"skills":["Every Divisor Row Recovers Its Original Arithmetic 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-20261110-r06-mobius-inversion-finite-divisors-0058","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad5d03a7534721107163f598751802b632536a93a385657aecb6669e6fe95957","family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106058,"semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":3,\"b\":-5,\"n\":8}"},"fields":{"title":"Every Divisor Row Recovers Its Original Arithmetic Value","stem":"Use positive integer n=8, exact integer arithmetic function f(k)=a*k+b with a=3, b=-5. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).","answer":{"kind":"expression","value":"(8, [[1, -2, -2, -2], [2, -1, 1, 1], [4, 6, 7, 7], [8, 25, 19, 19]], Yes, finite_divisor_poset_roundtrip)","explanation":"For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (8, [[1, -2, -2, -2], [2, -1, 1, 1], [4, 6, 7, 7], [8, 25, 19, 19]], Yes, finite_divisor_poset_roundtrip)."},"skills":["Every Divisor Row Recovers Its Original Arithmetic 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-20261110-r06-mobius-inversion-finite-divisors-0059","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8cce7a6bbd35e56583652def2053052a99d6c6e996b33934d5956819dcc52ec","family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106059,"semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":4,\"b\":-4,\"n\":9}"},"fields":{"title":"Every Divisor Row Recovers Its Original Arithmetic Value","stem":"Use positive integer n=9, exact integer arithmetic function f(k)=a*k+b with a=4, b=-4. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).","answer":{"kind":"expression","value":"(9, [[1, 0, 0, 0], [3, 8, 8, 8], [9, 40, 32, 32]], Yes, finite_divisor_poset_roundtrip)","explanation":"For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (9, [[1, 0, 0, 0], [3, 8, 8, 8], [9, 40, 32, 32]], Yes, finite_divisor_poset_roundtrip)."},"skills":["Every Divisor Row Recovers Its Original Arithmetic 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-20261110-r06-mobius-inversion-finite-divisors-0060","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e13faad5b7b0d7017509fdf69096756773f379e98990b4663d19fee6f784d2c","family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106060,"semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":5,\"b\":-3,\"n\":10}"},"fields":{"title":"Every Divisor Row Recovers Its Original Arithmetic Value","stem":"Use positive integer n=10, exact integer arithmetic function f(k)=a*k+b with a=5, b=-3. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).","answer":{"kind":"expression","value":"(10, [[1, 2, 2, 2], [2, 9, 7, 7], [5, 24, 22, 22], [10, 78, 47, 47]], Yes, finite_divisor_poset_roundtrip)","explanation":"For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (10, [[1, 2, 2, 2], [2, 9, 7, 7], [5, 24, 22, 22], [10, 78, 47, 47]], Yes, finite_divisor_poset_roundtrip)."},"skills":["Every Divisor Row Recovers Its Original Arithmetic 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-20261110-r06-mobius-inversion-finite-divisors-0061","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6192a581e631718b0b885bee7f3cd318d57b4b2ee4f27122cd0e76f595c8df4","family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106061,"semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":1,\"b\":-2,\"n\":11}"},"fields":{"title":"Every Divisor Row Recovers Its Original Arithmetic Value","stem":"Use positive integer n=11, exact integer arithmetic function f(k)=a*k+b with a=1, b=-2. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).","answer":{"kind":"expression","value":"(11, [[1, -1, -1, -1], [11, 8, 9, 9]], Yes, finite_divisor_poset_roundtrip)","explanation":"For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (11, [[1, -1, -1, -1], [11, 8, 9, 9]], Yes, finite_divisor_poset_roundtrip)."},"skills":["Every Divisor Row Recovers Its Original Arithmetic 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-20261110-r06-mobius-inversion-finite-divisors-0062","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36820977f082bd72b3623d2e16bc79add41b4136fa1c7c74fe7c63adee477b0b","family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106062,"semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":2,\"b\":-1,\"n\":12}"},"fields":{"title":"Every Divisor Row Recovers Its Original Arithmetic Value","stem":"Use positive integer n=12, exact integer arithmetic function f(k)=a*k+b with a=2, b=-1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).","answer":{"kind":"expression","value":"(12, [[1, 1, 1, 1], [2, 4, 3, 3], [3, 6, 5, 5], [4, 11, 7, 7], [6, 20, 11, 11], [12, 50, 23, 23]], Yes, finite_divisor_poset_roundtrip)","explanation":"For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (12, [[1, 1, 1, 1], [2, 4, 3, 3], [3, 6, 5, 5], [4, 11, 7, 7], [6, 20, 11, 11], [12, 50, 23, 23]], Yes, finite_divisor_poset_roundtrip)."},"skills":["Every Divisor Row Recovers Its Original Arithmetic 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-20261110-r06-mobius-inversion-finite-divisors-0063","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d8334c9d3d8c59e84a993a134531b1a211edb1e3d03aa9da60737838d5d9e45","family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106063,"semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":3,\"b\":0,\"n\":13}"},"fields":{"title":"Every Divisor Row Recovers Its Original Arithmetic Value","stem":"Use positive integer n=13, exact integer arithmetic function f(k)=a*k+b with a=3, b=0. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).","answer":{"kind":"expression","value":"(13, [[1, 3, 3, 3], [13, 42, 39, 39]], Yes, finite_divisor_poset_roundtrip)","explanation":"For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (13, [[1, 3, 3, 3], [13, 42, 39, 39]], Yes, finite_divisor_poset_roundtrip)."},"skills":["Every Divisor Row Recovers Its Original Arithmetic 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-20261110-r06-mobius-inversion-finite-divisors-0064","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b1521c510d1972e9d2bad02ef243f4b0b24fbcd0f6fa37904989e49abdbf705","family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106064,"semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":4,\"b\":1,\"n\":14}"},"fields":{"title":"Every Divisor Row Recovers Its Original Arithmetic Value","stem":"Use positive integer n=14, exact integer arithmetic function f(k)=a*k+b with a=4, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).","answer":{"kind":"expression","value":"(14, [[1, 5, 5, 5], [2, 14, 9, 9], [7, 34, 29, 29], [14, 100, 57, 57]], Yes, finite_divisor_poset_roundtrip)","explanation":"For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (14, [[1, 5, 5, 5], [2, 14, 9, 9], [7, 34, 29, 29], [14, 100, 57, 57]], Yes, finite_divisor_poset_roundtrip)."},"skills":["Every Divisor Row Recovers Its Original Arithmetic 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-20261110-r06-mobius-inversion-finite-divisors-0065","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00c3729e894c06571481ec96d3aa80c8c86fe1d90f0d9ad838445325e77ba3cc","family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106065,"semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":5,\"b\":2,\"n\":15}"},"fields":{"title":"Every Divisor Row Recovers Its Original Arithmetic Value","stem":"Use positive integer n=15, exact integer arithmetic function f(k)=a*k+b with a=5, b=2. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).","answer":{"kind":"expression","value":"(15, [[1, 7, 7, 7], [3, 24, 17, 17], [5, 34, 27, 27], [15, 128, 77, 77]], Yes, finite_divisor_poset_roundtrip)","explanation":"For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (15, [[1, 7, 7, 7], [3, 24, 17, 17], [5, 34, 27, 27], [15, 128, 77, 77]], Yes, finite_divisor_poset_roundtrip)."},"skills":["Every Divisor Row Recovers Its Original Arithmetic 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-20261110-r06-mobius-inversion-finite-divisors-0066","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8b6bb81b5688c250284847a6f6b6ef4d3ae74eaed83638f198130c404e6eb68","family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106066,"semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":1,\"b\":3,\"n\":16}"},"fields":{"title":"Every Divisor Row Recovers Its Original Arithmetic Value","stem":"Use positive integer n=16, exact integer arithmetic function f(k)=a*k+b with a=1, b=3. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).","answer":{"kind":"expression","value":"(16, [[1, 4, 4, 4], [2, 9, 5, 5], [4, 16, 7, 7], [8, 27, 11, 11], [16, 46, 19, 19]], Yes, finite_divisor_poset_roundtrip)","explanation":"For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (16, [[1, 4, 4, 4], [2, 9, 5, 5], [4, 16, 7, 7], [8, 27, 11, 11], [16, 46, 19, 19]], Yes, finite_divisor_poset_roundtrip)."},"skills":["Every Divisor Row Recovers Its Original Arithmetic 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-20261110-r06-mobius-inversion-finite-divisors-0067","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"920f23085f345a1325dd0496fb6d89578355bf2187c04d5bfee63e01b902ee0c","family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106067,"semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":2,\"b\":4,\"n\":17}"},"fields":{"title":"Every Divisor Row Recovers Its Original Arithmetic Value","stem":"Use positive integer n=17, exact integer arithmetic function f(k)=a*k+b with a=2, b=4. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).","answer":{"kind":"expression","value":"(17, [[1, 6, 6, 6], [17, 44, 38, 38]], Yes, finite_divisor_poset_roundtrip)","explanation":"For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (17, [[1, 6, 6, 6], [17, 44, 38, 38]], Yes, finite_divisor_poset_roundtrip)."},"skills":["Every Divisor Row Recovers Its Original Arithmetic 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-20261110-r06-mobius-inversion-finite-divisors-0068","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20e6fcc4a9c0023f8e6f56e936a2284727fb8ec1c945cf4ebdce8a25a5e157de","family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106068,"semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":3,\"b\":5,\"n\":18}"},"fields":{"title":"Every Divisor Row Recovers Its Original Arithmetic Value","stem":"Use positive integer n=18, exact integer arithmetic function f(k)=a*k+b with a=3, b=5. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).","answer":{"kind":"expression","value":"(18, [[1, 8, 8, 8], [2, 19, 11, 11], [3, 22, 14, 14], [6, 56, 23, 23], [9, 54, 32, 32], [18, 147, 59, 59]], Yes, finite_divisor_poset_roundtrip)","explanation":"For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (18, [[1, 8, 8, 8], [2, 19, 11, 11], [3, 22, 14, 14], [6, 56, 23, 23], [9, 54, 32, 32], [18, 147, 59, 59]], Yes, finite_divisor_poset_roundtrip)."},"skills":["Every Divisor Row Recovers Its Original Arithmetic 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-20261110-r06-mobius-inversion-finite-divisors-0069","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3fa141d489d1c2ea2ec2dd21a9ae1cfc22d89ffedf2dabef14153575142e6dc7","family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106069,"semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":4,\"b\":6,\"n\":19}"},"fields":{"title":"Every Divisor Row Recovers Its Original Arithmetic Value","stem":"Use positive integer n=19, exact integer arithmetic function f(k)=a*k+b with a=4, b=6. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).","answer":{"kind":"expression","value":"(19, [[1, 10, 10, 10], [19, 92, 82, 82]], Yes, finite_divisor_poset_roundtrip)","explanation":"For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (19, [[1, 10, 10, 10], [19, 92, 82, 82]], Yes, finite_divisor_poset_roundtrip)."},"skills":["Every Divisor Row Recovers Its Original Arithmetic 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-20261110-r06-mobius-inversion-finite-divisors-0070","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58ec1a5f4e29256188443aea1e663f24282e9d1aa8c5b8b58a722bf86648a6cf","family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106070,"semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":5,\"b\":7,\"n\":20}"},"fields":{"title":"Every Divisor Row Recovers Its Original Arithmetic Value","stem":"Use positive integer n=20, exact integer arithmetic function f(k)=a*k+b with a=5, b=7. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).","answer":{"kind":"expression","value":"(20, [[1, 12, 12, 12], [2, 29, 17, 17], [4, 56, 27, 27], [5, 44, 32, 32], [10, 118, 57, 57], [20, 252, 107, 107]], Yes, finite_divisor_poset_roundtrip)","explanation":"For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (20, [[1, 12, 12, 12], [2, 29, 17, 17], [4, 56, 27, 27], [5, 44, 32, 32], [10, 118, 57, 57], [20, 252, 107, 107]], Yes, finite_divisor_poset_roundtrip)."},"skills":["Every Divisor Row Recovers Its Original Arithmetic 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-20261110-r06-mobius-inversion-finite-divisors-0071","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b24bf941d78e1e08ad0d752f33871e3040ec7c312e47c1ba4bf0e0623bd092d3","family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106071,"semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":1,\"b\":8,\"n\":21}"},"fields":{"title":"Every Divisor Row Recovers Its Original Arithmetic Value","stem":"Use positive integer n=21, exact integer arithmetic function f(k)=a*k+b with a=1, b=8. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).","answer":{"kind":"expression","value":"(21, [[1, 9, 9, 9], [3, 20, 11, 11], [7, 24, 15, 15], [21, 64, 29, 29]], Yes, finite_divisor_poset_roundtrip)","explanation":"For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (21, [[1, 9, 9, 9], [3, 20, 11, 11], [7, 24, 15, 15], [21, 64, 29, 29]], Yes, finite_divisor_poset_roundtrip)."},"skills":["Every Divisor Row Recovers Its Original Arithmetic 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-20261110-r06-mobius-inversion-finite-divisors-0072","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2cfdbb7f5c1b66008c4611ef023a7cfd0a21206c7ab8ad532e8621497fdbb773","family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106072,"semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":2,\"b\":9,\"n\":22}"},"fields":{"title":"Every Divisor Row Recovers Its Original Arithmetic Value","stem":"Use positive integer n=22, exact integer arithmetic function f(k)=a*k+b with a=2, b=9. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).","answer":{"kind":"expression","value":"(22, [[1, 11, 11, 11], [2, 24, 13, 13], [11, 42, 31, 31], [22, 108, 53, 53]], Yes, finite_divisor_poset_roundtrip)","explanation":"For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (22, [[1, 11, 11, 11], [2, 24, 13, 13], [11, 42, 31, 31], [22, 108, 53, 53]], Yes, finite_divisor_poset_roundtrip)."},"skills":["Every Divisor Row Recovers Its Original Arithmetic 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-20261110-r06-mobius-inversion-finite-divisors-0073","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d44e1411b7f8a039331aeaba4d6745391162770894b990faf2a119fecabe6aea","family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106073,"semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":3,\"b\":10,\"n\":23}"},"fields":{"title":"Every Divisor Row Recovers Its Original Arithmetic Value","stem":"Use positive integer n=23, exact integer arithmetic function f(k)=a*k+b with a=3, b=10. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).","answer":{"kind":"expression","value":"(23, [[1, 13, 13, 13], [23, 92, 79, 79]], Yes, finite_divisor_poset_roundtrip)","explanation":"For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (23, [[1, 13, 13, 13], [23, 92, 79, 79]], Yes, finite_divisor_poset_roundtrip)."},"skills":["Every Divisor Row Recovers Its Original Arithmetic 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-20261110-r06-mobius-inversion-finite-divisors-0074","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e80ce7c614da51f165da13a06272e5a0e15a09ebfa13c48251cf7b07760a3849","family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106074,"semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":4,\"b\":11,\"n\":24}"},"fields":{"title":"Every Divisor Row Recovers Its Original Arithmetic Value","stem":"Use positive integer n=24, exact integer arithmetic function f(k)=a*k+b with a=4, b=11. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).","answer":{"kind":"expression","value":"(24, [[1, 15, 15, 15], [2, 34, 19, 19], [3, 38, 23, 23], [4, 61, 27, 27], [6, 92, 35, 35], [8, 104, 43, 43], [12, 178, 59, 59], [24, 328, 107, 107]], Yes, finite_divisor_poset_roundtrip)","explanation":"For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (24, [[1, 15, 15, 15], [2, 34, 19, 19], [3, 38, 23, 23], [4, 61, 27, 27], [6, 92, 35, 35], [8, 104, 43, 43], [12, 178, 59, 59], [24, 328, 107, 107]], Yes, finite_divisor_poset_roundtrip)."},"skills":["Every Divisor Row Recovers Its Original Arithmetic 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-20261110-r06-mobius-inversion-finite-divisors-0075","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bedf155c587bab20ceb58ee7f815ee737cc3039a290b430b415169035ba2f4f7","family_id":"qf-mod41-mobinv-roundtrip-table","seed":202611106075,"semantic_template":"qf-mod41-mobinv-roundtrip-table/{\"a\":5,\"b\":12,\"n\":25}"},"fields":{"title":"Every Divisor Row Recovers Its Original Arithmetic Value","stem":"Use positive integer n=25, exact integer arithmetic function f(k)=a*k+b with a=5, b=12. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, roundtrip_rows, all_equal, boundary).","answer":{"kind":"expression","value":"(25, [[1, 17, 17, 17], [5, 54, 37, 37], [25, 191, 137, 137]], Yes, finite_divisor_poset_roundtrip)","explanation":"For every divisor m of n, report [m,F(m),reconstructed f(m),original f(m)]. Keep the n=1 empty-prime-product/Mobius-one boundary. The exact result, retaining the requested order, is (25, [[1, 17, 17, 17], [5, 54, 37, 37], [25, 191, 137, 137]], Yes, finite_divisor_poset_roundtrip)."},"skills":["Every Divisor Row Recovers Its Original Arithmetic 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-20261110-r06-mobius-inversion-finite-divisors-0076","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c6fa2f851f83fed6d2161c6d136d2d4f5fb444bcca5e1ef5bca00ac941b4795","family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106076,"semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":1,\"b\":1,\"n\":6,\"omitted_divisor\":1}"},"fields":{"title":"An Omitted Divisor Must Be a Discriminative Missing-Term Witness","stem":"Use positive integer n=6, exact integer arithmetic function f(k)=a*k+b with a=1, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).","answer":{"kind":"expression","value":"(6, [1, 2, 3, 6], 1, [2, 3, 6], 7, -9, 7, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation":"Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (6, [1, 2, 3, 6], 1, [2, 3, 6], 7, -9, 7, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)."},"skills":["An Omitted Divisor Must Be a Discriminative Missing-Term Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0077","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5540811cba7ac26efe3545090bb417cae506c4f3f5110ee27bcaadfaeba117e4","family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106077,"semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":2,\"b\":1,\"n\":12,\"omitted_divisor\":1}"},"fields":{"title":"An Omitted Divisor Must Be a Discriminative Missing-Term Witness","stem":"Use positive integer n=12, exact integer arithmetic function f(k)=a*k+b with a=2, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).","answer":{"kind":"expression","value":"(12, [1, 2, 3, 4, 6, 12], 1, [2, 3, 4, 6, 12], 25, -37, 25, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation":"Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (12, [1, 2, 3, 4, 6, 12], 1, [2, 3, 4, 6, 12], 25, -37, 25, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)."},"skills":["An Omitted Divisor Must Be a Discriminative Missing-Term Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0078","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95d63d90a5192ab911a9b1601e33745f73ef48c8fad0f6c4fc8c414262c165dd","family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106078,"semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":3,\"b\":1,\"n\":20,\"omitted_divisor\":1}"},"fields":{"title":"An Omitted Divisor Must Be a Discriminative Missing-Term Witness","stem":"Use positive integer n=20, exact integer arithmetic function f(k)=a*k+b with a=3, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).","answer":{"kind":"expression","value":"(20, [1, 2, 4, 5, 10, 20], 1, [2, 4, 5, 10, 20], 61, -71, 61, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation":"Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (20, [1, 2, 4, 5, 10, 20], 1, [2, 4, 5, 10, 20], 61, -71, 61, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)."},"skills":["An Omitted Divisor Must Be a Discriminative Missing-Term Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0079","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b2ea23a4e96cda593f302b3d358085c8787a9dce404623f993ce60978319fb8","family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106079,"semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":4,\"b\":1,\"n\":30,\"omitted_divisor\":1}"},"fields":{"title":"An Omitted Divisor Must Be a Discriminative Missing-Term Witness","stem":"Use positive integer n=30, exact integer arithmetic function f(k)=a*k+b with a=4, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).","answer":{"kind":"expression","value":"(30, [1, 2, 3, 5, 6, 10, 15, 30], 1, [2, 3, 5, 6, 10, 15, 30], 121, -175, 121, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation":"Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (30, [1, 2, 3, 5, 6, 10, 15, 30], 1, [2, 3, 5, 6, 10, 15, 30], 121, -175, 121, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)."},"skills":["An Omitted Divisor Must Be a Discriminative Missing-Term Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0080","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce73346e98f8fdc562d6f5456d852a1738b428014220489b2052164be42deb6b","family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106080,"semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":5,\"b\":1,\"n\":42,\"omitted_divisor\":1}"},"fields":{"title":"An Omitted Divisor Must Be a Discriminative Missing-Term Witness","stem":"Use positive integer n=42, exact integer arithmetic function f(k)=a*k+b with a=5, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).","answer":{"kind":"expression","value":"(42, [1, 2, 3, 6, 7, 14, 21, 42], 1, [2, 3, 6, 7, 14, 21, 42], 211, -277, 211, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation":"Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (42, [1, 2, 3, 6, 7, 14, 21, 42], 1, [2, 3, 6, 7, 14, 21, 42], 211, -277, 211, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)."},"skills":["An Omitted Divisor Must Be a Discriminative Missing-Term Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0081","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2bb61cf7a712f74b843df381a414927782c1a87b49fb5d3ec8a9122ed80d7ac2","family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106081,"semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":6,\"b\":1,\"n\":56,\"omitted_divisor\":1}"},"fields":{"title":"An Omitted Divisor Must Be a Discriminative Missing-Term Witness","stem":"Use positive integer n=56, exact integer arithmetic function f(k)=a*k+b with a=6, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).","answer":{"kind":"expression","value":"(56, [1, 2, 4, 7, 8, 14, 28, 56], 1, [2, 4, 7, 8, 14, 28, 56], 337, -391, 337, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation":"Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (56, [1, 2, 4, 7, 8, 14, 28, 56], 1, [2, 4, 7, 8, 14, 28, 56], 337, -391, 337, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)."},"skills":["An Omitted Divisor Must Be a Discriminative Missing-Term Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0082","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b6dd1df151f72567bde36b1b7ac32f8c0045e4f8747e13a09ac70b84dc5d626","family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106082,"semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":7,\"b\":1,\"n\":72,\"omitted_divisor\":1}"},"fields":{"title":"An Omitted Divisor Must Be a Discriminative Missing-Term Witness","stem":"Use positive integer n=72, exact integer arithmetic function f(k)=a*k+b with a=7, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).","answer":{"kind":"expression","value":"(72, [1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72], 1, [2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72], 505, -872, 505, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation":"Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (72, [1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72], 1, [2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72], 505, -872, 505, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)."},"skills":["An Omitted Divisor Must Be a Discriminative Missing-Term Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0083","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21628cad4959cb98f74e9e80ebe57e778ad924b630d963ab1aad3f71655f83a2","family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106083,"semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":8,\"b\":1,\"n\":90,\"omitted_divisor\":1}"},"fields":{"title":"An Omitted Divisor Must Be a Discriminative Missing-Term Witness","stem":"Use positive integer n=90, exact integer arithmetic function f(k)=a*k+b with a=8, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).","answer":{"kind":"expression","value":"(90, [1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90], 1, [2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90], 721, -1163, 721, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation":"Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (90, [1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90], 1, [2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90], 721, -1163, 721, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)."},"skills":["An Omitted Divisor Must Be a Discriminative Missing-Term Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0084","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"558a684e635d67cb0a0526fb0beaecc55226f8defb81391144c91440549a114c","family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106084,"semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":9,\"b\":1,\"n\":110,\"omitted_divisor\":1}"},"fields":{"title":"An Omitted Divisor Must Be a Discriminative Missing-Term Witness","stem":"Use positive integer n=110, exact integer arithmetic function f(k)=a*k+b with a=9, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).","answer":{"kind":"expression","value":"(110, [1, 2, 5, 10, 11, 22, 55, 110], 1, [2, 5, 10, 11, 22, 55, 110], 991, -961, 991, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation":"Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (110, [1, 2, 5, 10, 11, 22, 55, 110], 1, [2, 5, 10, 11, 22, 55, 110], 991, -961, 991, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)."},"skills":["An Omitted Divisor Must Be a Discriminative Missing-Term Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0085","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b829f82283a4fd24d0aa9a546fc9a109e5eda62ed55f708954993f12d090324d","family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106085,"semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":10,\"b\":1,\"n\":132,\"omitted_divisor\":1}"},"fields":{"title":"An Omitted Divisor Must Be a Discriminative Missing-Term Witness","stem":"Use positive integer n=132, exact integer arithmetic function f(k)=a*k+b with a=10, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).","answer":{"kind":"expression","value":"(132, [1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132], 1, [2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132], 1321, -2051, 1321, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation":"Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (132, [1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132], 1, [2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132], 1321, -2051, 1321, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)."},"skills":["An Omitted Divisor Must Be a Discriminative Missing-Term Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0086","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22e9a31cde649814099c983e46ea8450f59cef468fe9be5f15c0624508c87356","family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106086,"semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":11,\"b\":1,\"n\":156,\"omitted_divisor\":1}"},"fields":{"title":"An Omitted Divisor Must Be a Discriminative Missing-Term Witness","stem":"Use positive integer n=156, exact integer arithmetic function f(k)=a*k+b with a=11, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).","answer":{"kind":"expression","value":"(156, [1, 2, 3, 4, 6, 12, 13, 26, 39, 52, 78, 156], 1, [2, 3, 4, 6, 12, 13, 26, 39, 52, 78, 156], 1717, -2607, 1717, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation":"Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (156, [1, 2, 3, 4, 6, 12, 13, 26, 39, 52, 78, 156], 1, [2, 3, 4, 6, 12, 13, 26, 39, 52, 78, 156], 1717, -2607, 1717, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)."},"skills":["An Omitted Divisor Must Be a Discriminative Missing-Term Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0087","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7a2e659418951babd87be333cb79af06b858ad7ff2aa632e174dc5448cf1cc3","family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106087,"semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":12,\"b\":1,\"n\":182,\"omitted_divisor\":1}"},"fields":{"title":"An Omitted Divisor Must Be a Discriminative Missing-Term Witness","stem":"Use positive integer n=182, exact integer arithmetic function f(k)=a*k+b with a=12, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).","answer":{"kind":"expression","value":"(182, [1, 2, 7, 13, 14, 26, 91, 182], 1, [2, 7, 13, 14, 26, 91, 182], 2185, -1855, 2185, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation":"Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (182, [1, 2, 7, 13, 14, 26, 91, 182], 1, [2, 7, 13, 14, 26, 91, 182], 2185, -1855, 2185, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)."},"skills":["An Omitted Divisor Must Be a Discriminative Missing-Term Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0088","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3f4d631dba76b038992ecfed116f1134d600f0892fd95c28cdcd55044ba37ed","family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106088,"semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":13,\"b\":1,\"n\":210,\"omitted_divisor\":1}"},"fields":{"title":"An Omitted Divisor Must Be a Discriminative Missing-Term Witness","stem":"Use positive integer n=210, exact integer arithmetic function f(k)=a*k+b with a=13, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).","answer":{"kind":"expression","value":"(210, [1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210], 1, [2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210], 2731, -4773, 2731, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation":"Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (210, [1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210], 1, [2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210], 2731, -4773, 2731, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)."},"skills":["An Omitted Divisor Must Be a Discriminative Missing-Term Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0089","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"683bbc5f1012f231d5e027f7f0e486f8804ca067958417064583967e4e3a12f6","family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106089,"semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":14,\"b\":1,\"n\":240,\"omitted_divisor\":1}"},"fields":{"title":"An Omitted Divisor Must Be a Discriminative Missing-Term Witness","stem":"Use positive integer n=240, exact integer arithmetic function f(k)=a*k+b with a=14, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).","answer":{"kind":"expression","value":"(240, [1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240], 1, [2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240], 3361, -7075, 3361, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation":"Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (240, [1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240], 1, [2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240], 3361, -7075, 3361, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)."},"skills":["An Omitted Divisor Must Be a Discriminative Missing-Term Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0090","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77fdcbf6e70243d35c8ed59d0c7c5bfa407fa3666b411bd977f1e6e3f4057bde","family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106090,"semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":15,\"b\":1,\"n\":272,\"omitted_divisor\":1}"},"fields":{"title":"An Omitted Divisor Must Be a Discriminative Missing-Term Witness","stem":"Use positive integer n=272, exact integer arithmetic function f(k)=a*k+b with a=15, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).","answer":{"kind":"expression","value":"(272, [1, 2, 4, 8, 16, 17, 34, 68, 136, 272], 1, [2, 4, 8, 16, 17, 34, 68, 136, 272], 4081, -4299, 4081, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation":"Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (272, [1, 2, 4, 8, 16, 17, 34, 68, 136, 272], 1, [2, 4, 8, 16, 17, 34, 68, 136, 272], 4081, -4299, 4081, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)."},"skills":["An Omitted Divisor Must Be a Discriminative Missing-Term Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0091","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62c73fbf4386d6074f4622f793a5a7564b264870b5554267616523a692fd000c","family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106091,"semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":16,\"b\":1,\"n\":306,\"omitted_divisor\":1}"},"fields":{"title":"An Omitted Divisor Must Be a Discriminative Missing-Term Witness","stem":"Use positive integer n=306, exact integer arithmetic function f(k)=a*k+b with a=16, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).","answer":{"kind":"expression","value":"(306, [1, 2, 3, 6, 9, 17, 18, 34, 51, 102, 153, 306], 1, [2, 3, 6, 9, 17, 18, 34, 51, 102, 153, 306], 4897, -6347, 4897, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation":"Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (306, [1, 2, 3, 6, 9, 17, 18, 34, 51, 102, 153, 306], 1, [2, 3, 6, 9, 17, 18, 34, 51, 102, 153, 306], 4897, -6347, 4897, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)."},"skills":["An Omitted Divisor Must Be a Discriminative Missing-Term Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0092","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51d8d5f5cac014bfa7a066d4a116dcb7cc9b4458f5619e5826416ea285ea1975","family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106092,"semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":17,\"b\":1,\"n\":342,\"omitted_divisor\":1}"},"fields":{"title":"An Omitted Divisor Must Be a Discriminative Missing-Term Witness","stem":"Use positive integer n=342, exact integer arithmetic function f(k)=a*k+b with a=17, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).","answer":{"kind":"expression","value":"(342, [1, 2, 3, 6, 9, 18, 19, 38, 57, 114, 171, 342], 1, [2, 3, 6, 9, 18, 19, 38, 57, 114, 171, 342], 5815, -7457, 5815, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation":"Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (342, [1, 2, 3, 6, 9, 18, 19, 38, 57, 114, 171, 342], 1, [2, 3, 6, 9, 18, 19, 38, 57, 114, 171, 342], 5815, -7457, 5815, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)."},"skills":["An Omitted Divisor Must Be a Discriminative Missing-Term Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0093","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"148127e59a0ac2f7f993fbd6cbff1bd8639c54a554162772126273fa83c2e330","family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106093,"semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":18,\"b\":1,\"n\":380,\"omitted_divisor\":1}"},"fields":{"title":"An Omitted Divisor Must Be a Discriminative Missing-Term Witness","stem":"Use positive integer n=380, exact integer arithmetic function f(k)=a*k+b with a=18, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).","answer":{"kind":"expression","value":"(380, [1, 2, 4, 5, 10, 19, 20, 38, 76, 95, 190, 380], 1, [2, 4, 5, 10, 19, 20, 38, 76, 95, 190, 380], 6841, -8291, 6841, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation":"Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (380, [1, 2, 4, 5, 10, 19, 20, 38, 76, 95, 190, 380], 1, [2, 4, 5, 10, 19, 20, 38, 76, 95, 190, 380], 6841, -8291, 6841, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)."},"skills":["An Omitted Divisor Must Be a Discriminative Missing-Term Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0094","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7806e4ff3d8277bd94ac33a0f0e8730891b188bfb72572ca8aafed07af72349e","family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106094,"semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":19,\"b\":1,\"n\":420,\"omitted_divisor\":1}"},"fields":{"title":"An Omitted Divisor Must Be a Discriminative Missing-Term Witness","stem":"Use positive integer n=420, exact integer arithmetic function f(k)=a*k+b with a=19, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).","answer":{"kind":"expression","value":"(420, [1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420], 1, [2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420], 7981, -17579, 7981, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation":"Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (420, [1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420], 1, [2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420], 7981, -17579, 7981, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)."},"skills":["An Omitted Divisor Must Be a Discriminative Missing-Term Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0095","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dddf2a7931dc250d7b88cb300d71e45113ffd09679958f4339c1fcfa7a826586","family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106095,"semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":20,\"b\":1,\"n\":462,\"omitted_divisor\":1}"},"fields":{"title":"An Omitted Divisor Must Be a Discriminative Missing-Term Witness","stem":"Use positive integer n=462, exact integer arithmetic function f(k)=a*k+b with a=20, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).","answer":{"kind":"expression","value":"(462, [1, 2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462], 1, [2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462], 9241, -13815, 9241, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation":"Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (462, [1, 2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462], 1, [2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462], 9241, -13815, 9241, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)."},"skills":["An Omitted Divisor Must Be a Discriminative Missing-Term Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0096","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30b7cb89a7716401aa1dbb4c1c72e4bb53973136fc34cf5fb3d3ee6df65325e4","family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106096,"semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":21,\"b\":1,\"n\":506,\"omitted_divisor\":1}"},"fields":{"title":"An Omitted Divisor Must Be a Discriminative Missing-Term Witness","stem":"Use positive integer n=506, exact integer arithmetic function f(k)=a*k+b with a=21, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).","answer":{"kind":"expression","value":"(506, [1, 2, 11, 22, 23, 46, 253, 506], 1, [2, 11, 22, 23, 46, 253, 506], 10627, -7525, 10627, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation":"Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (506, [1, 2, 11, 22, 23, 46, 253, 506], 1, [2, 11, 22, 23, 46, 253, 506], 10627, -7525, 10627, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)."},"skills":["An Omitted Divisor Must Be a Discriminative Missing-Term Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0097","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e146e6c08d7cff1d825de34b540c3f385b865ef7e03a29b5274c47249fbada00","family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106097,"semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":22,\"b\":1,\"n\":552,\"omitted_divisor\":1}"},"fields":{"title":"An Omitted Divisor Must Be a Discriminative Missing-Term Witness","stem":"Use positive integer n=552, exact integer arithmetic function f(k)=a*k+b with a=22, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).","answer":{"kind":"expression","value":"(552, [1, 2, 3, 4, 6, 8, 12, 23, 24, 46, 69, 92, 138, 184, 276, 552], 1, [2, 3, 4, 6, 8, 12, 23, 24, 46, 69, 92, 138, 184, 276, 552], 12145, -19551, 12145, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation":"Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (552, [1, 2, 3, 4, 6, 8, 12, 23, 24, 46, 69, 92, 138, 184, 276, 552], 1, [2, 3, 4, 6, 8, 12, 23, 24, 46, 69, 92, 138, 184, 276, 552], 12145, -19551, 12145, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)."},"skills":["An Omitted Divisor Must Be a Discriminative Missing-Term Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0098","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed894edff06c7e0027a6f7a63280722071ef19d923e4e5f03124a26ce0a6a996","family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106098,"semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":23,\"b\":1,\"n\":600,\"omitted_divisor\":1}"},"fields":{"title":"An Omitted Divisor Must Be a Discriminative Missing-Term Witness","stem":"Use positive integer n=600, exact integer arithmetic function f(k)=a*k+b with a=23, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).","answer":{"kind":"expression","value":"(600, [1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 25, 30, 40, 50, 60, 75, 100, 120, 150, 200, 300, 600], 1, [2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 25, 30, 40, 50, 60, 75, 100, 120, 150, 200, 300, 600], 13801, -29003, 13801, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation":"Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (600, [1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 25, 30, 40, 50, 60, 75, 100, 120, 150, 200, 300, 600], 1, [2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 25, 30, 40, 50, 60, 75, 100, 120, 150, 200, 300, 600], 13801, -29003, 13801, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)."},"skills":["An Omitted Divisor Must Be a Discriminative Missing-Term Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0099","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37610b19b5e3627dca01785986b6ec6e92d2d74fb74440a4f8bd6c431f0a0dfe","family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106099,"semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":24,\"b\":1,\"n\":650,\"omitted_divisor\":1}"},"fields":{"title":"An Omitted Divisor Must Be a Discriminative Missing-Term Witness","stem":"Use positive integer n=650, exact integer arithmetic function f(k)=a*k+b with a=24, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).","answer":{"kind":"expression","value":"(650, [1, 2, 5, 10, 13, 25, 26, 50, 65, 130, 325, 650], 1, [2, 5, 10, 13, 25, 26, 50, 65, 130, 325, 650], 15601, -15659, 15601, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation":"Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (650, [1, 2, 5, 10, 13, 25, 26, 50, 65, 130, 325, 650], 1, [2, 5, 10, 13, 25, 26, 50, 65, 130, 325, 650], 15601, -15659, 15601, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)."},"skills":["An Omitted Divisor Must Be a Discriminative Missing-Term Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261110-r06-mobius-inversion-finite-divisors-0100","batch_id":"batch-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c8c6f3057c0dcaf3f07e4b509df351720ab4c7ce8a5f6d017b8b9deae512fd5","family_id":"qf-mod41-mobinv-incomplete-divisor-boundary","seed":202611106100,"semantic_template":"qf-mod41-mobinv-incomplete-divisor-boundary/{\"a\":25,\"b\":1,\"n\":702,\"omitted_divisor\":1}"},"fields":{"title":"An Omitted Divisor Must Be a Discriminative Missing-Term Witness","stem":"Use positive integer n=702, exact integer arithmetic function f(k)=a*k+b with a=25, b=1. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Omit divisor=1. Give (n, all_divisors, omitted_divisor, partial_divisors, complete_value, partial_value, f_n, incomplete_certificate, boundary).","answer":{"kind":"expression","value":"(702, [1, 2, 3, 6, 9, 13, 18, 26, 27, 39, 54, 78, 117, 234, 351, 702], 1, [2, 3, 6, 9, 13, 18, 26, 27, 39, 54, 78, 117, 234, 351, 702], 17551, -24465, 17551, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)","explanation":"Require the omitted value actually to divide n and the incomplete inverse to differ from f(n). Retain full/partial divisor lists and both sums. The fixed No refuses an incomplete certificate, not the correct complete inverse. The exact result, retaining the requested order, is (702, [1, 2, 3, 6, 9, 13, 18, 26, 27, 39, 54, 78, 117, 234, 351, 702], 1, [2, 3, 6, 9, 13, 18, 26, 27, 39, 54, 78, 117, 234, 351, 702], 17551, -24465, 17551, No, all_positive_divisors_and_consistent_d_to_n_over_d_indexing_required)."},"skills":["An Omitted Divisor Must Be a Discriminative Missing-Term Witness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}