{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0001","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d0c4561476f8a1e7a1171686a24c4c254e6ac5c4e0213c8c0766e54c2fab77a","family_id":"qf-mod39-upf-factorization-audit","seed":202611086001,"semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,2,3],\"n\":12}"},"fields":{"title":"Prime Factors and a Complete Product Are Separate Validity Checks","stem":"Use positive integer n=12>1 and candidate factors=[2, 3, 2] in supplied order. Every factor must be prime and their complete exact product must equal n. A matching product of composite factors alone is not a valid prime factorization. Give (n, factors, all_factors_prime, factor_product, valid_factorization, certificate).","answer":{"kind":"expression","value":"(12, [2, 3, 2], Yes, 12, Yes, existence_requires_complete_prime_product)","explanation":"Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (12, [2, 3, 2], Yes, 12, Yes, existence_requires_complete_prime_product)."},"skills":["Prime Factors and a Complete Product Are Separate Validity Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0002","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"26c029015237b7b4d1cdea625d54fef4641c6bb81ca026ee6ef4824a7d9143c4","family_id":"qf-mod39-upf-factorization-audit","seed":202611086002,"semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,3],\"n\":18}"},"fields":{"title":"Prime Factors and a Complete Product Are Separate Validity Checks","stem":"Use positive integer n=18>1 and candidate factors=[2, 3, 3] in supplied order. Every factor must be prime and their complete exact product must equal n. A matching product of composite factors alone is not a valid prime factorization. Give (n, factors, all_factors_prime, factor_product, valid_factorization, certificate).","answer":{"kind":"expression","value":"(18, [2, 3, 3], Yes, 18, Yes, existence_requires_complete_prime_product)","explanation":"Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (18, [2, 3, 3], Yes, 18, Yes, existence_requires_complete_prime_product)."},"skills":["Prime Factors and a Complete Product Are Separate Validity Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0003","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f5fc66f787a549bf54e42da963171709bc256f0c5bbbb804b08cac51df5ed61","family_id":"qf-mod39-upf-factorization-audit","seed":202611086003,"semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,5],\"n\":30}"},"fields":{"title":"Prime Factors and a Complete Product Are Separate Validity Checks","stem":"Use positive integer n=30>1 and candidate factors=[2, 3, 5] in supplied order. Every factor must be prime and their complete exact product must equal n. A matching product of composite factors alone is not a valid prime factorization. Give (n, factors, all_factors_prime, factor_product, valid_factorization, certificate).","answer":{"kind":"expression","value":"(30, [2, 3, 5], Yes, 30, Yes, existence_requires_complete_prime_product)","explanation":"Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (30, [2, 3, 5], Yes, 30, Yes, existence_requires_complete_prime_product)."},"skills":["Prime Factors and a Complete Product Are Separate Validity Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0004","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5608f960356589a42c087b909de15a47e9a00ddc61dbb23de23e6b5cb6fe0aad","family_id":"qf-mod39-upf-factorization-audit","seed":202611086004,"semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,7],\"n\":42}"},"fields":{"title":"Prime Factors and a Complete Product Are Separate Validity Checks","stem":"Use positive integer n=42>1 and candidate factors=[2, 3, 7] in supplied order. Every factor must be prime and their complete exact product must equal n. A matching product of composite factors alone is not a valid prime factorization. Give (n, factors, all_factors_prime, factor_product, valid_factorization, certificate).","answer":{"kind":"expression","value":"(42, [2, 3, 7], Yes, 42, Yes, existence_requires_complete_prime_product)","explanation":"Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (42, [2, 3, 7], Yes, 42, Yes, existence_requires_complete_prime_product)."},"skills":["Prime Factors and a Complete Product Are Separate Validity Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0005","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"161dc8721e87773c90dd5b4fedb3da21f82df22fac8ba9e590739feb856afa28","family_id":"qf-mod39-upf-factorization-audit","seed":202611086005,"semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,11],\"n\":66}"},"fields":{"title":"Prime Factors and a Complete Product Are Separate Validity Checks","stem":"Use positive integer n=66>1 and candidate factors=[2, 3, 11] in supplied order. Every factor must be prime and their complete exact product must equal n. A matching product of composite factors alone is not a valid prime factorization. Give (n, factors, all_factors_prime, factor_product, valid_factorization, certificate).","answer":{"kind":"expression","value":"(66, [2, 3, 11], Yes, 66, Yes, existence_requires_complete_prime_product)","explanation":"Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (66, [2, 3, 11], Yes, 66, Yes, existence_requires_complete_prime_product)."},"skills":["Prime Factors and a Complete Product Are Separate Validity Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0006","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5dc5dfba8c672dd875934e79e4b08bb6ba45745a8d663923eaa00cb807ec8a70","family_id":"qf-mod39-upf-factorization-audit","seed":202611086006,"semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,13],\"n\":78}"},"fields":{"title":"Prime Factors and a Complete Product Are Separate Validity Checks","stem":"Use positive integer n=78>1 and candidate factors=[2, 3, 13] in supplied order. Every factor must be prime and their complete exact product must equal n. A matching product of composite factors alone is not a valid prime factorization. Give (n, factors, all_factors_prime, factor_product, valid_factorization, certificate).","answer":{"kind":"expression","value":"(78, [2, 3, 13], Yes, 78, Yes, existence_requires_complete_prime_product)","explanation":"Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (78, [2, 3, 13], Yes, 78, Yes, existence_requires_complete_prime_product)."},"skills":["Prime Factors and a Complete Product Are Separate Validity Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0007","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0fd9c734b6dd94ea616fac74a4286ca5cc4dad9875ca72b86a617e685a0dd75c","family_id":"qf-mod39-upf-factorization-audit","seed":202611086007,"semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,17],\"n\":102}"},"fields":{"title":"Prime Factors and a Complete Product Are Separate Validity Checks","stem":"Use positive integer n=102>1 and candidate factors=[2, 3, 17] in supplied order. Every factor must be prime and their complete exact product must equal n. A matching product of composite factors alone is not a valid prime factorization. Give (n, factors, all_factors_prime, factor_product, valid_factorization, certificate).","answer":{"kind":"expression","value":"(102, [2, 3, 17], Yes, 102, Yes, existence_requires_complete_prime_product)","explanation":"Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (102, [2, 3, 17], Yes, 102, Yes, existence_requires_complete_prime_product)."},"skills":["Prime Factors and a Complete Product Are Separate Validity Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0008","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e75673ad2844e811ebf84119f98c143bf1d16d28fda9fa1859c4a809bd632b60","family_id":"qf-mod39-upf-factorization-audit","seed":202611086008,"semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,19],\"n\":114}"},"fields":{"title":"Prime Factors and a Complete Product Are Separate Validity Checks","stem":"Use positive integer n=114>1 and candidate factors=[2, 3, 19] in supplied order. Every factor must be prime and their complete exact product must equal n. A matching product of composite factors alone is not a valid prime factorization. Give (n, factors, all_factors_prime, factor_product, valid_factorization, certificate).","answer":{"kind":"expression","value":"(114, [2, 3, 19], Yes, 114, Yes, existence_requires_complete_prime_product)","explanation":"Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (114, [2, 3, 19], Yes, 114, Yes, existence_requires_complete_prime_product)."},"skills":["Prime Factors and a Complete Product Are Separate Validity Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0009","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01167e2d4bb77ff6b5f7cc90a16adf41f5c08908ce5583d79989e2a3807f8a6e","family_id":"qf-mod39-upf-factorization-audit","seed":202611086009,"semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,23],\"n\":138}"},"fields":{"title":"Prime Factors and a Complete Product Are Separate Validity Checks","stem":"Use positive integer n=138>1 and candidate factors=[2, 3, 23] in supplied order. Every factor must be prime and their complete exact product must equal n. A matching product of composite factors alone is not a valid prime factorization. Give (n, factors, all_factors_prime, factor_product, valid_factorization, certificate).","answer":{"kind":"expression","value":"(138, [2, 3, 23], Yes, 138, Yes, existence_requires_complete_prime_product)","explanation":"Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (138, [2, 3, 23], Yes, 138, Yes, existence_requires_complete_prime_product)."},"skills":["Prime Factors and a Complete Product Are Separate Validity Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0010","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02b450eda30f40b24e64eff2c0f3dbc38f388577f35aaff462c7292354ad9fbe","family_id":"qf-mod39-upf-factorization-audit","seed":202611086010,"semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,29],\"n\":174}"},"fields":{"title":"Prime Factors and a Complete Product Are Separate Validity Checks","stem":"Use positive integer n=174>1 and candidate factors=[2, 3, 29] in supplied order. Every factor must be prime and their complete exact product must equal n. A matching product of composite factors alone is not a valid prime factorization. Give (n, factors, all_factors_prime, factor_product, valid_factorization, certificate).","answer":{"kind":"expression","value":"(174, [2, 3, 29], Yes, 174, Yes, existence_requires_complete_prime_product)","explanation":"Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (174, [2, 3, 29], Yes, 174, Yes, existence_requires_complete_prime_product)."},"skills":["Prime Factors and a Complete Product Are Separate Validity Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0011","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66a064d4fd54091f1aca4ffa4373af307c8cee6c21b70ade01f8bf2c4f134bda","family_id":"qf-mod39-upf-factorization-audit","seed":202611086011,"semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,31],\"n\":186}"},"fields":{"title":"Prime Factors and a Complete Product Are Separate Validity Checks","stem":"Use positive integer n=186>1 and candidate factors=[2, 3, 31] in supplied order. Every factor must be prime and their complete exact product must equal n. A matching product of composite factors alone is not a valid prime factorization. Give (n, factors, all_factors_prime, factor_product, valid_factorization, certificate).","answer":{"kind":"expression","value":"(186, [2, 3, 31], Yes, 186, Yes, existence_requires_complete_prime_product)","explanation":"Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (186, [2, 3, 31], Yes, 186, Yes, existence_requires_complete_prime_product)."},"skills":["Prime Factors and a Complete Product Are Separate Validity Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0012","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46d605966c80cffedef07c21f3d826e491674a1cc835cbdf086c53384c07de0e","family_id":"qf-mod39-upf-factorization-audit","seed":202611086012,"semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,37],\"n\":222}"},"fields":{"title":"Prime Factors and a Complete Product Are Separate Validity Checks","stem":"Use positive integer n=222>1 and candidate factors=[2, 3, 37] in supplied order. Every factor must be prime and their complete exact product must equal n. A matching product of composite factors alone is not a valid prime factorization. Give (n, factors, all_factors_prime, factor_product, valid_factorization, certificate).","answer":{"kind":"expression","value":"(222, [2, 3, 37], Yes, 222, Yes, existence_requires_complete_prime_product)","explanation":"Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (222, [2, 3, 37], Yes, 222, Yes, existence_requires_complete_prime_product)."},"skills":["Prime Factors and a Complete Product Are Separate Validity Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0013","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f439a4dc811f212d15c9f7c1491be90df3fa6df01cbb2152a3c5b22b908f31b9","family_id":"qf-mod39-upf-factorization-audit","seed":202611086013,"semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,41],\"n\":246}"},"fields":{"title":"Prime Factors and a Complete Product Are Separate Validity Checks","stem":"Use positive integer n=246>1 and candidate factors=[2, 3, 41] in supplied order. Every factor must be prime and their complete exact product must equal n. A matching product of composite factors alone is not a valid prime factorization. Give (n, factors, all_factors_prime, factor_product, valid_factorization, certificate).","answer":{"kind":"expression","value":"(246, [2, 3, 41], Yes, 246, Yes, existence_requires_complete_prime_product)","explanation":"Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (246, [2, 3, 41], Yes, 246, Yes, existence_requires_complete_prime_product)."},"skills":["Prime Factors and a Complete Product Are Separate Validity Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0014","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d4b405693eeb50641b68cdffe8560907d2bc1b1f07a2ca0c4941de068369072","family_id":"qf-mod39-upf-factorization-audit","seed":202611086014,"semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,43],\"n\":258}"},"fields":{"title":"Prime Factors and a Complete Product Are Separate Validity Checks","stem":"Use positive integer n=258>1 and candidate factors=[2, 3, 43] in supplied order. Every factor must be prime and their complete exact product must equal n. A matching product of composite factors alone is not a valid prime factorization. Give (n, factors, all_factors_prime, factor_product, valid_factorization, certificate).","answer":{"kind":"expression","value":"(258, [2, 3, 43], Yes, 258, Yes, existence_requires_complete_prime_product)","explanation":"Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (258, [2, 3, 43], Yes, 258, Yes, existence_requires_complete_prime_product)."},"skills":["Prime Factors and a Complete Product Are Separate Validity Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0015","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ceb97098c8d3e561e8d3a1caa6d99fd89d20603359ac3523469ffa51ff5e19c8","family_id":"qf-mod39-upf-factorization-audit","seed":202611086015,"semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,47],\"n\":282}"},"fields":{"title":"Prime Factors and a Complete Product Are Separate Validity Checks","stem":"Use positive integer n=282>1 and candidate factors=[2, 3, 47] in supplied order. Every factor must be prime and their complete exact product must equal n. A matching product of composite factors alone is not a valid prime factorization. Give (n, factors, all_factors_prime, factor_product, valid_factorization, certificate).","answer":{"kind":"expression","value":"(282, [2, 3, 47], Yes, 282, Yes, existence_requires_complete_prime_product)","explanation":"Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (282, [2, 3, 47], Yes, 282, Yes, existence_requires_complete_prime_product)."},"skills":["Prime Factors and a Complete Product Are Separate Validity Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0016","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8faea635dce867b28625bbdac1f76fb0692b24616d26f16d169fc56a83bd484f","family_id":"qf-mod39-upf-factorization-audit","seed":202611086016,"semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,53],\"n\":318}"},"fields":{"title":"Prime Factors and a Complete Product Are Separate Validity Checks","stem":"Use positive integer n=318>1 and candidate factors=[2, 3, 53] in supplied order. Every factor must be prime and their complete exact product must equal n. A matching product of composite factors alone is not a valid prime factorization. Give (n, factors, all_factors_prime, factor_product, valid_factorization, certificate).","answer":{"kind":"expression","value":"(318, [2, 3, 53], Yes, 318, Yes, existence_requires_complete_prime_product)","explanation":"Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (318, [2, 3, 53], Yes, 318, Yes, existence_requires_complete_prime_product)."},"skills":["Prime Factors and a Complete Product Are Separate Validity Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0017","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7559753ea05167f9510afb8ed01431101344e8ff7529358693348c35e5683c7","family_id":"qf-mod39-upf-factorization-audit","seed":202611086017,"semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,59],\"n\":354}"},"fields":{"title":"Prime Factors and a Complete Product Are Separate Validity Checks","stem":"Use positive integer n=354>1 and candidate factors=[2, 3, 59] in supplied order. Every factor must be prime and their complete exact product must equal n. A matching product of composite factors alone is not a valid prime factorization. Give (n, factors, all_factors_prime, factor_product, valid_factorization, certificate).","answer":{"kind":"expression","value":"(354, [2, 3, 59], Yes, 354, Yes, existence_requires_complete_prime_product)","explanation":"Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (354, [2, 3, 59], Yes, 354, Yes, existence_requires_complete_prime_product)."},"skills":["Prime Factors and a Complete Product Are Separate Validity Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0018","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16154605ef5e184ee62df8c23313ef74c156aaa87b969dda5f67143e379085b2","family_id":"qf-mod39-upf-factorization-audit","seed":202611086018,"semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,61],\"n\":366}"},"fields":{"title":"Prime Factors and a Complete Product Are Separate Validity Checks","stem":"Use positive integer n=366>1 and candidate factors=[2, 3, 61] in supplied order. Every factor must be prime and their complete exact product must equal n. A matching product of composite factors alone is not a valid prime factorization. Give (n, factors, all_factors_prime, factor_product, valid_factorization, certificate).","answer":{"kind":"expression","value":"(366, [2, 3, 61], Yes, 366, Yes, existence_requires_complete_prime_product)","explanation":"Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (366, [2, 3, 61], Yes, 366, Yes, existence_requires_complete_prime_product)."},"skills":["Prime Factors and a Complete Product Are Separate Validity Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0019","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7cc7aa1d383d793adfdb375ae862737c76bc0c032d4c8302be31d27b7885b499","family_id":"qf-mod39-upf-factorization-audit","seed":202611086019,"semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,67],\"n\":402}"},"fields":{"title":"Prime Factors and a Complete Product Are Separate Validity Checks","stem":"Use positive integer n=402>1 and candidate factors=[2, 3, 67] in supplied order. Every factor must be prime and their complete exact product must equal n. A matching product of composite factors alone is not a valid prime factorization. Give (n, factors, all_factors_prime, factor_product, valid_factorization, certificate).","answer":{"kind":"expression","value":"(402, [2, 3, 67], Yes, 402, Yes, existence_requires_complete_prime_product)","explanation":"Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (402, [2, 3, 67], Yes, 402, Yes, existence_requires_complete_prime_product)."},"skills":["Prime Factors and a Complete Product Are Separate Validity Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0020","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4679a25f994fe10b11ba293bb3c438a79901ca14a46504fcfb0977078f7be8d","family_id":"qf-mod39-upf-factorization-audit","seed":202611086020,"semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,71],\"n\":426}"},"fields":{"title":"Prime Factors and a Complete Product Are Separate Validity Checks","stem":"Use positive integer n=426>1 and candidate factors=[2, 3, 71] in supplied order. Every factor must be prime and their complete exact product must equal n. A matching product of composite factors alone is not a valid prime factorization. Give (n, factors, all_factors_prime, factor_product, valid_factorization, certificate).","answer":{"kind":"expression","value":"(426, [2, 3, 71], Yes, 426, Yes, existence_requires_complete_prime_product)","explanation":"Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (426, [2, 3, 71], Yes, 426, Yes, existence_requires_complete_prime_product)."},"skills":["Prime Factors and a Complete Product Are Separate Validity Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0021","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5ded231cb7c7b433d28763d272ddeee1868869afa53c3d3d5db7aa1586ab14e","family_id":"qf-mod39-upf-factorization-audit","seed":202611086021,"semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,73],\"n\":438}"},"fields":{"title":"Prime Factors and a Complete Product Are Separate Validity Checks","stem":"Use positive integer n=438>1 and candidate factors=[2, 3, 73] in supplied order. Every factor must be prime and their complete exact product must equal n. A matching product of composite factors alone is not a valid prime factorization. Give (n, factors, all_factors_prime, factor_product, valid_factorization, certificate).","answer":{"kind":"expression","value":"(438, [2, 3, 73], Yes, 438, Yes, existence_requires_complete_prime_product)","explanation":"Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (438, [2, 3, 73], Yes, 438, Yes, existence_requires_complete_prime_product)."},"skills":["Prime Factors and a Complete Product Are Separate Validity Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0022","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41498ffafa1fab933461f7095a0b5badc70f1173c4bbc154c697b17ae0f91ac5","family_id":"qf-mod39-upf-factorization-audit","seed":202611086022,"semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,79],\"n\":474}"},"fields":{"title":"Prime Factors and a Complete Product Are Separate Validity Checks","stem":"Use positive integer n=474>1 and candidate factors=[2, 3, 79] in supplied order. Every factor must be prime and their complete exact product must equal n. A matching product of composite factors alone is not a valid prime factorization. Give (n, factors, all_factors_prime, factor_product, valid_factorization, certificate).","answer":{"kind":"expression","value":"(474, [2, 3, 79], Yes, 474, Yes, existence_requires_complete_prime_product)","explanation":"Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (474, [2, 3, 79], Yes, 474, Yes, existence_requires_complete_prime_product)."},"skills":["Prime Factors and a Complete Product Are Separate Validity Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0023","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"428bb313f28041bccdc055e28be8e9581d78bb87a720acb5f02e04a0b76499fb","family_id":"qf-mod39-upf-factorization-audit","seed":202611086023,"semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,83],\"n\":498}"},"fields":{"title":"Prime Factors and a Complete Product Are Separate Validity Checks","stem":"Use positive integer n=498>1 and candidate factors=[2, 3, 83] in supplied order. Every factor must be prime and their complete exact product must equal n. A matching product of composite factors alone is not a valid prime factorization. Give (n, factors, all_factors_prime, factor_product, valid_factorization, certificate).","answer":{"kind":"expression","value":"(498, [2, 3, 83], Yes, 498, Yes, existence_requires_complete_prime_product)","explanation":"Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (498, [2, 3, 83], Yes, 498, Yes, existence_requires_complete_prime_product)."},"skills":["Prime Factors and a Complete Product Are Separate Validity Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0024","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73b08554085254a5bf03a17c9dbf61e3d0af183adb2de6dd95b44a6b4486cc24","family_id":"qf-mod39-upf-factorization-audit","seed":202611086024,"semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,89],\"n\":534}"},"fields":{"title":"Prime Factors and a Complete Product Are Separate Validity Checks","stem":"Use positive integer n=534>1 and candidate factors=[2, 3, 89] in supplied order. Every factor must be prime and their complete exact product must equal n. A matching product of composite factors alone is not a valid prime factorization. Give (n, factors, all_factors_prime, factor_product, valid_factorization, certificate).","answer":{"kind":"expression","value":"(534, [2, 3, 89], Yes, 534, Yes, existence_requires_complete_prime_product)","explanation":"Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (534, [2, 3, 89], Yes, 534, Yes, existence_requires_complete_prime_product)."},"skills":["Prime Factors and a Complete Product Are Separate Validity Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0025","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dcc575a72d0b9259a039cccc3eb10346ccaf5f1a32ac9c8bd7280db2494526ff","family_id":"qf-mod39-upf-factorization-audit","seed":202611086025,"semantic_template":"qf-mod39-upf-factorization-audit/{\"factors\":[2,3,97],\"n\":582}"},"fields":{"title":"Prime Factors and a Complete Product Are Separate Validity Checks","stem":"Use positive integer n=582>1 and candidate factors=[2, 3, 97] in supplied order. Every factor must be prime and their complete exact product must equal n. A matching product of composite factors alone is not a valid prime factorization. Give (n, factors, all_factors_prime, factor_product, valid_factorization, certificate).","answer":{"kind":"expression","value":"(582, [2, 3, 97], Yes, 582, Yes, existence_requires_complete_prime_product)","explanation":"Retain original factor order, every-factor primality, the complete product and joint validity. A valid factorization supplies existence for this integer; uniqueness needs its separate theorem obligations. Unsafe product arithmetic is rejected before projection. The exact result, retaining the requested order, is (582, [2, 3, 97], Yes, 582, Yes, existence_requires_complete_prime_product)."},"skills":["Prime Factors and a Complete Product Are Separate Validity Checks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0026","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c9fbaeba74484acb4920e9e21ad3c591038504798e52f29c9cedd10c5c13134","family_id":"qf-mod39-upf-canonical-factorization","seed":202611086026,"semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":12}"},"fields":{"title":"Complete Trial Division Produces the Canonical Prime Multiset","stem":"Use positive integer n=12>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).","answer":{"kind":"expression","value":"(12, [2, 2, 3], [[2, 2], [3, 1]], 12, Yes, complete_trial_division_range)","explanation":"Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (12, [2, 2, 3], [[2, 2], [3, 1]], 12, Yes, complete_trial_division_range)."},"skills":["Complete Trial Division Produces the Canonical Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0027","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85e8b2273d14a74fe5a9600fd84200001b630eb6c4a6ec5f93809b1affb68f6b","family_id":"qf-mod39-upf-canonical-factorization","seed":202611086027,"semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":18}"},"fields":{"title":"Complete Trial Division Produces the Canonical Prime Multiset","stem":"Use positive integer n=18>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).","answer":{"kind":"expression","value":"(18, [2, 3, 3], [[2, 1], [3, 2]], 18, Yes, complete_trial_division_range)","explanation":"Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (18, [2, 3, 3], [[2, 1], [3, 2]], 18, Yes, complete_trial_division_range)."},"skills":["Complete Trial Division Produces the Canonical Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0028","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5435c2c4d40ed9a049cc5d62fbff7c3c2d00a6eb201e80629e9f925526a92bd0","family_id":"qf-mod39-upf-canonical-factorization","seed":202611086028,"semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":30}"},"fields":{"title":"Complete Trial Division Produces the Canonical Prime Multiset","stem":"Use positive integer n=30>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).","answer":{"kind":"expression","value":"(30, [2, 3, 5], [[2, 1], [3, 1], [5, 1]], 30, Yes, complete_trial_division_range)","explanation":"Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (30, [2, 3, 5], [[2, 1], [3, 1], [5, 1]], 30, Yes, complete_trial_division_range)."},"skills":["Complete Trial Division Produces the Canonical Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0029","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0fea0538b5862f192f48e666866ec5264a108bf1fb297455be4f554784d9f5ce","family_id":"qf-mod39-upf-canonical-factorization","seed":202611086029,"semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":42}"},"fields":{"title":"Complete Trial Division Produces the Canonical Prime Multiset","stem":"Use positive integer n=42>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).","answer":{"kind":"expression","value":"(42, [2, 3, 7], [[2, 1], [3, 1], [7, 1]], 42, Yes, complete_trial_division_range)","explanation":"Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (42, [2, 3, 7], [[2, 1], [3, 1], [7, 1]], 42, Yes, complete_trial_division_range)."},"skills":["Complete Trial Division Produces the Canonical Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0030","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c327b5d1b9e196d909a64b49200acae30f05de3f20be25fa4a2343fa7d927c62","family_id":"qf-mod39-upf-canonical-factorization","seed":202611086030,"semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":66}"},"fields":{"title":"Complete Trial Division Produces the Canonical Prime Multiset","stem":"Use positive integer n=66>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).","answer":{"kind":"expression","value":"(66, [2, 3, 11], [[2, 1], [3, 1], [11, 1]], 66, Yes, complete_trial_division_range)","explanation":"Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (66, [2, 3, 11], [[2, 1], [3, 1], [11, 1]], 66, Yes, complete_trial_division_range)."},"skills":["Complete Trial Division Produces the Canonical Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0031","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"259ef178854726b7e2ebc230020cc4aa3218582817f71a9649b5e8660a4599ea","family_id":"qf-mod39-upf-canonical-factorization","seed":202611086031,"semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":78}"},"fields":{"title":"Complete Trial Division Produces the Canonical Prime Multiset","stem":"Use positive integer n=78>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).","answer":{"kind":"expression","value":"(78, [2, 3, 13], [[2, 1], [3, 1], [13, 1]], 78, Yes, complete_trial_division_range)","explanation":"Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (78, [2, 3, 13], [[2, 1], [3, 1], [13, 1]], 78, Yes, complete_trial_division_range)."},"skills":["Complete Trial Division Produces the Canonical Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0032","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2aede629c06ada2c79a1b68fc99aa6f8be2d6024b869815c9f1c01130eb2728","family_id":"qf-mod39-upf-canonical-factorization","seed":202611086032,"semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":102}"},"fields":{"title":"Complete Trial Division Produces the Canonical Prime Multiset","stem":"Use positive integer n=102>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).","answer":{"kind":"expression","value":"(102, [2, 3, 17], [[2, 1], [3, 1], [17, 1]], 102, Yes, complete_trial_division_range)","explanation":"Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (102, [2, 3, 17], [[2, 1], [3, 1], [17, 1]], 102, Yes, complete_trial_division_range)."},"skills":["Complete Trial Division Produces the Canonical Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0033","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"009dcb75f4223170c0d7258ca28ade4368d288ef6c165ee1a6fdb0db1a45b6f3","family_id":"qf-mod39-upf-canonical-factorization","seed":202611086033,"semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":114}"},"fields":{"title":"Complete Trial Division Produces the Canonical Prime Multiset","stem":"Use positive integer n=114>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).","answer":{"kind":"expression","value":"(114, [2, 3, 19], [[2, 1], [3, 1], [19, 1]], 114, Yes, complete_trial_division_range)","explanation":"Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (114, [2, 3, 19], [[2, 1], [3, 1], [19, 1]], 114, Yes, complete_trial_division_range)."},"skills":["Complete Trial Division Produces the Canonical Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0034","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8d595e8f321c33b2dbe2de666e0d3cfece096a7b31b885c2a416c92296bd248","family_id":"qf-mod39-upf-canonical-factorization","seed":202611086034,"semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":138}"},"fields":{"title":"Complete Trial Division Produces the Canonical Prime Multiset","stem":"Use positive integer n=138>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).","answer":{"kind":"expression","value":"(138, [2, 3, 23], [[2, 1], [3, 1], [23, 1]], 138, Yes, complete_trial_division_range)","explanation":"Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (138, [2, 3, 23], [[2, 1], [3, 1], [23, 1]], 138, Yes, complete_trial_division_range)."},"skills":["Complete Trial Division Produces the Canonical Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0035","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7fc7d64f1ed16337c613f3572154b036c2a2ec721958ce1bdf8b9b66972988b4","family_id":"qf-mod39-upf-canonical-factorization","seed":202611086035,"semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":174}"},"fields":{"title":"Complete Trial Division Produces the Canonical Prime Multiset","stem":"Use positive integer n=174>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).","answer":{"kind":"expression","value":"(174, [2, 3, 29], [[2, 1], [3, 1], [29, 1]], 174, Yes, complete_trial_division_range)","explanation":"Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (174, [2, 3, 29], [[2, 1], [3, 1], [29, 1]], 174, Yes, complete_trial_division_range)."},"skills":["Complete Trial Division Produces the Canonical Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0036","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a4752642fe5aa7e985c74d45c9075fa8089fe94265c92dc50088403c045664e","family_id":"qf-mod39-upf-canonical-factorization","seed":202611086036,"semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":186}"},"fields":{"title":"Complete Trial Division Produces the Canonical Prime Multiset","stem":"Use positive integer n=186>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).","answer":{"kind":"expression","value":"(186, [2, 3, 31], [[2, 1], [3, 1], [31, 1]], 186, Yes, complete_trial_division_range)","explanation":"Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (186, [2, 3, 31], [[2, 1], [3, 1], [31, 1]], 186, Yes, complete_trial_division_range)."},"skills":["Complete Trial Division Produces the Canonical Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0037","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ad6a69118c100e47391b095925298375a51a0d0d75d3197dbccd17c232206c1","family_id":"qf-mod39-upf-canonical-factorization","seed":202611086037,"semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":222}"},"fields":{"title":"Complete Trial Division Produces the Canonical Prime Multiset","stem":"Use positive integer n=222>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).","answer":{"kind":"expression","value":"(222, [2, 3, 37], [[2, 1], [3, 1], [37, 1]], 222, Yes, complete_trial_division_range)","explanation":"Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (222, [2, 3, 37], [[2, 1], [3, 1], [37, 1]], 222, Yes, complete_trial_division_range)."},"skills":["Complete Trial Division Produces the Canonical Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0038","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89e0e3d35b823b0f36b3d492fde6d17941567c0d76de455c9bc2ebf5bafa6f11","family_id":"qf-mod39-upf-canonical-factorization","seed":202611086038,"semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":246}"},"fields":{"title":"Complete Trial Division Produces the Canonical Prime Multiset","stem":"Use positive integer n=246>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).","answer":{"kind":"expression","value":"(246, [2, 3, 41], [[2, 1], [3, 1], [41, 1]], 246, Yes, complete_trial_division_range)","explanation":"Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (246, [2, 3, 41], [[2, 1], [3, 1], [41, 1]], 246, Yes, complete_trial_division_range)."},"skills":["Complete Trial Division Produces the Canonical Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0039","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"afc983758a66d26b88c7183613abab1e8ed505daddd0256de15e7cc2e113000e","family_id":"qf-mod39-upf-canonical-factorization","seed":202611086039,"semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":258}"},"fields":{"title":"Complete Trial Division Produces the Canonical Prime Multiset","stem":"Use positive integer n=258>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).","answer":{"kind":"expression","value":"(258, [2, 3, 43], [[2, 1], [3, 1], [43, 1]], 258, Yes, complete_trial_division_range)","explanation":"Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (258, [2, 3, 43], [[2, 1], [3, 1], [43, 1]], 258, Yes, complete_trial_division_range)."},"skills":["Complete Trial Division Produces the Canonical Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0040","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"729918874a69de50075ec955bc416af18caa002e4c6b5896a323cf30de1b4e25","family_id":"qf-mod39-upf-canonical-factorization","seed":202611086040,"semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":282}"},"fields":{"title":"Complete Trial Division Produces the Canonical Prime Multiset","stem":"Use positive integer n=282>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).","answer":{"kind":"expression","value":"(282, [2, 3, 47], [[2, 1], [3, 1], [47, 1]], 282, Yes, complete_trial_division_range)","explanation":"Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (282, [2, 3, 47], [[2, 1], [3, 1], [47, 1]], 282, Yes, complete_trial_division_range)."},"skills":["Complete Trial Division Produces the Canonical Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0041","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16a861c2c74c116daef4a24e07d725f3f11b1404ccebb01e66a0786f532a9d09","family_id":"qf-mod39-upf-canonical-factorization","seed":202611086041,"semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":318}"},"fields":{"title":"Complete Trial Division Produces the Canonical Prime Multiset","stem":"Use positive integer n=318>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).","answer":{"kind":"expression","value":"(318, [2, 3, 53], [[2, 1], [3, 1], [53, 1]], 318, Yes, complete_trial_division_range)","explanation":"Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (318, [2, 3, 53], [[2, 1], [3, 1], [53, 1]], 318, Yes, complete_trial_division_range)."},"skills":["Complete Trial Division Produces the Canonical Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0042","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2faaea4aeba80a93882e9e71864968c1d801f0c56ac3a691e70ef274ddfdf3a8","family_id":"qf-mod39-upf-canonical-factorization","seed":202611086042,"semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":354}"},"fields":{"title":"Complete Trial Division Produces the Canonical Prime Multiset","stem":"Use positive integer n=354>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).","answer":{"kind":"expression","value":"(354, [2, 3, 59], [[2, 1], [3, 1], [59, 1]], 354, Yes, complete_trial_division_range)","explanation":"Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (354, [2, 3, 59], [[2, 1], [3, 1], [59, 1]], 354, Yes, complete_trial_division_range)."},"skills":["Complete Trial Division Produces the Canonical Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0043","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe67f907efacc3f1e524b7d2d6983f8a2a6a89ac3385393ccea19d03b27652cd","family_id":"qf-mod39-upf-canonical-factorization","seed":202611086043,"semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":366}"},"fields":{"title":"Complete Trial Division Produces the Canonical Prime Multiset","stem":"Use positive integer n=366>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).","answer":{"kind":"expression","value":"(366, [2, 3, 61], [[2, 1], [3, 1], [61, 1]], 366, Yes, complete_trial_division_range)","explanation":"Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (366, [2, 3, 61], [[2, 1], [3, 1], [61, 1]], 366, Yes, complete_trial_division_range)."},"skills":["Complete Trial Division Produces the Canonical Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0044","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b283c3facf5c6d175ce96a35510786bcc1be5bd59c605ce6f8e85cddb41deaa","family_id":"qf-mod39-upf-canonical-factorization","seed":202611086044,"semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":402}"},"fields":{"title":"Complete Trial Division Produces the Canonical Prime Multiset","stem":"Use positive integer n=402>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).","answer":{"kind":"expression","value":"(402, [2, 3, 67], [[2, 1], [3, 1], [67, 1]], 402, Yes, complete_trial_division_range)","explanation":"Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (402, [2, 3, 67], [[2, 1], [3, 1], [67, 1]], 402, Yes, complete_trial_division_range)."},"skills":["Complete Trial Division Produces the Canonical Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0045","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf49a9ff7c23d69f1226d3bb9c246936cdb2ceca0bac40bc14df32f99ab01c94","family_id":"qf-mod39-upf-canonical-factorization","seed":202611086045,"semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":426}"},"fields":{"title":"Complete Trial Division Produces the Canonical Prime Multiset","stem":"Use positive integer n=426>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).","answer":{"kind":"expression","value":"(426, [2, 3, 71], [[2, 1], [3, 1], [71, 1]], 426, Yes, complete_trial_division_range)","explanation":"Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (426, [2, 3, 71], [[2, 1], [3, 1], [71, 1]], 426, Yes, complete_trial_division_range)."},"skills":["Complete Trial Division Produces the Canonical Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0046","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7faa895ee3dccdbcafe26d6fd25c6feb6113461e8ad4db2f2a91a4b4caf5051","family_id":"qf-mod39-upf-canonical-factorization","seed":202611086046,"semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":438}"},"fields":{"title":"Complete Trial Division Produces the Canonical Prime Multiset","stem":"Use positive integer n=438>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).","answer":{"kind":"expression","value":"(438, [2, 3, 73], [[2, 1], [3, 1], [73, 1]], 438, Yes, complete_trial_division_range)","explanation":"Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (438, [2, 3, 73], [[2, 1], [3, 1], [73, 1]], 438, Yes, complete_trial_division_range)."},"skills":["Complete Trial Division Produces the Canonical Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0047","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8221dbb3670d22263f6b269e561f81342f7b7f5e7cc7ed59a35a5be9d2a3e219","family_id":"qf-mod39-upf-canonical-factorization","seed":202611086047,"semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":474}"},"fields":{"title":"Complete Trial Division Produces the Canonical Prime Multiset","stem":"Use positive integer n=474>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).","answer":{"kind":"expression","value":"(474, [2, 3, 79], [[2, 1], [3, 1], [79, 1]], 474, Yes, complete_trial_division_range)","explanation":"Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (474, [2, 3, 79], [[2, 1], [3, 1], [79, 1]], 474, Yes, complete_trial_division_range)."},"skills":["Complete Trial Division Produces the Canonical Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0048","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d607b8b25c896fa921d172c97074121d8a87497c152742a1be6b42776a1fc34","family_id":"qf-mod39-upf-canonical-factorization","seed":202611086048,"semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":498}"},"fields":{"title":"Complete Trial Division Produces the Canonical Prime Multiset","stem":"Use positive integer n=498>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).","answer":{"kind":"expression","value":"(498, [2, 3, 83], [[2, 1], [3, 1], [83, 1]], 498, Yes, complete_trial_division_range)","explanation":"Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (498, [2, 3, 83], [[2, 1], [3, 1], [83, 1]], 498, Yes, complete_trial_division_range)."},"skills":["Complete Trial Division Produces the Canonical Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0049","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"285e4552edbec13a4ad79163a276d59223db9af213c75a0f501578cce40725be","family_id":"qf-mod39-upf-canonical-factorization","seed":202611086049,"semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":534}"},"fields":{"title":"Complete Trial Division Produces the Canonical Prime Multiset","stem":"Use positive integer n=534>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).","answer":{"kind":"expression","value":"(534, [2, 3, 89], [[2, 1], [3, 1], [89, 1]], 534, Yes, complete_trial_division_range)","explanation":"Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (534, [2, 3, 89], [[2, 1], [3, 1], [89, 1]], 534, Yes, complete_trial_division_range)."},"skills":["Complete Trial Division Produces the Canonical Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0050","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23f47fd573781224ccdb5f62c6f8ffa01d1186c1da1e3b77c327b1725d167b7d","family_id":"qf-mod39-upf-canonical-factorization","seed":202611086050,"semantic_template":"qf-mod39-upf-canonical-factorization/{\"n\":582}"},"fields":{"title":"Complete Trial Division Produces the Canonical Prime Multiset","stem":"Use positive integer n=582>1. Give (n, prime_multiset, exponents, product, complete_factorization_certificate, certificate).","answer":{"kind":"expression","value":"(582, [2, 3, 97], [[2, 1], [3, 1], [97, 1]], 582, Yes, complete_trial_division_range)","explanation":"Divide by every candidate divisor through the square root of the remaining factor, retaining the final prime remainder. Report the increasingly ordered prime multiset, prime/exponent counts and exact full product. This complete bounded computation is not the general existence/uniqueness proof. The exact result, retaining the requested order, is (582, [2, 3, 97], [[2, 1], [3, 1], [97, 1]], 582, Yes, complete_trial_division_range)."},"skills":["Complete Trial Division Produces the Canonical Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0051","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cba4c812249b43fc1b9486c96a963561f171dbd8e1901e97ae397cc560d8cb1d","family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086051,"semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,2,3],\"factors_right\":[2,2,3],\"n\":12}"},"fields":{"title":"Only Two Valid Prime Products Can Supply This Uniqueness Certificate","stem":"Use positive integer n=12>1, candidate factors_left=[2, 3, 2] and factors_right=[2, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, [2, 2, 3], [2, 2, 3], [[2, Yes, 0, 2, Yes], [2, Yes, 0, 2, Yes], [3, Yes, 2, 3, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation":"Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 2, 3], [2, 2, 3], [[2, Yes, 0, 2, Yes], [2, Yes, 0, 2, Yes], [3, Yes, 2, 3, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)."},"skills":["Only Two Valid Prime Products Can Supply This Uniqueness Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0052","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3cbc8c0560663500f87da8348e91c0091bae85d51d3b5836cd73c2d8efaf5182","family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086052,"semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,3],\"factors_right\":[2,3,3],\"n\":18}"},"fields":{"title":"Only Two Valid Prime Products Can Supply This Uniqueness Certificate","stem":"Use positive integer n=18>1, candidate factors_left=[2, 3, 3] and factors_right=[3, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, [2, 3, 3], [2, 3, 3], [[2, Yes, 1, 2, Yes], [3, Yes, 0, 3, Yes], [3, Yes, 0, 3, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation":"Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 3], [2, 3, 3], [[2, Yes, 1, 2, Yes], [3, Yes, 0, 3, Yes], [3, Yes, 0, 3, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)."},"skills":["Only Two Valid Prime Products Can Supply This Uniqueness Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0053","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47f7a1be104d7ceeb7af46429d8c6aad9977b8106c8672fe6d3187dc1d834ef7","family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086053,"semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,5],\"factors_right\":[2,3,5],\"n\":30}"},"fields":{"title":"Only Two Valid Prime Products Can Supply This Uniqueness Certificate","stem":"Use positive integer n=30>1, candidate factors_left=[2, 3, 5] and factors_right=[5, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, [2, 3, 5], [2, 3, 5], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [5, Yes, 0, 5, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation":"Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 5], [2, 3, 5], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [5, Yes, 0, 5, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)."},"skills":["Only Two Valid Prime Products Can Supply This Uniqueness Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0054","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fecbdae9febef1de5b65226a6e902c8b08443fdbfa3d25e196b15b0af65645d1","family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086054,"semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,7],\"factors_right\":[2,3,7],\"n\":42}"},"fields":{"title":"Only Two Valid Prime Products Can Supply This Uniqueness Certificate","stem":"Use positive integer n=42>1, candidate factors_left=[2, 3, 7] and factors_right=[7, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, [2, 3, 7], [2, 3, 7], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [7, Yes, 0, 7, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation":"Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 7], [2, 3, 7], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [7, Yes, 0, 7, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)."},"skills":["Only Two Valid Prime Products Can Supply This Uniqueness Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0055","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3ca88b1e17d83aef69e37c6421f9290d21f90626f819f8e8cda21fa5af61abc","family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086055,"semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,11],\"factors_right\":[2,3,11],\"n\":66}"},"fields":{"title":"Only Two Valid Prime Products Can Supply This Uniqueness Certificate","stem":"Use positive integer n=66>1, candidate factors_left=[2, 3, 11] and factors_right=[11, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, [2, 3, 11], [2, 3, 11], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [11, Yes, 0, 11, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation":"Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 11], [2, 3, 11], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [11, Yes, 0, 11, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)."},"skills":["Only Two Valid Prime Products Can Supply This Uniqueness Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0056","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50248dc1c263aae4d64c6d88de9fc36ca9b5820a8d9c725fbc5dec4860690dce","family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086056,"semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,13],\"factors_right\":[2,3,13],\"n\":78}"},"fields":{"title":"Only Two Valid Prime Products Can Supply This Uniqueness Certificate","stem":"Use positive integer n=78>1, candidate factors_left=[2, 3, 13] and factors_right=[13, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, [2, 3, 13], [2, 3, 13], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [13, Yes, 0, 13, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation":"Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 13], [2, 3, 13], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [13, Yes, 0, 13, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)."},"skills":["Only Two Valid Prime Products Can Supply This Uniqueness Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0057","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6be26f285626956275df278a2988fb580e3e59eef9b9053007be8eee4c6fa00","family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086057,"semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,17],\"factors_right\":[2,3,17],\"n\":102}"},"fields":{"title":"Only Two Valid Prime Products Can Supply This Uniqueness Certificate","stem":"Use positive integer n=102>1, candidate factors_left=[2, 3, 17] and factors_right=[17, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, [2, 3, 17], [2, 3, 17], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [17, Yes, 0, 17, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation":"Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 17], [2, 3, 17], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [17, Yes, 0, 17, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)."},"skills":["Only Two Valid Prime Products Can Supply This Uniqueness Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0058","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad71598d290b24ffbf6e60e04df1f0d4004d356b7437ce6d19712b61a8685935","family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086058,"semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,19],\"factors_right\":[2,3,19],\"n\":114}"},"fields":{"title":"Only Two Valid Prime Products Can Supply This Uniqueness Certificate","stem":"Use positive integer n=114>1, candidate factors_left=[2, 3, 19] and factors_right=[19, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, [2, 3, 19], [2, 3, 19], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [19, Yes, 0, 19, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation":"Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 19], [2, 3, 19], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [19, Yes, 0, 19, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)."},"skills":["Only Two Valid Prime Products Can Supply This Uniqueness Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0059","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64a396591545b34660a6cb88a15959b406aea052bc13f4c27d06bd07d66903bd","family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086059,"semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,23],\"factors_right\":[2,3,23],\"n\":138}"},"fields":{"title":"Only Two Valid Prime Products Can Supply This Uniqueness Certificate","stem":"Use positive integer n=138>1, candidate factors_left=[2, 3, 23] and factors_right=[23, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, [2, 3, 23], [2, 3, 23], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [23, Yes, 0, 23, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation":"Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 23], [2, 3, 23], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [23, Yes, 0, 23, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)."},"skills":["Only Two Valid Prime Products Can Supply This Uniqueness Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0060","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75c96a6ea7f0a3ecb46ce4033d2204d257ff879740b4d1ca6323a0d5e6291594","family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086060,"semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,29],\"factors_right\":[2,3,29],\"n\":174}"},"fields":{"title":"Only Two Valid Prime Products Can Supply This Uniqueness Certificate","stem":"Use positive integer n=174>1, candidate factors_left=[2, 3, 29] and factors_right=[29, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, [2, 3, 29], [2, 3, 29], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [29, Yes, 0, 29, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation":"Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 29], [2, 3, 29], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [29, Yes, 0, 29, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)."},"skills":["Only Two Valid Prime Products Can Supply This Uniqueness Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0061","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f380740008c96062806f1e323e13288e09e4f5d6d24fa5341b2663bfe32294bf","family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086061,"semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,31],\"factors_right\":[2,3,31],\"n\":186}"},"fields":{"title":"Only Two Valid Prime Products Can Supply This Uniqueness Certificate","stem":"Use positive integer n=186>1, candidate factors_left=[2, 3, 31] and factors_right=[31, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, [2, 3, 31], [2, 3, 31], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [31, Yes, 0, 31, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation":"Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 31], [2, 3, 31], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [31, Yes, 0, 31, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)."},"skills":["Only Two Valid Prime Products Can Supply This Uniqueness Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0062","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa54c71910bcf7b5a8ef2f882c54f3c4539c468a03ae2c10bfc1ea88d81f0eb0","family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086062,"semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,37],\"factors_right\":[2,3,37],\"n\":222}"},"fields":{"title":"Only Two Valid Prime Products Can Supply This Uniqueness Certificate","stem":"Use positive integer n=222>1, candidate factors_left=[2, 3, 37] and factors_right=[37, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, [2, 3, 37], [2, 3, 37], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [37, Yes, 0, 37, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation":"Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 37], [2, 3, 37], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [37, Yes, 0, 37, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)."},"skills":["Only Two Valid Prime Products Can Supply This Uniqueness Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0063","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6cb9ec83244997bbe84d98faa15c0542c1938284971993114a4833969e847f30","family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086063,"semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,41],\"factors_right\":[2,3,41],\"n\":246}"},"fields":{"title":"Only Two Valid Prime Products Can Supply This Uniqueness Certificate","stem":"Use positive integer n=246>1, candidate factors_left=[2, 3, 41] and factors_right=[41, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, [2, 3, 41], [2, 3, 41], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [41, Yes, 0, 41, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation":"Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 41], [2, 3, 41], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [41, Yes, 0, 41, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)."},"skills":["Only Two Valid Prime Products Can Supply This Uniqueness Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0064","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47632a8e21da553592710e0841a34f54a66bd7da7382e4091cba24edff270a23","family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086064,"semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,43],\"factors_right\":[2,3,43],\"n\":258}"},"fields":{"title":"Only Two Valid Prime Products Can Supply This Uniqueness Certificate","stem":"Use positive integer n=258>1, candidate factors_left=[2, 3, 43] and factors_right=[43, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, [2, 3, 43], [2, 3, 43], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [43, Yes, 0, 43, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation":"Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 43], [2, 3, 43], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [43, Yes, 0, 43, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)."},"skills":["Only Two Valid Prime Products Can Supply This Uniqueness Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0065","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2939a397714a361ac397d9317a7042fed3020cc37df6e5522d7632e68f7e555","family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086065,"semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,47],\"factors_right\":[2,3,47],\"n\":282}"},"fields":{"title":"Only Two Valid Prime Products Can Supply This Uniqueness Certificate","stem":"Use positive integer n=282>1, candidate factors_left=[2, 3, 47] and factors_right=[47, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, [2, 3, 47], [2, 3, 47], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [47, Yes, 0, 47, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation":"Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 47], [2, 3, 47], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [47, Yes, 0, 47, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)."},"skills":["Only Two Valid Prime Products Can Supply This Uniqueness Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0066","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8fd9828a92b6480193c4610b6d9376369cec9875c1067aa1b6aaf504a1347159","family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086066,"semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,53],\"factors_right\":[2,3,53],\"n\":318}"},"fields":{"title":"Only Two Valid Prime Products Can Supply This Uniqueness Certificate","stem":"Use positive integer n=318>1, candidate factors_left=[2, 3, 53] and factors_right=[53, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, [2, 3, 53], [2, 3, 53], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [53, Yes, 0, 53, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation":"Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 53], [2, 3, 53], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [53, Yes, 0, 53, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)."},"skills":["Only Two Valid Prime Products Can Supply This Uniqueness Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0067","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3f414f082a6ea7791ce5264dad9990fe1a24ba1f2e1c251115a4116c0e37923","family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086067,"semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,59],\"factors_right\":[2,3,59],\"n\":354}"},"fields":{"title":"Only Two Valid Prime Products Can Supply This Uniqueness Certificate","stem":"Use positive integer n=354>1, candidate factors_left=[2, 3, 59] and factors_right=[59, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, [2, 3, 59], [2, 3, 59], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [59, Yes, 0, 59, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation":"Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 59], [2, 3, 59], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [59, Yes, 0, 59, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)."},"skills":["Only Two Valid Prime Products Can Supply This Uniqueness Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0068","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7220ac9798b146e989079d15d107dcf50569234b56ae8f337b73e881acdb799a","family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086068,"semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,61],\"factors_right\":[2,3,61],\"n\":366}"},"fields":{"title":"Only Two Valid Prime Products Can Supply This Uniqueness Certificate","stem":"Use positive integer n=366>1, candidate factors_left=[2, 3, 61] and factors_right=[61, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, [2, 3, 61], [2, 3, 61], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [61, Yes, 0, 61, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation":"Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 61], [2, 3, 61], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [61, Yes, 0, 61, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)."},"skills":["Only Two Valid Prime Products Can Supply This Uniqueness Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0069","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36e288a30131666cdec8d5f24b65d5e6a268421e2f3cae137070110088548dd6","family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086069,"semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,67],\"factors_right\":[2,3,67],\"n\":402}"},"fields":{"title":"Only Two Valid Prime Products Can Supply This Uniqueness Certificate","stem":"Use positive integer n=402>1, candidate factors_left=[2, 3, 67] and factors_right=[67, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, [2, 3, 67], [2, 3, 67], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [67, Yes, 0, 67, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation":"Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 67], [2, 3, 67], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [67, Yes, 0, 67, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)."},"skills":["Only Two Valid Prime Products Can Supply This Uniqueness Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0070","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12bfa0593ecfabd0817f3546d267fe0e5a8b9cd22499715765a61acf0871bc2e","family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086070,"semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,71],\"factors_right\":[2,3,71],\"n\":426}"},"fields":{"title":"Only Two Valid Prime Products Can Supply This Uniqueness Certificate","stem":"Use positive integer n=426>1, candidate factors_left=[2, 3, 71] and factors_right=[71, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, [2, 3, 71], [2, 3, 71], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [71, Yes, 0, 71, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation":"Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 71], [2, 3, 71], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [71, Yes, 0, 71, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)."},"skills":["Only Two Valid Prime Products Can Supply This Uniqueness Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0071","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79fdedcf45fef7838c9d9df81506304c88aae0c780411d098a3695a6951cde06","family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086071,"semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,73],\"factors_right\":[2,3,73],\"n\":438}"},"fields":{"title":"Only Two Valid Prime Products Can Supply This Uniqueness Certificate","stem":"Use positive integer n=438>1, candidate factors_left=[2, 3, 73] and factors_right=[73, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, [2, 3, 73], [2, 3, 73], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [73, Yes, 0, 73, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation":"Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 73], [2, 3, 73], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [73, Yes, 0, 73, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)."},"skills":["Only Two Valid Prime Products Can Supply This Uniqueness Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0072","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea893d1419eeaebd7f9204916e251d8d81df3e72ee8e4989bf24c59ce4a8962a","family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086072,"semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,79],\"factors_right\":[2,3,79],\"n\":474}"},"fields":{"title":"Only Two Valid Prime Products Can Supply This Uniqueness Certificate","stem":"Use positive integer n=474>1, candidate factors_left=[2, 3, 79] and factors_right=[79, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, [2, 3, 79], [2, 3, 79], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [79, Yes, 0, 79, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation":"Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 79], [2, 3, 79], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [79, Yes, 0, 79, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)."},"skills":["Only Two Valid Prime Products Can Supply This Uniqueness Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0073","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4696581b47aa9a4c80bcc6efb8452bd5fbcd8b4b688eeb268a03029e993b7350","family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086073,"semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,83],\"factors_right\":[2,3,83],\"n\":498}"},"fields":{"title":"Only Two Valid Prime Products Can Supply This Uniqueness Certificate","stem":"Use positive integer n=498>1, candidate factors_left=[2, 3, 83] and factors_right=[83, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, [2, 3, 83], [2, 3, 83], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [83, Yes, 0, 83, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation":"Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 83], [2, 3, 83], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [83, Yes, 0, 83, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)."},"skills":["Only Two Valid Prime Products Can Supply This Uniqueness Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0074","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1399d56965b197dae1850662e40b0f0d44f82ad552d41dbe765fbe1aea78cae","family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086074,"semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,89],\"factors_right\":[2,3,89],\"n\":534}"},"fields":{"title":"Only Two Valid Prime Products Can Supply This Uniqueness Certificate","stem":"Use positive integer n=534>1, candidate factors_left=[2, 3, 89] and factors_right=[89, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, [2, 3, 89], [2, 3, 89], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [89, Yes, 0, 89, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation":"Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 89], [2, 3, 89], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [89, Yes, 0, 89, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)."},"skills":["Only Two Valid Prime Products Can Supply This Uniqueness Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0075","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"326f83568c6a59be4c2f957d106e72b3083ccfe07b54b429e16a1b49e93d6a89","family_id":"qf-mod39-upf-uniqueness-audit","seed":202611086075,"semantic_template":"qf-mod39-upf-uniqueness-audit/{\"factors_left\":[2,3,97],\"factors_right\":[2,3,97],\"n\":582}"},"fields":{"title":"Only Two Valid Prime Products Can Supply This Uniqueness Certificate","stem":"Use positive integer n=582>1, candidate factors_left=[2, 3, 97] and factors_right=[97, 2, 3], retaining both original orders. Give (left_valid, right_valid, sorted_left, sorted_right, divisibility_rows, uniqueness_certificate, certificate).","answer":{"kind":"expression","value":"(Yes, Yes, [2, 3, 97], [2, 3, 97], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [97, Yes, 0, 97, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)","explanation":"Validate primality and complete product on both sides before the joint uniqueness flag. Sort the prime multisets to compare multiplicities. For each sorted left factor retain the source first right-input factor divisible by it and its index; this local witness may be reused for repeated primes and is not an occurrence-by-occurrence matching. Invalid composite-factor rows do not assert Euclid lemma for composites. The exact result, retaining the requested order, is (Yes, Yes, [2, 3, 97], [2, 3, 97], [[2, Yes, 1, 2, Yes], [3, Yes, 2, 3, Yes], [97, Yes, 0, 97, Yes]], Yes, Euclid_lemma_witness_p_divides_product_implies_p_divides_a_prime_factor)."},"skills":["Only Two Valid Prime Products Can Supply This Uniqueness Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0076","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ee4e3cd6cb7ccc26d2df09f4e439335800d96fa461b96fc02d0d00b013d74bc","family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086076,"semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":0}"},"fields":{"title":"Separate Zero, Sign Units and the Positive Prime Multiset","stem":"Use original bounded integer n=0. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).","answer":{"kind":"expression","value":"(0, No, null, [], zero_has_no_prime_factorization)","explanation":"Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (0, No, null, [], zero_has_no_prime_factorization)."},"skills":["Separate Zero, Sign Units and the Positive Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0077","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"de45f352e5e8e4d208206ca8b754cf767a988be9cfe36761f5e8572c233968ec","family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086077,"semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":1}"},"fields":{"title":"Separate Zero, Sign Units and the Positive Prime Multiset","stem":"Use original bounded integer n=1. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).","answer":{"kind":"expression","value":"(1, No, 1, [], units_have_empty_prime_multiset)","explanation":"Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (1, No, 1, [], units_have_empty_prime_multiset)."},"skills":["Separate Zero, Sign Units and the Positive Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0078","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0f1ded881d68138a07295cfbcb15d1f531d9bc70452316f00f69e6105b00662","family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086078,"semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-1}"},"fields":{"title":"Separate Zero, Sign Units and the Positive Prime Multiset","stem":"Use original bounded integer n=-1. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).","answer":{"kind":"expression","value":"(-1, No, -1, [], units_have_empty_prime_multiset)","explanation":"Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-1, No, -1, [], units_have_empty_prime_multiset)."},"skills":["Separate Zero, Sign Units and the Positive Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0079","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c94715d8861f37909fc0838d4b7c5d2b2c514e86cc75db12c4a000c712e213a1","family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086079,"semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-42}"},"fields":{"title":"Separate Zero, Sign Units and the Positive Prime Multiset","stem":"Use original bounded integer n=-42. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).","answer":{"kind":"expression","value":"(-42, No, -1, [2, 3, 7], integer_sign_separated_from_positive_prime_multiset)","explanation":"Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-42, No, -1, [2, 3, 7], integer_sign_separated_from_positive_prime_multiset)."},"skills":["Separate Zero, Sign Units and the Positive Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0080","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0644d748d6e355befced1cba8c917311eb5dacc26bfd48b6ab19d7c172517ce","family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086080,"semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-66}"},"fields":{"title":"Separate Zero, Sign Units and the Positive Prime Multiset","stem":"Use original bounded integer n=-66. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).","answer":{"kind":"expression","value":"(-66, No, -1, [2, 3, 11], integer_sign_separated_from_positive_prime_multiset)","explanation":"Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-66, No, -1, [2, 3, 11], integer_sign_separated_from_positive_prime_multiset)."},"skills":["Separate Zero, Sign Units and the Positive Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0081","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1df0bf110ae4357595bd63570ff3ae60ac4f6049e9e33048c74fe94a29865cca","family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086081,"semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-78}"},"fields":{"title":"Separate Zero, Sign Units and the Positive Prime Multiset","stem":"Use original bounded integer n=-78. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).","answer":{"kind":"expression","value":"(-78, No, -1, [2, 3, 13], integer_sign_separated_from_positive_prime_multiset)","explanation":"Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-78, No, -1, [2, 3, 13], integer_sign_separated_from_positive_prime_multiset)."},"skills":["Separate Zero, Sign Units and the Positive Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0082","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6a68ba70f4ea26704b6fad4b4d3cd3860506439c8bc574bd615b6f6d2af601e","family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086082,"semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-102}"},"fields":{"title":"Separate Zero, Sign Units and the Positive Prime Multiset","stem":"Use original bounded integer n=-102. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).","answer":{"kind":"expression","value":"(-102, No, -1, [2, 3, 17], integer_sign_separated_from_positive_prime_multiset)","explanation":"Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-102, No, -1, [2, 3, 17], integer_sign_separated_from_positive_prime_multiset)."},"skills":["Separate Zero, Sign Units and the Positive Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0083","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf1e704e44a9a0a26d82bfc76f2ac89cd9d8e33138c2dde0cd42c5a1e74c2524","family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086083,"semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-114}"},"fields":{"title":"Separate Zero, Sign Units and the Positive Prime Multiset","stem":"Use original bounded integer n=-114. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).","answer":{"kind":"expression","value":"(-114, No, -1, [2, 3, 19], integer_sign_separated_from_positive_prime_multiset)","explanation":"Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-114, No, -1, [2, 3, 19], integer_sign_separated_from_positive_prime_multiset)."},"skills":["Separate Zero, Sign Units and the Positive Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0084","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e46fa36f2956b64f238154d89140ee9ca2456043269fac160b86ac864d961920","family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086084,"semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-138}"},"fields":{"title":"Separate Zero, Sign Units and the Positive Prime Multiset","stem":"Use original bounded integer n=-138. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).","answer":{"kind":"expression","value":"(-138, No, -1, [2, 3, 23], integer_sign_separated_from_positive_prime_multiset)","explanation":"Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-138, No, -1, [2, 3, 23], integer_sign_separated_from_positive_prime_multiset)."},"skills":["Separate Zero, Sign Units and the Positive Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0085","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"590a485e3ac271ffa14dce84ddef25caae0c4e212f75fb9ecdd189c1d4e822f9","family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086085,"semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-174}"},"fields":{"title":"Separate Zero, Sign Units and the Positive Prime Multiset","stem":"Use original bounded integer n=-174. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).","answer":{"kind":"expression","value":"(-174, No, -1, [2, 3, 29], integer_sign_separated_from_positive_prime_multiset)","explanation":"Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-174, No, -1, [2, 3, 29], integer_sign_separated_from_positive_prime_multiset)."},"skills":["Separate Zero, Sign Units and the Positive Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0086","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d056f4da6e20529acd997314bab183aefbee76c1ab556b33826f3e1e6c94f7d","family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086086,"semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-186}"},"fields":{"title":"Separate Zero, Sign Units and the Positive Prime Multiset","stem":"Use original bounded integer n=-186. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).","answer":{"kind":"expression","value":"(-186, No, -1, [2, 3, 31], integer_sign_separated_from_positive_prime_multiset)","explanation":"Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-186, No, -1, [2, 3, 31], integer_sign_separated_from_positive_prime_multiset)."},"skills":["Separate Zero, Sign Units and the Positive Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0087","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e315372ecc1adf8584504462145e9a2d609fe7df4fa9dd284f67a2be1a659afb","family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086087,"semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-222}"},"fields":{"title":"Separate Zero, Sign Units and the Positive Prime Multiset","stem":"Use original bounded integer n=-222. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).","answer":{"kind":"expression","value":"(-222, No, -1, [2, 3, 37], integer_sign_separated_from_positive_prime_multiset)","explanation":"Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-222, No, -1, [2, 3, 37], integer_sign_separated_from_positive_prime_multiset)."},"skills":["Separate Zero, Sign Units and the Positive Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0088","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49c90b7cf3ec5eb97483feb3d2df21334337a2f0cf0a97fe10b690d2702d3f3e","family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086088,"semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-246}"},"fields":{"title":"Separate Zero, Sign Units and the Positive Prime Multiset","stem":"Use original bounded integer n=-246. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).","answer":{"kind":"expression","value":"(-246, No, -1, [2, 3, 41], integer_sign_separated_from_positive_prime_multiset)","explanation":"Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-246, No, -1, [2, 3, 41], integer_sign_separated_from_positive_prime_multiset)."},"skills":["Separate Zero, Sign Units and the Positive Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0089","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc5047242ee86bf353665c37d07bc999a482612cecf5fbb9c6d652b0740a7f83","family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086089,"semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-258}"},"fields":{"title":"Separate Zero, Sign Units and the Positive Prime Multiset","stem":"Use original bounded integer n=-258. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).","answer":{"kind":"expression","value":"(-258, No, -1, [2, 3, 43], integer_sign_separated_from_positive_prime_multiset)","explanation":"Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-258, No, -1, [2, 3, 43], integer_sign_separated_from_positive_prime_multiset)."},"skills":["Separate Zero, Sign Units and the Positive Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0090","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5d33c2b7e4f2eff65d5321571485e6468ae38eef806986eb60ff3c1fee53a7d","family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086090,"semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-282}"},"fields":{"title":"Separate Zero, Sign Units and the Positive Prime Multiset","stem":"Use original bounded integer n=-282. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).","answer":{"kind":"expression","value":"(-282, No, -1, [2, 3, 47], integer_sign_separated_from_positive_prime_multiset)","explanation":"Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-282, No, -1, [2, 3, 47], integer_sign_separated_from_positive_prime_multiset)."},"skills":["Separate Zero, Sign Units and the Positive Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0091","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ba7a5188175cbbf25f6db7dfd9f9330f6c50d439fd643d4cd5681dd690c69cc","family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086091,"semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-318}"},"fields":{"title":"Separate Zero, Sign Units and the Positive Prime Multiset","stem":"Use original bounded integer n=-318. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).","answer":{"kind":"expression","value":"(-318, No, -1, [2, 3, 53], integer_sign_separated_from_positive_prime_multiset)","explanation":"Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-318, No, -1, [2, 3, 53], integer_sign_separated_from_positive_prime_multiset)."},"skills":["Separate Zero, Sign Units and the Positive Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0092","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c278317383aeb773d2e6c524f212efea8139022cd99ce429a1b4260770d1b10","family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086092,"semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-354}"},"fields":{"title":"Separate Zero, Sign Units and the Positive Prime Multiset","stem":"Use original bounded integer n=-354. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).","answer":{"kind":"expression","value":"(-354, No, -1, [2, 3, 59], integer_sign_separated_from_positive_prime_multiset)","explanation":"Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-354, No, -1, [2, 3, 59], integer_sign_separated_from_positive_prime_multiset)."},"skills":["Separate Zero, Sign Units and the Positive Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0093","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b60f3c567f058c55bb54ba38702853ebb0fed88cb8d881cf31a367d9afc2ca6","family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086093,"semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-366}"},"fields":{"title":"Separate Zero, Sign Units and the Positive Prime Multiset","stem":"Use original bounded integer n=-366. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).","answer":{"kind":"expression","value":"(-366, No, -1, [2, 3, 61], integer_sign_separated_from_positive_prime_multiset)","explanation":"Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-366, No, -1, [2, 3, 61], integer_sign_separated_from_positive_prime_multiset)."},"skills":["Separate Zero, Sign Units and the Positive Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0094","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"167a3b352f7b257d796f5d922a49cc4a2233100fb974f01f1c6bbe4507a4fb3f","family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086094,"semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-402}"},"fields":{"title":"Separate Zero, Sign Units and the Positive Prime Multiset","stem":"Use original bounded integer n=-402. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).","answer":{"kind":"expression","value":"(-402, No, -1, [2, 3, 67], integer_sign_separated_from_positive_prime_multiset)","explanation":"Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-402, No, -1, [2, 3, 67], integer_sign_separated_from_positive_prime_multiset)."},"skills":["Separate Zero, Sign Units and the Positive Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0095","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a19b3273570dccd5bf21ec66f94cf8d81232b502babb1c020beb499245043bc","family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086095,"semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-426}"},"fields":{"title":"Separate Zero, Sign Units and the Positive Prime Multiset","stem":"Use original bounded integer n=-426. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).","answer":{"kind":"expression","value":"(-426, No, -1, [2, 3, 71], integer_sign_separated_from_positive_prime_multiset)","explanation":"Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-426, No, -1, [2, 3, 71], integer_sign_separated_from_positive_prime_multiset)."},"skills":["Separate Zero, Sign Units and the Positive Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0096","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb99be7f0c7dd81ecbcaea811767c44c65e5db3fc0fbd3515e7ff7213c490c49","family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086096,"semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-438}"},"fields":{"title":"Separate Zero, Sign Units and the Positive Prime Multiset","stem":"Use original bounded integer n=-438. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).","answer":{"kind":"expression","value":"(-438, No, -1, [2, 3, 73], integer_sign_separated_from_positive_prime_multiset)","explanation":"Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-438, No, -1, [2, 3, 73], integer_sign_separated_from_positive_prime_multiset)."},"skills":["Separate Zero, Sign Units and the Positive Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0097","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1e0550f17b600b0ce0e7a5ae57f3b1dad90303b3ee3e3faf09715f9ca33573d","family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086097,"semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-474}"},"fields":{"title":"Separate Zero, Sign Units and the Positive Prime Multiset","stem":"Use original bounded integer n=-474. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).","answer":{"kind":"expression","value":"(-474, No, -1, [2, 3, 79], integer_sign_separated_from_positive_prime_multiset)","explanation":"Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-474, No, -1, [2, 3, 79], integer_sign_separated_from_positive_prime_multiset)."},"skills":["Separate Zero, Sign Units and the Positive Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0098","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7c63218ed641c8f400d0180afaa2802eabc16e226a90a82aca2936c861f55c46","family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086098,"semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-498}"},"fields":{"title":"Separate Zero, Sign Units and the Positive Prime Multiset","stem":"Use original bounded integer n=-498. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).","answer":{"kind":"expression","value":"(-498, No, -1, [2, 3, 83], integer_sign_separated_from_positive_prime_multiset)","explanation":"Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-498, No, -1, [2, 3, 83], integer_sign_separated_from_positive_prime_multiset)."},"skills":["Separate Zero, Sign Units and the Positive Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0099","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c929cdb0c2247b15241b83616fc78e520915349acd662c328f261395a61a2581","family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086099,"semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-534}"},"fields":{"title":"Separate Zero, Sign Units and the Positive Prime Multiset","stem":"Use original bounded integer n=-534. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).","answer":{"kind":"expression","value":"(-534, No, -1, [2, 3, 89], integer_sign_separated_from_positive_prime_multiset)","explanation":"Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-534, No, -1, [2, 3, 89], integer_sign_separated_from_positive_prime_multiset)."},"skills":["Separate Zero, Sign Units and the Positive Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r06-unique-prime-factorization-0100","batch_id":"batch-20261014-advance-20261108-r06-unique-prime-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e969103c1791d534b36b71e4b50da1211b67f47216c39c9a160d846ead48b33d","family_id":"qf-mod39-upf-unit-sign-boundary","seed":202611086100,"semantic_template":"qf-mod39-upf-unit-sign-boundary/{\"n\":-582}"},"fields":{"title":"Separate Zero, Sign Units and the Positive Prime Multiset","stem":"Use original bounded integer n=-582. Give (n, ordinary_positive_form_claim, sign_unit, absolute_prime_multiset, boundary).","answer":{"kind":"expression","value":"(-582, No, -1, [2, 3, 97], integer_sign_separated_from_positive_prime_multiset)","explanation":"Retain the fixed No nonclaim for extending the ordinary positive form without these distinctions. It is not a positivity/compositeness test. Zero has null sign and no prime factorization; units±1 have an empty prime multiset; otherwise separate sign±1 from the prime factorization of |n|, including sign+1 for a positive input. The exact result, retaining the requested order, is (-582, No, -1, [2, 3, 97], integer_sign_separated_from_positive_prime_multiset)."},"skills":["Separate Zero, Sign Units and the Positive Prime Multiset"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}