{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0001","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d922c6829ebe42cda72ee76b62c3d080c15d70a09dffeba8b78054b19c1951b","family_id":"qf-mod43-legendre-classification-table","seed":202611126001,"semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":3}"},"fields":{"title":"Legendre Classes Include Zero before Nonzero Residue Classification","stem":"Use bounded odd prime p=3 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).","answer":{"kind":"expression","value":"(3, [[0, 0], [1, 1], [2, -1]], Yes, zero_residue_nonresidue_partition_is_complete)","explanation":"Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (3, [[0, 0], [1, 1], [2, -1]], Yes, zero_residue_nonresidue_partition_is_complete)."},"skills":["Legendre Classes Include Zero before Nonzero Residue Classification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0002","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3044dc6240a9c131bb17352027b354db24e13e15214e2b98597b0b5e44761a8b","family_id":"qf-mod43-legendre-classification-table","seed":202611126002,"semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":5}"},"fields":{"title":"Legendre Classes Include Zero before Nonzero Residue Classification","stem":"Use bounded odd prime p=5 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).","answer":{"kind":"expression","value":"(5, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1]], Yes, zero_residue_nonresidue_partition_is_complete)","explanation":"Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (5, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1]], Yes, zero_residue_nonresidue_partition_is_complete)."},"skills":["Legendre Classes Include Zero before Nonzero Residue Classification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0003","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4d056a0ddd2007edb2f8e536aaaf756c78ea5a4c081a82ca91c439944faa1f6","family_id":"qf-mod43-legendre-classification-table","seed":202611126003,"semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":7}"},"fields":{"title":"Legendre Classes Include Zero before Nonzero Residue Classification","stem":"Use bounded odd prime p=7 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).","answer":{"kind":"expression","value":"(7, [[0, 0], [1, 1], [2, 1], [3, -1], [4, 1], [5, -1], [6, -1]], Yes, zero_residue_nonresidue_partition_is_complete)","explanation":"Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (7, [[0, 0], [1, 1], [2, 1], [3, -1], [4, 1], [5, -1], [6, -1]], Yes, zero_residue_nonresidue_partition_is_complete)."},"skills":["Legendre Classes Include Zero before Nonzero Residue Classification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0004","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1e2205f92123a60565efbd5f94b4b81a4628baba70ee2197555ea2394899a76","family_id":"qf-mod43-legendre-classification-table","seed":202611126004,"semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":11}"},"fields":{"title":"Legendre Classes Include Zero before Nonzero Residue Classification","stem":"Use bounded odd prime p=11 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).","answer":{"kind":"expression","value":"(11, [[0, 0], [1, 1], [2, -1], [3, 1], [4, 1], [5, 1], [6, -1], [7, -1], [8, -1], [9, 1], [10, -1]], Yes, zero_residue_nonresidue_partition_is_complete)","explanation":"Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (11, [[0, 0], [1, 1], [2, -1], [3, 1], [4, 1], [5, 1], [6, -1], [7, -1], [8, -1], [9, 1], [10, -1]], Yes, zero_residue_nonresidue_partition_is_complete)."},"skills":["Legendre Classes Include Zero before Nonzero Residue Classification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0005","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"647986f599b3703cab0945f140e5b25dff3fa48f4046622ab956fa6dd6605e84","family_id":"qf-mod43-legendre-classification-table","seed":202611126005,"semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":13}"},"fields":{"title":"Legendre Classes Include Zero before Nonzero Residue Classification","stem":"Use bounded odd prime p=13 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).","answer":{"kind":"expression","value":"(13, [[0, 0], [1, 1], [2, -1], [3, 1], [4, 1], [5, -1], [6, -1], [7, -1], [8, -1], [9, 1], [10, 1], [11, -1], [12, 1]], Yes, zero_residue_nonresidue_partition_is_complete)","explanation":"Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (13, [[0, 0], [1, 1], [2, -1], [3, 1], [4, 1], [5, -1], [6, -1], [7, -1], [8, -1], [9, 1], [10, 1], [11, -1], [12, 1]], Yes, zero_residue_nonresidue_partition_is_complete)."},"skills":["Legendre Classes Include Zero before Nonzero Residue Classification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0006","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e784f65f5878a8b35b44a5a5aa578bf6e5c186affd452cc6909d7de652e9e45","family_id":"qf-mod43-legendre-classification-table","seed":202611126006,"semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":17}"},"fields":{"title":"Legendre Classes Include Zero before Nonzero Residue Classification","stem":"Use bounded odd prime p=17 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).","answer":{"kind":"expression","value":"(17, [[0, 0], [1, 1], [2, 1], [3, -1], [4, 1], [5, -1], [6, -1], [7, -1], [8, 1], [9, 1], [10, -1], [11, -1], [12, -1], [13, 1], [14, -1], [15, 1], [16, 1]], Yes, zero_residue_nonresidue_partition_is_complete)","explanation":"Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (17, [[0, 0], [1, 1], [2, 1], [3, -1], [4, 1], [5, -1], [6, -1], [7, -1], [8, 1], [9, 1], [10, -1], [11, -1], [12, -1], [13, 1], [14, -1], [15, 1], [16, 1]], Yes, zero_residue_nonresidue_partition_is_complete)."},"skills":["Legendre Classes Include Zero before Nonzero Residue Classification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0007","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8442f7b4cb82a93c40811d3ea0f00e8caa198e1ba43dc445b775d31ae477098","family_id":"qf-mod43-legendre-classification-table","seed":202611126007,"semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":19}"},"fields":{"title":"Legendre Classes Include Zero before Nonzero Residue Classification","stem":"Use bounded odd prime p=19 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).","answer":{"kind":"expression","value":"(19, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1], [5, 1], [6, 1], [7, 1], [8, -1], [9, 1], [10, -1], [11, 1], [12, -1], [13, -1], [14, -1], [15, -1], [16, 1], [17, 1], [18, -1]], Yes, zero_residue_nonresidue_partition_is_complete)","explanation":"Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (19, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1], [5, 1], [6, 1], [7, 1], [8, -1], [9, 1], [10, -1], [11, 1], [12, -1], [13, -1], [14, -1], [15, -1], [16, 1], [17, 1], [18, -1]], Yes, zero_residue_nonresidue_partition_is_complete)."},"skills":["Legendre Classes Include Zero before Nonzero Residue Classification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0008","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"868be48bfdf841f6b3c59980c37c8c01e54b55ed3b6e007f25533def9173f804","family_id":"qf-mod43-legendre-classification-table","seed":202611126008,"semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":23}"},"fields":{"title":"Legendre Classes Include Zero before Nonzero Residue Classification","stem":"Use bounded odd prime p=23 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).","answer":{"kind":"expression","value":"(23, [[0, 0], [1, 1], [2, 1], [3, 1], [4, 1], [5, -1], [6, 1], [7, -1], [8, 1], [9, 1], [10, -1], [11, -1], [12, 1], [13, 1], [14, -1], [15, -1], [16, 1], [17, -1], [18, 1], [19, -1], [20, -1], [21, -1], [22, -1]], Yes, zero_residue_nonresidue_partition_is_complete)","explanation":"Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (23, [[0, 0], [1, 1], [2, 1], [3, 1], [4, 1], [5, -1], [6, 1], [7, -1], [8, 1], [9, 1], [10, -1], [11, -1], [12, 1], [13, 1], [14, -1], [15, -1], [16, 1], [17, -1], [18, 1], [19, -1], [20, -1], [21, -1], [22, -1]], Yes, zero_residue_nonresidue_partition_is_complete)."},"skills":["Legendre Classes Include Zero before Nonzero Residue Classification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0009","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20e3abd041b3da33904af9412135ff3674a275181ed1f5663288dd55656e6ca9","family_id":"qf-mod43-legendre-classification-table","seed":202611126009,"semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":29}"},"fields":{"title":"Legendre Classes Include Zero before Nonzero Residue Classification","stem":"Use bounded odd prime p=29 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).","answer":{"kind":"expression","value":"(29, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1], [5, 1], [6, 1], [7, 1], [8, -1], [9, 1], [10, -1], [11, -1], [12, -1], [13, 1], [14, -1], [15, -1], [16, 1], [17, -1], [18, -1], [19, -1], [20, 1], [21, -1], [22, 1], [23, 1], [24, 1], [25, 1], [26, -1], [27, -1], [28, 1]], Yes, zero_residue_nonresidue_partition_is_complete)","explanation":"Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (29, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1], [5, 1], [6, 1], [7, 1], [8, -1], [9, 1], [10, -1], [11, -1], [12, -1], [13, 1], [14, -1], [15, -1], [16, 1], [17, -1], [18, -1], [19, -1], [20, 1], [21, -1], [22, 1], [23, 1], [24, 1], [25, 1], [26, -1], [27, -1], [28, 1]], Yes, zero_residue_nonresidue_partition_is_complete)."},"skills":["Legendre Classes Include Zero before Nonzero Residue Classification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0010","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18c3b2596d8db024363383ae45079be26c5d80e25e4101605e572425cb6cc20e","family_id":"qf-mod43-legendre-classification-table","seed":202611126010,"semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":31}"},"fields":{"title":"Legendre Classes Include Zero before Nonzero Residue Classification","stem":"Use bounded odd prime p=31 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).","answer":{"kind":"expression","value":"(31, [[0, 0], [1, 1], [2, 1], [3, -1], [4, 1], [5, 1], [6, -1], [7, 1], [8, 1], [9, 1], [10, 1], [11, -1], [12, -1], [13, -1], [14, 1], [15, -1], [16, 1], [17, -1], [18, 1], [19, 1], [20, 1], [21, -1], [22, -1], [23, -1], [24, -1], [25, 1], [26, -1], [27, -1], [28, 1], [29, -1], [30, -1]], Yes, zero_residue_nonresidue_partition_is_complete)","explanation":"Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (31, [[0, 0], [1, 1], [2, 1], [3, -1], [4, 1], [5, 1], [6, -1], [7, 1], [8, 1], [9, 1], [10, 1], [11, -1], [12, -1], [13, -1], [14, 1], [15, -1], [16, 1], [17, -1], [18, 1], [19, 1], [20, 1], [21, -1], [22, -1], [23, -1], [24, -1], [25, 1], [26, -1], [27, -1], [28, 1], [29, -1], [30, -1]], Yes, zero_residue_nonresidue_partition_is_complete)."},"skills":["Legendre Classes Include Zero before Nonzero Residue Classification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0011","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7eb63b1625c109407162d9a0e8ea9872283e06045d15a18a12c38bc3054820f1","family_id":"qf-mod43-legendre-classification-table","seed":202611126011,"semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":37}"},"fields":{"title":"Legendre Classes Include Zero before Nonzero Residue Classification","stem":"Use bounded odd prime p=37 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).","answer":{"kind":"expression","value":"(37, [[0, 0], [1, 1], [2, -1], [3, 1], [4, 1], [5, -1], [6, -1], [7, 1], [8, -1], [9, 1], [10, 1], [11, 1], [12, 1], [13, -1], [14, -1], [15, -1], [16, 1], [17, -1], [18, -1], [19, -1], [20, -1], [21, 1], [22, -1], [23, -1], [24, -1], [25, 1], [26, 1], [27, 1], [28, 1], [29, -1], [30, 1], [31, -1], [32, -1], [33, 1], [34, 1], [35, -1], [36, 1]], Yes, zero_residue_nonresidue_partition_is_complete)","explanation":"Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (37, [[0, 0], [1, 1], [2, -1], [3, 1], [4, 1], [5, -1], [6, -1], [7, 1], [8, -1], [9, 1], [10, 1], [11, 1], [12, 1], [13, -1], [14, -1], [15, -1], [16, 1], [17, -1], [18, -1], [19, -1], [20, -1], [21, 1], [22, -1], [23, -1], [24, -1], [25, 1], [26, 1], [27, 1], [28, 1], [29, -1], [30, 1], [31, -1], [32, -1], [33, 1], [34, 1], [35, -1], [36, 1]], Yes, zero_residue_nonresidue_partition_is_complete)."},"skills":["Legendre Classes Include Zero before Nonzero Residue Classification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0012","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b213410772eeaced2113d4fe19a2e192a8995c7aa5839c9f09e8229b46620899","family_id":"qf-mod43-legendre-classification-table","seed":202611126012,"semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":41}"},"fields":{"title":"Legendre Classes Include Zero before Nonzero Residue Classification","stem":"Use bounded odd prime p=41 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).","answer":{"kind":"expression","value":"(41, [[0, 0], [1, 1], [2, 1], [3, -1], [4, 1], [5, 1], [6, -1], [7, -1], [8, 1], [9, 1], [10, 1], [11, -1], [12, -1], [13, -1], [14, -1], [15, -1], [16, 1], [17, -1], [18, 1], [19, -1], [20, 1], [21, 1], [22, -1], [23, 1], [24, -1], [25, 1], [26, -1], [27, -1], [28, -1], [29, -1], [30, -1], [31, 1], [32, 1], [33, 1], [34, -1], [35, -1], [36, 1], [37, 1], [38, -1], [39, 1], [40, 1]], Yes, zero_residue_nonresidue_partition_is_complete)","explanation":"Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (41, [[0, 0], [1, 1], [2, 1], [3, -1], [4, 1], [5, 1], [6, -1], [7, -1], [8, 1], [9, 1], [10, 1], [11, -1], [12, -1], [13, -1], [14, -1], [15, -1], [16, 1], [17, -1], [18, 1], [19, -1], [20, 1], [21, 1], [22, -1], [23, 1], [24, -1], [25, 1], [26, -1], [27, -1], [28, -1], [29, -1], [30, -1], [31, 1], [32, 1], [33, 1], [34, -1], [35, -1], [36, 1], [37, 1], [38, -1], [39, 1], [40, 1]], Yes, zero_residue_nonresidue_partition_is_complete)."},"skills":["Legendre Classes Include Zero before Nonzero Residue Classification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0013","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0bb1bb31a89f5f6977a6e1c601b7fcb0306d3bbfd9345e7e2c8324b8cb484e5a","family_id":"qf-mod43-legendre-classification-table","seed":202611126013,"semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":43}"},"fields":{"title":"Legendre Classes Include Zero before Nonzero Residue Classification","stem":"Use bounded odd prime p=43 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).","answer":{"kind":"expression","value":"(43, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1], [5, -1], [6, 1], [7, -1], [8, -1], [9, 1], [10, 1], [11, 1], [12, -1], [13, 1], [14, 1], [15, 1], [16, 1], [17, 1], [18, -1], [19, -1], [20, -1], [21, 1], [22, -1], [23, 1], [24, 1], [25, 1], [26, -1], [27, -1], [28, -1], [29, -1], [30, -1], [31, 1], [32, -1], [33, -1], [34, -1], [35, 1], [36, 1], [37, -1], [38, 1], [39, -1], [40, 1], [41, 1], [42, -1]], Yes, zero_residue_nonresidue_partition_is_complete)","explanation":"Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (43, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1], [5, -1], [6, 1], [7, -1], [8, -1], [9, 1], [10, 1], [11, 1], [12, -1], [13, 1], [14, 1], [15, 1], [16, 1], [17, 1], [18, -1], [19, -1], [20, -1], [21, 1], [22, -1], [23, 1], [24, 1], [25, 1], [26, -1], [27, -1], [28, -1], [29, -1], [30, -1], [31, 1], [32, -1], [33, -1], [34, -1], [35, 1], [36, 1], [37, -1], [38, 1], [39, -1], [40, 1], [41, 1], [42, -1]], Yes, zero_residue_nonresidue_partition_is_complete)."},"skills":["Legendre Classes Include Zero before Nonzero Residue Classification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0014","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"94cabfb4ec8c654efa2268ac71d5155f72f989cb7c1a31252952d61c0c42c28c","family_id":"qf-mod43-legendre-classification-table","seed":202611126014,"semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":47}"},"fields":{"title":"Legendre Classes Include Zero before Nonzero Residue Classification","stem":"Use bounded odd prime p=47 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).","answer":{"kind":"expression","value":"(47, [[0, 0], [1, 1], [2, 1], [3, 1], [4, 1], [5, -1], [6, 1], [7, 1], [8, 1], [9, 1], [10, -1], [11, -1], [12, 1], [13, -1], [14, 1], [15, -1], [16, 1], [17, 1], [18, 1], [19, -1], [20, -1], [21, 1], [22, -1], [23, -1], [24, 1], [25, 1], [26, -1], [27, 1], [28, 1], [29, -1], [30, -1], [31, -1], [32, 1], [33, -1], [34, 1], [35, -1], [36, 1], [37, 1], [38, -1], [39, -1], [40, -1], [41, -1], [42, 1], [43, -1], [44, -1], [45, -1], [46, -1]], Yes, zero_residue_nonresidue_partition_is_complete)","explanation":"Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (47, [[0, 0], [1, 1], [2, 1], [3, 1], [4, 1], [5, -1], [6, 1], [7, 1], [8, 1], [9, 1], [10, -1], [11, -1], [12, 1], [13, -1], [14, 1], [15, -1], [16, 1], [17, 1], [18, 1], [19, -1], [20, -1], [21, 1], [22, -1], [23, -1], [24, 1], [25, 1], [26, -1], [27, 1], [28, 1], [29, -1], [30, -1], [31, -1], [32, 1], [33, -1], [34, 1], [35, -1], [36, 1], [37, 1], [38, -1], [39, -1], [40, -1], [41, -1], [42, 1], [43, -1], [44, -1], [45, -1], [46, -1]], Yes, zero_residue_nonresidue_partition_is_complete)."},"skills":["Legendre Classes Include Zero before Nonzero Residue Classification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0015","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a03da8e6f3dbd1b07f6807eb24e49d13739cde28478bc063047fddffd244f474","family_id":"qf-mod43-legendre-classification-table","seed":202611126015,"semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":53}"},"fields":{"title":"Legendre Classes Include Zero before Nonzero Residue Classification","stem":"Use bounded odd prime p=53 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).","answer":{"kind":"expression","value":"(53, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1], [5, -1], [6, 1], [7, 1], [8, -1], [9, 1], [10, 1], [11, 1], [12, -1], [13, 1], [14, -1], [15, 1], [16, 1], [17, 1], [18, -1], [19, -1], [20, -1], [21, -1], [22, -1], [23, -1], [24, 1], [25, 1], [26, -1], [27, -1], [28, 1], [29, 1], [30, -1], [31, -1], [32, -1], [33, -1], [34, -1], [35, -1], [36, 1], [37, 1], [38, 1], [39, -1], [40, 1], [41, -1], [42, 1], [43, 1], [44, 1], [45, -1], [46, 1], [47, 1], [48, -1], [49, 1], [50, -1], [51, -1], [52, 1]], Yes, zero_residue_nonresidue_partition_is_complete)","explanation":"Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (53, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1], [5, -1], [6, 1], [7, 1], [8, -1], [9, 1], [10, 1], [11, 1], [12, -1], [13, 1], [14, -1], [15, 1], [16, 1], [17, 1], [18, -1], [19, -1], [20, -1], [21, -1], [22, -1], [23, -1], [24, 1], [25, 1], [26, -1], [27, -1], [28, 1], [29, 1], [30, -1], [31, -1], [32, -1], [33, -1], [34, -1], [35, -1], [36, 1], [37, 1], [38, 1], [39, -1], [40, 1], [41, -1], [42, 1], [43, 1], [44, 1], [45, -1], [46, 1], [47, 1], [48, -1], [49, 1], [50, -1], [51, -1], [52, 1]], Yes, zero_residue_nonresidue_partition_is_complete)."},"skills":["Legendre Classes Include Zero before Nonzero Residue Classification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0016","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1bba3140289430d1654ee2713d58cbe8972b394a820c6628fb7543a35ee38c8","family_id":"qf-mod43-legendre-classification-table","seed":202611126016,"semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":59}"},"fields":{"title":"Legendre Classes Include Zero before Nonzero Residue Classification","stem":"Use bounded odd prime p=59 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).","answer":{"kind":"expression","value":"(59, [[0, 0], [1, 1], [2, -1], [3, 1], [4, 1], [5, 1], [6, -1], [7, 1], [8, -1], [9, 1], [10, -1], [11, -1], [12, 1], [13, -1], [14, -1], [15, 1], [16, 1], [17, 1], [18, -1], [19, 1], [20, 1], [21, 1], [22, 1], [23, -1], [24, -1], [25, 1], [26, 1], [27, 1], [28, 1], [29, 1], [30, -1], [31, -1], [32, -1], [33, -1], [34, -1], [35, 1], [36, 1], [37, -1], [38, -1], [39, -1], [40, -1], [41, 1], [42, -1], [43, -1], [44, -1], [45, 1], [46, 1], [47, -1], [48, 1], [49, 1], [50, -1], [51, 1], [52, -1], [53, 1], [54, -1], [55, -1], [56, -1], [57, 1], [58, -1]], Yes, zero_residue_nonresidue_partition_is_complete)","explanation":"Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (59, [[0, 0], [1, 1], [2, -1], [3, 1], [4, 1], [5, 1], [6, -1], [7, 1], [8, -1], [9, 1], [10, -1], [11, -1], [12, 1], [13, -1], [14, -1], [15, 1], [16, 1], [17, 1], [18, -1], [19, 1], [20, 1], [21, 1], [22, 1], [23, -1], [24, -1], [25, 1], [26, 1], [27, 1], [28, 1], [29, 1], [30, -1], [31, -1], [32, -1], [33, -1], [34, -1], [35, 1], [36, 1], [37, -1], [38, -1], [39, -1], [40, -1], [41, 1], [42, -1], [43, -1], [44, -1], [45, 1], [46, 1], [47, -1], [48, 1], [49, 1], [50, -1], [51, 1], [52, -1], [53, 1], [54, -1], [55, -1], [56, -1], [57, 1], [58, -1]], Yes, zero_residue_nonresidue_partition_is_complete)."},"skills":["Legendre Classes Include Zero before Nonzero Residue Classification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0017","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a966303e2d9655337a823c9c516705713e05800b03c62488d0acaeda62a9a738","family_id":"qf-mod43-legendre-classification-table","seed":202611126017,"semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":61}"},"fields":{"title":"Legendre Classes Include Zero before Nonzero Residue Classification","stem":"Use bounded odd prime p=61 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).","answer":{"kind":"expression","value":"(61, [[0, 0], [1, 1], [2, -1], [3, 1], [4, 1], [5, 1], [6, -1], [7, -1], [8, -1], [9, 1], [10, -1], [11, -1], [12, 1], [13, 1], [14, 1], [15, 1], [16, 1], [17, -1], [18, -1], [19, 1], [20, 1], [21, -1], [22, 1], [23, -1], [24, -1], [25, 1], [26, -1], [27, 1], [28, -1], [29, -1], [30, -1], [31, -1], [32, -1], [33, -1], [34, 1], [35, -1], [36, 1], [37, -1], [38, -1], [39, 1], [40, -1], [41, 1], [42, 1], [43, -1], [44, -1], [45, 1], [46, 1], [47, 1], [48, 1], [49, 1], [50, -1], [51, -1], [52, 1], [53, -1], [54, -1], [55, -1], [56, 1], [57, 1], [58, 1], [59, -1], [60, 1]], Yes, zero_residue_nonresidue_partition_is_complete)","explanation":"Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (61, [[0, 0], [1, 1], [2, -1], [3, 1], [4, 1], [5, 1], [6, -1], [7, -1], [8, -1], [9, 1], [10, -1], [11, -1], [12, 1], [13, 1], [14, 1], [15, 1], [16, 1], [17, -1], [18, -1], [19, 1], [20, 1], [21, -1], [22, 1], [23, -1], [24, -1], [25, 1], [26, -1], [27, 1], [28, -1], [29, -1], [30, -1], [31, -1], [32, -1], [33, -1], [34, 1], [35, -1], [36, 1], [37, -1], [38, -1], [39, 1], [40, -1], [41, 1], [42, 1], [43, -1], [44, -1], [45, 1], [46, 1], [47, 1], [48, 1], [49, 1], [50, -1], [51, -1], [52, 1], [53, -1], [54, -1], [55, -1], [56, 1], [57, 1], [58, 1], [59, -1], [60, 1]], Yes, zero_residue_nonresidue_partition_is_complete)."},"skills":["Legendre Classes Include Zero before Nonzero Residue Classification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0018","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"853a701d0dcbd5e64e851678d8ad98848cb1e0bea140bdf7af849cfae83d887a","family_id":"qf-mod43-legendre-classification-table","seed":202611126018,"semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":67}"},"fields":{"title":"Legendre Classes Include Zero before Nonzero Residue Classification","stem":"Use bounded odd prime p=67 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).","answer":{"kind":"expression","value":"(67, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1], [5, -1], [6, 1], [7, -1], [8, -1], [9, 1], [10, 1], [11, -1], [12, -1], [13, -1], [14, 1], [15, 1], [16, 1], [17, 1], [18, -1], [19, 1], [20, -1], [21, 1], [22, 1], [23, 1], [24, 1], [25, 1], [26, 1], [27, -1], [28, -1], [29, 1], [30, -1], [31, -1], [32, -1], [33, 1], [34, -1], [35, 1], [36, 1], [37, 1], [38, -1], [39, 1], [40, 1], [41, -1], [42, -1], [43, -1], [44, -1], [45, -1], [46, -1], [47, 1], [48, -1], [49, 1], [50, -1], [51, -1], [52, -1], [53, -1], [54, 1], [55, 1], [56, 1], [57, -1], [58, -1], [59, 1], [60, 1], [61, -1], [62, 1], [63, -1], [64, 1], [65, 1], [66, -1]], Yes, zero_residue_nonresidue_partition_is_complete)","explanation":"Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (67, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1], [5, -1], [6, 1], [7, -1], [8, -1], [9, 1], [10, 1], [11, -1], [12, -1], [13, -1], [14, 1], [15, 1], [16, 1], [17, 1], [18, -1], [19, 1], [20, -1], [21, 1], [22, 1], [23, 1], [24, 1], [25, 1], [26, 1], [27, -1], [28, -1], [29, 1], [30, -1], [31, -1], [32, -1], [33, 1], [34, -1], [35, 1], [36, 1], [37, 1], [38, -1], [39, 1], [40, 1], [41, -1], [42, -1], [43, -1], [44, -1], [45, -1], [46, -1], [47, 1], [48, -1], [49, 1], [50, -1], [51, -1], [52, -1], [53, -1], [54, 1], [55, 1], [56, 1], [57, -1], [58, -1], [59, 1], [60, 1], [61, -1], [62, 1], [63, -1], [64, 1], [65, 1], [66, -1]], Yes, zero_residue_nonresidue_partition_is_complete)."},"skills":["Legendre Classes Include Zero before Nonzero Residue Classification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0019","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ccd79cb3167598e8e9a1a9ec1a4e89bb248a9f7f8ba7117170d77137d9aaaab","family_id":"qf-mod43-legendre-classification-table","seed":202611126019,"semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":71}"},"fields":{"title":"Legendre Classes Include Zero before Nonzero Residue Classification","stem":"Use bounded odd prime p=71 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).","answer":{"kind":"expression","value":"(71, [[0, 0], [1, 1], [2, 1], [3, 1], [4, 1], [5, 1], [6, 1], [7, -1], [8, 1], [9, 1], [10, 1], [11, -1], [12, 1], [13, -1], [14, -1], [15, 1], [16, 1], [17, -1], [18, 1], [19, 1], [20, 1], [21, -1], [22, -1], [23, -1], [24, 1], [25, 1], [26, -1], [27, 1], [28, -1], [29, 1], [30, 1], [31, -1], [32, 1], [33, -1], [34, -1], [35, -1], [36, 1], [37, 1], [38, 1], [39, -1], [40, 1], [41, -1], [42, -1], [43, 1], [44, -1], [45, 1], [46, -1], [47, -1], [48, 1], [49, 1], [50, 1], [51, -1], [52, -1], [53, -1], [54, 1], [55, -1], [56, -1], [57, 1], [58, 1], [59, -1], [60, 1], [61, -1], [62, -1], [63, -1], [64, 1], [65, -1], [66, -1], [67, -1], [68, -1], [69, -1], [70, -1]], Yes, zero_residue_nonresidue_partition_is_complete)","explanation":"Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (71, [[0, 0], [1, 1], [2, 1], [3, 1], [4, 1], [5, 1], [6, 1], [7, -1], [8, 1], [9, 1], [10, 1], [11, -1], [12, 1], [13, -1], [14, -1], [15, 1], [16, 1], [17, -1], [18, 1], [19, 1], [20, 1], [21, -1], [22, -1], [23, -1], [24, 1], [25, 1], [26, -1], [27, 1], [28, -1], [29, 1], [30, 1], [31, -1], [32, 1], [33, -1], [34, -1], [35, -1], [36, 1], [37, 1], [38, 1], [39, -1], [40, 1], [41, -1], [42, -1], [43, 1], [44, -1], [45, 1], [46, -1], [47, -1], [48, 1], [49, 1], [50, 1], [51, -1], [52, -1], [53, -1], [54, 1], [55, -1], [56, -1], [57, 1], [58, 1], [59, -1], [60, 1], [61, -1], [62, -1], [63, -1], [64, 1], [65, -1], [66, -1], [67, -1], [68, -1], [69, -1], [70, -1]], Yes, zero_residue_nonresidue_partition_is_complete)."},"skills":["Legendre Classes Include Zero before Nonzero Residue Classification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0020","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c466ba320c83d01e5e56c03808f574d706a84441957df5620a0441d909423bbc","family_id":"qf-mod43-legendre-classification-table","seed":202611126020,"semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":73}"},"fields":{"title":"Legendre Classes Include Zero before Nonzero Residue Classification","stem":"Use bounded odd prime p=73 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).","answer":{"kind":"expression","value":"(73, [[0, 0], [1, 1], [2, 1], [3, 1], [4, 1], [5, -1], [6, 1], [7, -1], [8, 1], [9, 1], [10, -1], [11, -1], [12, 1], [13, -1], [14, -1], [15, -1], [16, 1], [17, -1], [18, 1], [19, 1], [20, -1], [21, -1], [22, -1], [23, 1], [24, 1], [25, 1], [26, -1], [27, 1], [28, -1], [29, -1], [30, -1], [31, -1], [32, 1], [33, -1], [34, -1], [35, 1], [36, 1], [37, 1], [38, 1], [39, -1], [40, -1], [41, 1], [42, -1], [43, -1], [44, -1], [45, -1], [46, 1], [47, -1], [48, 1], [49, 1], [50, 1], [51, -1], [52, -1], [53, -1], [54, 1], [55, 1], [56, -1], [57, 1], [58, -1], [59, -1], [60, -1], [61, 1], [62, -1], [63, -1], [64, 1], [65, 1], [66, -1], [67, 1], [68, -1], [69, 1], [70, 1], [71, 1], [72, 1]], Yes, zero_residue_nonresidue_partition_is_complete)","explanation":"Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (73, [[0, 0], [1, 1], [2, 1], [3, 1], [4, 1], [5, -1], [6, 1], [7, -1], [8, 1], [9, 1], [10, -1], [11, -1], [12, 1], [13, -1], [14, -1], [15, -1], [16, 1], [17, -1], [18, 1], [19, 1], [20, -1], [21, -1], [22, -1], [23, 1], [24, 1], [25, 1], [26, -1], [27, 1], [28, -1], [29, -1], [30, -1], [31, -1], [32, 1], [33, -1], [34, -1], [35, 1], [36, 1], [37, 1], [38, 1], [39, -1], [40, -1], [41, 1], [42, -1], [43, -1], [44, -1], [45, -1], [46, 1], [47, -1], [48, 1], [49, 1], [50, 1], [51, -1], [52, -1], [53, -1], [54, 1], [55, 1], [56, -1], [57, 1], [58, -1], [59, -1], [60, -1], [61, 1], [62, -1], [63, -1], [64, 1], [65, 1], [66, -1], [67, 1], [68, -1], [69, 1], [70, 1], [71, 1], [72, 1]], Yes, zero_residue_nonresidue_partition_is_complete)."},"skills":["Legendre Classes Include Zero before Nonzero Residue Classification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0021","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53a0a56f83d411abb38b8b898ef7ba31cc67990afa322347dea32d04692c73fd","family_id":"qf-mod43-legendre-classification-table","seed":202611126021,"semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":79}"},"fields":{"title":"Legendre Classes Include Zero before Nonzero Residue Classification","stem":"Use bounded odd prime p=79 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).","answer":{"kind":"expression","value":"(79, [[0, 0], [1, 1], [2, 1], [3, -1], [4, 1], [5, 1], [6, -1], [7, -1], [8, 1], [9, 1], [10, 1], [11, 1], [12, -1], [13, 1], [14, -1], [15, -1], [16, 1], [17, -1], [18, 1], [19, 1], [20, 1], [21, 1], [22, 1], [23, 1], [24, -1], [25, 1], [26, 1], [27, -1], [28, -1], [29, -1], [30, -1], [31, 1], [32, 1], [33, -1], [34, -1], [35, -1], [36, 1], [37, -1], [38, 1], [39, -1], [40, 1], [41, -1], [42, 1], [43, -1], [44, 1], [45, 1], [46, 1], [47, -1], [48, -1], [49, 1], [50, 1], [51, 1], [52, 1], [53, -1], [54, -1], [55, 1], [56, -1], [57, -1], [58, -1], [59, -1], [60, -1], [61, -1], [62, 1], [63, -1], [64, 1], [65, 1], [66, -1], [67, 1], [68, -1], [69, -1], [70, -1], [71, -1], [72, 1], [73, 1], [74, -1], [75, -1], [76, 1], [77, -1], [78, -1]], Yes, zero_residue_nonresidue_partition_is_complete)","explanation":"Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (79, [[0, 0], [1, 1], [2, 1], [3, -1], [4, 1], [5, 1], [6, -1], [7, -1], [8, 1], [9, 1], [10, 1], [11, 1], [12, -1], [13, 1], [14, -1], [15, -1], [16, 1], [17, -1], [18, 1], [19, 1], [20, 1], [21, 1], [22, 1], [23, 1], [24, -1], [25, 1], [26, 1], [27, -1], [28, -1], [29, -1], [30, -1], [31, 1], [32, 1], [33, -1], [34, -1], [35, -1], [36, 1], [37, -1], [38, 1], [39, -1], [40, 1], [41, -1], [42, 1], [43, -1], [44, 1], [45, 1], [46, 1], [47, -1], [48, -1], [49, 1], [50, 1], [51, 1], [52, 1], [53, -1], [54, -1], [55, 1], [56, -1], [57, -1], [58, -1], [59, -1], [60, -1], [61, -1], [62, 1], [63, -1], [64, 1], [65, 1], [66, -1], [67, 1], [68, -1], [69, -1], [70, -1], [71, -1], [72, 1], [73, 1], [74, -1], [75, -1], [76, 1], [77, -1], [78, -1]], Yes, zero_residue_nonresidue_partition_is_complete)."},"skills":["Legendre Classes Include Zero before Nonzero Residue Classification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0022","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b776c21ab1cde83c7973b259ff227bbc4ec92e970b532e268923de26140fc9d1","family_id":"qf-mod43-legendre-classification-table","seed":202611126022,"semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":83}"},"fields":{"title":"Legendre Classes Include Zero before Nonzero Residue Classification","stem":"Use bounded odd prime p=83 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).","answer":{"kind":"expression","value":"(83, [[0, 0], [1, 1], [2, -1], [3, 1], [4, 1], [5, -1], [6, -1], [7, 1], [8, -1], [9, 1], [10, 1], [11, 1], [12, 1], [13, -1], [14, -1], [15, -1], [16, 1], [17, 1], [18, -1], [19, -1], [20, -1], [21, 1], [22, -1], [23, 1], [24, -1], [25, 1], [26, 1], [27, 1], [28, 1], [29, 1], [30, 1], [31, 1], [32, -1], [33, 1], [34, -1], [35, -1], [36, 1], [37, 1], [38, 1], [39, -1], [40, 1], [41, 1], [42, -1], [43, -1], [44, 1], [45, -1], [46, -1], [47, -1], [48, 1], [49, 1], [50, -1], [51, 1], [52, -1], [53, -1], [54, -1], [55, -1], [56, -1], [57, -1], [58, -1], [59, 1], [60, -1], [61, 1], [62, -1], [63, 1], [64, 1], [65, 1], [66, -1], [67, -1], [68, 1], [69, 1], [70, 1], [71, -1], [72, -1], [73, -1], [74, -1], [75, 1], [76, -1], [77, 1], [78, 1], [79, -1], [80, -1], [81, 1], [82, -1]], Yes, zero_residue_nonresidue_partition_is_complete)","explanation":"Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (83, [[0, 0], [1, 1], [2, -1], [3, 1], [4, 1], [5, -1], [6, -1], [7, 1], [8, -1], [9, 1], [10, 1], [11, 1], [12, 1], [13, -1], [14, -1], [15, -1], [16, 1], [17, 1], [18, -1], [19, -1], [20, -1], [21, 1], [22, -1], [23, 1], [24, -1], [25, 1], [26, 1], [27, 1], [28, 1], [29, 1], [30, 1], [31, 1], [32, -1], [33, 1], [34, -1], [35, -1], [36, 1], [37, 1], [38, 1], [39, -1], [40, 1], [41, 1], [42, -1], [43, -1], [44, 1], [45, -1], [46, -1], [47, -1], [48, 1], [49, 1], [50, -1], [51, 1], [52, -1], [53, -1], [54, -1], [55, -1], [56, -1], [57, -1], [58, -1], [59, 1], [60, -1], [61, 1], [62, -1], [63, 1], [64, 1], [65, 1], [66, -1], [67, -1], [68, 1], [69, 1], [70, 1], [71, -1], [72, -1], [73, -1], [74, -1], [75, 1], [76, -1], [77, 1], [78, 1], [79, -1], [80, -1], [81, 1], [82, -1]], Yes, zero_residue_nonresidue_partition_is_complete)."},"skills":["Legendre Classes Include Zero before Nonzero Residue Classification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0023","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d10140cb5dbcaea56b00120be86f18d949f183bc7130b06ebcfca819c7d49a0","family_id":"qf-mod43-legendre-classification-table","seed":202611126023,"semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":89}"},"fields":{"title":"Legendre Classes Include Zero before Nonzero Residue Classification","stem":"Use bounded odd prime p=89 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).","answer":{"kind":"expression","value":"(89, [[0, 0], [1, 1], [2, 1], [3, -1], [4, 1], [5, 1], [6, -1], [7, -1], [8, 1], [9, 1], [10, 1], [11, 1], [12, -1], [13, -1], [14, -1], [15, -1], [16, 1], [17, 1], [18, 1], [19, -1], [20, 1], [21, 1], [22, 1], [23, -1], [24, -1], [25, 1], [26, -1], [27, -1], [28, -1], [29, -1], [30, -1], [31, -1], [32, 1], [33, -1], [34, 1], [35, -1], [36, 1], [37, -1], [38, -1], [39, 1], [40, 1], [41, -1], [42, 1], [43, -1], [44, 1], [45, 1], [46, -1], [47, 1], [48, -1], [49, 1], [50, 1], [51, -1], [52, -1], [53, 1], [54, -1], [55, 1], [56, -1], [57, 1], [58, -1], [59, -1], [60, -1], [61, -1], [62, -1], [63, -1], [64, 1], [65, -1], [66, -1], [67, 1], [68, 1], [69, 1], [70, -1], [71, 1], [72, 1], [73, 1], [74, -1], [75, -1], [76, -1], [77, -1], [78, 1], [79, 1], [80, 1], [81, 1], [82, -1], [83, -1], [84, 1], [85, 1], [86, -1], [87, 1], [88, 1]], Yes, zero_residue_nonresidue_partition_is_complete)","explanation":"Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (89, [[0, 0], [1, 1], [2, 1], [3, -1], [4, 1], [5, 1], [6, -1], [7, -1], [8, 1], [9, 1], [10, 1], [11, 1], [12, -1], [13, -1], [14, -1], [15, -1], [16, 1], [17, 1], [18, 1], [19, -1], [20, 1], [21, 1], [22, 1], [23, -1], [24, -1], [25, 1], [26, -1], [27, -1], [28, -1], [29, -1], [30, -1], [31, -1], [32, 1], [33, -1], [34, 1], [35, -1], [36, 1], [37, -1], [38, -1], [39, 1], [40, 1], [41, -1], [42, 1], [43, -1], [44, 1], [45, 1], [46, -1], [47, 1], [48, -1], [49, 1], [50, 1], [51, -1], [52, -1], [53, 1], [54, -1], [55, 1], [56, -1], [57, 1], [58, -1], [59, -1], [60, -1], [61, -1], [62, -1], [63, -1], [64, 1], [65, -1], [66, -1], [67, 1], [68, 1], [69, 1], [70, -1], [71, 1], [72, 1], [73, 1], [74, -1], [75, -1], [76, -1], [77, -1], [78, 1], [79, 1], [80, 1], [81, 1], [82, -1], [83, -1], [84, 1], [85, 1], [86, -1], [87, 1], [88, 1]], Yes, zero_residue_nonresidue_partition_is_complete)."},"skills":["Legendre Classes Include Zero before Nonzero Residue Classification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0024","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13487234bb4c20cc88467fc1b9e58df9f741a1502e57133501c712408498e214","family_id":"qf-mod43-legendre-classification-table","seed":202611126024,"semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":97}"},"fields":{"title":"Legendre Classes Include Zero before Nonzero Residue Classification","stem":"Use bounded odd prime p=97 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).","answer":{"kind":"expression","value":"(97, [[0, 0], [1, 1], [2, 1], [3, 1], [4, 1], [5, -1], [6, 1], [7, -1], [8, 1], [9, 1], [10, -1], [11, 1], [12, 1], [13, -1], [14, -1], [15, -1], [16, 1], [17, -1], [18, 1], [19, -1], [20, -1], [21, -1], [22, 1], [23, -1], [24, 1], [25, 1], [26, -1], [27, 1], [28, -1], [29, -1], [30, -1], [31, 1], [32, 1], [33, 1], [34, -1], [35, 1], [36, 1], [37, -1], [38, -1], [39, -1], [40, -1], [41, -1], [42, -1], [43, 1], [44, 1], [45, -1], [46, -1], [47, 1], [48, 1], [49, 1], [50, 1], [51, -1], [52, -1], [53, 1], [54, 1], [55, -1], [56, -1], [57, -1], [58, -1], [59, -1], [60, -1], [61, 1], [62, 1], [63, -1], [64, 1], [65, 1], [66, 1], [67, -1], [68, -1], [69, -1], [70, 1], [71, -1], [72, 1], [73, 1], [74, -1], [75, 1], [76, -1], [77, -1], [78, -1], [79, 1], [80, -1], [81, 1], [82, -1], [83, -1], [84, -1], [85, 1], [86, 1], [87, -1], [88, 1], [89, 1], [90, -1], [91, 1], [92, -1], [93, 1], [94, 1], [95, 1], [96, 1]], Yes, zero_residue_nonresidue_partition_is_complete)","explanation":"Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (97, [[0, 0], [1, 1], [2, 1], [3, 1], [4, 1], [5, -1], [6, 1], [7, -1], [8, 1], [9, 1], [10, -1], [11, 1], [12, 1], [13, -1], [14, -1], [15, -1], [16, 1], [17, -1], [18, 1], [19, -1], [20, -1], [21, -1], [22, 1], [23, -1], [24, 1], [25, 1], [26, -1], [27, 1], [28, -1], [29, -1], [30, -1], [31, 1], [32, 1], [33, 1], [34, -1], [35, 1], [36, 1], [37, -1], [38, -1], [39, -1], [40, -1], [41, -1], [42, -1], [43, 1], [44, 1], [45, -1], [46, -1], [47, 1], [48, 1], [49, 1], [50, 1], [51, -1], [52, -1], [53, 1], [54, 1], [55, -1], [56, -1], [57, -1], [58, -1], [59, -1], [60, -1], [61, 1], [62, 1], [63, -1], [64, 1], [65, 1], [66, 1], [67, -1], [68, -1], [69, -1], [70, 1], [71, -1], [72, 1], [73, 1], [74, -1], [75, 1], [76, -1], [77, -1], [78, -1], [79, 1], [80, -1], [81, 1], [82, -1], [83, -1], [84, -1], [85, 1], [86, 1], [87, -1], [88, 1], [89, 1], [90, -1], [91, 1], [92, -1], [93, 1], [94, 1], [95, 1], [96, 1]], Yes, zero_residue_nonresidue_partition_is_complete)."},"skills":["Legendre Classes Include Zero before Nonzero Residue Classification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0025","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab7f776e9df9fd27c022e861a61b244ebbce9ff828f15d5c9cf1e28205951fd3","family_id":"qf-mod43-legendre-classification-table","seed":202611126025,"semantic_template":"qf-mod43-legendre-classification-table/{\"prime\":101}"},"fields":{"title":"Legendre Classes Include Zero before Nonzero Residue Classification","stem":"Use bounded odd prime p=101 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).","answer":{"kind":"expression","value":"(101, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1], [5, 1], [6, 1], [7, -1], [8, -1], [9, 1], [10, -1], [11, -1], [12, -1], [13, 1], [14, 1], [15, -1], [16, 1], [17, 1], [18, -1], [19, 1], [20, 1], [21, 1], [22, 1], [23, 1], [24, 1], [25, 1], [26, -1], [27, -1], [28, -1], [29, -1], [30, 1], [31, 1], [32, -1], [33, 1], [34, -1], [35, -1], [36, 1], [37, 1], [38, -1], [39, -1], [40, -1], [41, -1], [42, -1], [43, 1], [44, -1], [45, 1], [46, -1], [47, 1], [48, -1], [49, 1], [50, -1], [51, -1], [52, 1], [53, -1], [54, 1], [55, -1], [56, 1], [57, -1], [58, 1], [59, -1], [60, -1], [61, -1], [62, -1], [63, -1], [64, 1], [65, 1], [66, -1], [67, -1], [68, 1], [69, -1], [70, 1], [71, 1], [72, -1], [73, -1], [74, -1], [75, -1], [76, 1], [77, 1], [78, 1], [79, 1], [80, 1], [81, 1], [82, 1], [83, -1], [84, 1], [85, 1], [86, -1], [87, 1], [88, 1], [89, -1], [90, -1], [91, -1], [92, 1], [93, -1], [94, -1], [95, 1], [96, 1], [97, 1], [98, -1], [99, -1], [100, 1]], Yes, zero_residue_nonresidue_partition_is_complete)","explanation":"Classify every normalized class0 through p−1: zero gets0, a nonzero square gets1, another nonzero class gets−1. Completeness comes from the full table; do not pre-use Euler criterion or a few multiplicativity samples. The exact result, retaining the requested order, is (101, [[0, 0], [1, 1], [2, -1], [3, -1], [4, 1], [5, 1], [6, 1], [7, -1], [8, -1], [9, 1], [10, -1], [11, -1], [12, -1], [13, 1], [14, 1], [15, -1], [16, 1], [17, 1], [18, -1], [19, 1], [20, 1], [21, 1], [22, 1], [23, 1], [24, 1], [25, 1], [26, -1], [27, -1], [28, -1], [29, -1], [30, 1], [31, 1], [32, -1], [33, 1], [34, -1], [35, -1], [36, 1], [37, 1], [38, -1], [39, -1], [40, -1], [41, -1], [42, -1], [43, 1], [44, -1], [45, 1], [46, -1], [47, 1], [48, -1], [49, 1], [50, -1], [51, -1], [52, 1], [53, -1], [54, 1], [55, -1], [56, 1], [57, -1], [58, 1], [59, -1], [60, -1], [61, -1], [62, -1], [63, -1], [64, 1], [65, 1], [66, -1], [67, -1], [68, 1], [69, -1], [70, 1], [71, 1], [72, -1], [73, -1], [74, -1], [75, -1], [76, 1], [77, 1], [78, 1], [79, 1], [80, 1], [81, 1], [82, 1], [83, -1], [84, 1], [85, 1], [86, -1], [87, 1], [88, 1], [89, -1], [90, -1], [91, -1], [92, 1], [93, -1], [94, -1], [95, 1], [96, 1], [97, 1], [98, -1], [99, -1], [100, 1]], Yes, zero_residue_nonresidue_partition_is_complete)."},"skills":["Legendre Classes Include Zero before Nonzero Residue Classification"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0026","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c02558076526dde71de3984e3f3654a385c78f3bec1543c7375903f9f33f647f","family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126026,"semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":1,\"prime\":3}"},"fields":{"title":"A Complete Root Table Distinguishes Symbol Plus One from Minus One","stem":"Use odd prime p=3, original a=1. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).","answer":{"kind":"expression","value":"(3, 1, 1, [1, 2], 1, complete_nonzero_square_root_witness)","explanation":"Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (3, 1, 1, [1, 2], 1, complete_nonzero_square_root_witness)."},"skills":["A Complete Root Table Distinguishes Symbol Plus One from Minus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0027","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61339214698ecf62369ac70444971843443fc4794a4507675181069c06c7fe46","family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126027,"semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":2,\"prime\":5}"},"fields":{"title":"A Complete Root Table Distinguishes Symbol Plus One from Minus One","stem":"Use odd prime p=5, original a=2. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).","answer":{"kind":"expression","value":"(5, 2, 2, [], -1, complete_square_table_has_no_root)","explanation":"Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (5, 2, 2, [], -1, complete_square_table_has_no_root)."},"skills":["A Complete Root Table Distinguishes Symbol Plus One from Minus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0028","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"de18da93b1809e738e914254a234ef1b88a5896b90cabd774ecb023905c2ada3","family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126028,"semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":3,\"prime\":7}"},"fields":{"title":"A Complete Root Table Distinguishes Symbol Plus One from Minus One","stem":"Use odd prime p=7, original a=3. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).","answer":{"kind":"expression","value":"(7, 3, 3, [], -1, complete_square_table_has_no_root)","explanation":"Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (7, 3, 3, [], -1, complete_square_table_has_no_root)."},"skills":["A Complete Root Table Distinguishes Symbol Plus One from Minus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0029","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c9415b2bb4b8326a7156b9019d5f3462d25dbefc67ecd1124ab6abb14f906ea","family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126029,"semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":4,\"prime\":11}"},"fields":{"title":"A Complete Root Table Distinguishes Symbol Plus One from Minus One","stem":"Use odd prime p=11, original a=4. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).","answer":{"kind":"expression","value":"(11, 4, 4, [2, 9], 1, complete_nonzero_square_root_witness)","explanation":"Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (11, 4, 4, [2, 9], 1, complete_nonzero_square_root_witness)."},"skills":["A Complete Root Table Distinguishes Symbol Plus One from Minus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0030","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"023b833d60d025c1fb371aaa3a838691432d2b814eede6b75af8d0cb624999f2","family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126030,"semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":5,\"prime\":13}"},"fields":{"title":"A Complete Root Table Distinguishes Symbol Plus One from Minus One","stem":"Use odd prime p=13, original a=5. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).","answer":{"kind":"expression","value":"(13, 5, 5, [], -1, complete_square_table_has_no_root)","explanation":"Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (13, 5, 5, [], -1, complete_square_table_has_no_root)."},"skills":["A Complete Root Table Distinguishes Symbol Plus One from Minus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0031","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"294def91f32cd7985903b8a6dbbee468ab6126fdde6dcbafd8462f7ade086ee7","family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126031,"semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":6,\"prime\":17}"},"fields":{"title":"A Complete Root Table Distinguishes Symbol Plus One from Minus One","stem":"Use odd prime p=17, original a=6. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).","answer":{"kind":"expression","value":"(17, 6, 6, [], -1, complete_square_table_has_no_root)","explanation":"Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (17, 6, 6, [], -1, complete_square_table_has_no_root)."},"skills":["A Complete Root Table Distinguishes Symbol Plus One from Minus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0032","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bea63131929a0aac169e04a99f1041738403ebbe267e6c420ed6e9d2bd3d8b3f","family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126032,"semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":7,\"prime\":19}"},"fields":{"title":"A Complete Root Table Distinguishes Symbol Plus One from Minus One","stem":"Use odd prime p=19, original a=7. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).","answer":{"kind":"expression","value":"(19, 7, 7, [8, 11], 1, complete_nonzero_square_root_witness)","explanation":"Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (19, 7, 7, [8, 11], 1, complete_nonzero_square_root_witness)."},"skills":["A Complete Root Table Distinguishes Symbol Plus One from Minus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0033","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9af3ec491ba03395f6dd222792611b41316d8e3a98825fe6ae7181e7e72e4412","family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126033,"semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":8,\"prime\":23}"},"fields":{"title":"A Complete Root Table Distinguishes Symbol Plus One from Minus One","stem":"Use odd prime p=23, original a=8. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).","answer":{"kind":"expression","value":"(23, 8, 8, [10, 13], 1, complete_nonzero_square_root_witness)","explanation":"Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (23, 8, 8, [10, 13], 1, complete_nonzero_square_root_witness)."},"skills":["A Complete Root Table Distinguishes Symbol Plus One from Minus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0034","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef7086e01452718c80d0dd700921a3677cb203f86f34a563a811e899db335949","family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126034,"semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":9,\"prime\":29}"},"fields":{"title":"A Complete Root Table Distinguishes Symbol Plus One from Minus One","stem":"Use odd prime p=29, original a=9. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).","answer":{"kind":"expression","value":"(29, 9, 9, [3, 26], 1, complete_nonzero_square_root_witness)","explanation":"Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (29, 9, 9, [3, 26], 1, complete_nonzero_square_root_witness)."},"skills":["A Complete Root Table Distinguishes Symbol Plus One from Minus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0035","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4acd8f3e925d114e904beffefdc4880bcaadb163f349dbff6e0b774702d80543","family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126035,"semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":10,\"prime\":31}"},"fields":{"title":"A Complete Root Table Distinguishes Symbol Plus One from Minus One","stem":"Use odd prime p=31, original a=10. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).","answer":{"kind":"expression","value":"(31, 10, 10, [14, 17], 1, complete_nonzero_square_root_witness)","explanation":"Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (31, 10, 10, [14, 17], 1, complete_nonzero_square_root_witness)."},"skills":["A Complete Root Table Distinguishes Symbol Plus One from Minus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0036","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9d67453e1ea69f97d2d76a75973ea685ad1c77532395853b551590a1f035e3d","family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126036,"semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":11,\"prime\":37}"},"fields":{"title":"A Complete Root Table Distinguishes Symbol Plus One from Minus One","stem":"Use odd prime p=37, original a=11. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).","answer":{"kind":"expression","value":"(37, 11, 11, [14, 23], 1, complete_nonzero_square_root_witness)","explanation":"Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (37, 11, 11, [14, 23], 1, complete_nonzero_square_root_witness)."},"skills":["A Complete Root Table Distinguishes Symbol Plus One from Minus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0037","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13c27ceeac993e2735ec69073f641d829d24769ffc9bff3866e2b9b13a6b9b1f","family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126037,"semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":12,\"prime\":41}"},"fields":{"title":"A Complete Root Table Distinguishes Symbol Plus One from Minus One","stem":"Use odd prime p=41, original a=12. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).","answer":{"kind":"expression","value":"(41, 12, 12, [], -1, complete_square_table_has_no_root)","explanation":"Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (41, 12, 12, [], -1, complete_square_table_has_no_root)."},"skills":["A Complete Root Table Distinguishes Symbol Plus One from Minus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0038","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac9efba0f0ef47d5d4ca4d5d767cece05df2951d1fc9a7ce1dc215fc21ab0962","family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126038,"semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":13,\"prime\":43}"},"fields":{"title":"A Complete Root Table Distinguishes Symbol Plus One from Minus One","stem":"Use odd prime p=43, original a=13. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).","answer":{"kind":"expression","value":"(43, 13, 13, [20, 23], 1, complete_nonzero_square_root_witness)","explanation":"Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (43, 13, 13, [20, 23], 1, complete_nonzero_square_root_witness)."},"skills":["A Complete Root Table Distinguishes Symbol Plus One from Minus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0039","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"108d068c040388762b4db0f21dcf5acbc94256fd5478e68dcb08e1d799d84ccf","family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126039,"semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":14,\"prime\":47}"},"fields":{"title":"A Complete Root Table Distinguishes Symbol Plus One from Minus One","stem":"Use odd prime p=47, original a=14. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).","answer":{"kind":"expression","value":"(47, 14, 14, [22, 25], 1, complete_nonzero_square_root_witness)","explanation":"Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (47, 14, 14, [22, 25], 1, complete_nonzero_square_root_witness)."},"skills":["A Complete Root Table Distinguishes Symbol Plus One from Minus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0040","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2cba1a9572ef0e9eb0b1e6f3e83796ddbd875065b131910dc84f0d737795b8c7","family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126040,"semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":15,\"prime\":53}"},"fields":{"title":"A Complete Root Table Distinguishes Symbol Plus One from Minus One","stem":"Use odd prime p=53, original a=15. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).","answer":{"kind":"expression","value":"(53, 15, 15, [11, 42], 1, complete_nonzero_square_root_witness)","explanation":"Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (53, 15, 15, [11, 42], 1, complete_nonzero_square_root_witness)."},"skills":["A Complete Root Table Distinguishes Symbol Plus One from Minus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0041","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0280e04ae029e66f4773854450ff8da49fc1ac03327f1b77637cdd0e5fa49a8","family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126041,"semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":16,\"prime\":59}"},"fields":{"title":"A Complete Root Table Distinguishes Symbol Plus One from Minus One","stem":"Use odd prime p=59, original a=16. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).","answer":{"kind":"expression","value":"(59, 16, 16, [4, 55], 1, complete_nonzero_square_root_witness)","explanation":"Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (59, 16, 16, [4, 55], 1, complete_nonzero_square_root_witness)."},"skills":["A Complete Root Table Distinguishes Symbol Plus One from Minus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0042","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7d814023f9549fe91bb4a6732b659d35661cc1187139518f8cea14ce5e1874d","family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126042,"semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":17,\"prime\":61}"},"fields":{"title":"A Complete Root Table Distinguishes Symbol Plus One from Minus One","stem":"Use odd prime p=61, original a=17. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).","answer":{"kind":"expression","value":"(61, 17, 17, [], -1, complete_square_table_has_no_root)","explanation":"Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (61, 17, 17, [], -1, complete_square_table_has_no_root)."},"skills":["A Complete Root Table Distinguishes Symbol Plus One from Minus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0043","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b8ed65b07eed7cf69bc6cbd95b05f56d22fdebc41f99c3651107da58267a716","family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126043,"semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":18,\"prime\":67}"},"fields":{"title":"A Complete Root Table Distinguishes Symbol Plus One from Minus One","stem":"Use odd prime p=67, original a=18. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).","answer":{"kind":"expression","value":"(67, 18, 18, [], -1, complete_square_table_has_no_root)","explanation":"Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (67, 18, 18, [], -1, complete_square_table_has_no_root)."},"skills":["A Complete Root Table Distinguishes Symbol Plus One from Minus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0044","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"87bb1ae097bda60ba4d4cb5e39d9c973b348329628a91e282bf4372246bee78d","family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126044,"semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":19,\"prime\":71}"},"fields":{"title":"A Complete Root Table Distinguishes Symbol Plus One from Minus One","stem":"Use odd prime p=71, original a=19. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).","answer":{"kind":"expression","value":"(71, 19, 19, [27, 44], 1, complete_nonzero_square_root_witness)","explanation":"Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (71, 19, 19, [27, 44], 1, complete_nonzero_square_root_witness)."},"skills":["A Complete Root Table Distinguishes Symbol Plus One from Minus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0045","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95341c232855c896b0c1976bb2650159e3efe3521300bee7d5f6dce7dbc149ec","family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126045,"semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":20,\"prime\":73}"},"fields":{"title":"A Complete Root Table Distinguishes Symbol Plus One from Minus One","stem":"Use odd prime p=73, original a=20. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).","answer":{"kind":"expression","value":"(73, 20, 20, [], -1, complete_square_table_has_no_root)","explanation":"Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (73, 20, 20, [], -1, complete_square_table_has_no_root)."},"skills":["A Complete Root Table Distinguishes Symbol Plus One from Minus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0046","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e21a8d04ef6bdcf87542b383d4228b5a2a2084282a8d3b697d712ae5db0266ec","family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126046,"semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":21,\"prime\":79}"},"fields":{"title":"A Complete Root Table Distinguishes Symbol Plus One from Minus One","stem":"Use odd prime p=79, original a=21. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).","answer":{"kind":"expression","value":"(79, 21, 21, [10, 69], 1, complete_nonzero_square_root_witness)","explanation":"Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (79, 21, 21, [10, 69], 1, complete_nonzero_square_root_witness)."},"skills":["A Complete Root Table Distinguishes Symbol Plus One from Minus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0047","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b27d495de8af23c1ba9b72bc7033e41ecc2621389d603b5023507ca367037724","family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126047,"semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":22,\"prime\":83}"},"fields":{"title":"A Complete Root Table Distinguishes Symbol Plus One from Minus One","stem":"Use odd prime p=83, original a=22. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).","answer":{"kind":"expression","value":"(83, 22, 22, [], -1, complete_square_table_has_no_root)","explanation":"Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (83, 22, 22, [], -1, complete_square_table_has_no_root)."},"skills":["A Complete Root Table Distinguishes Symbol Plus One from Minus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0048","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef2b5189f2e1644f7b6b763160f7f15350b1a5f5f9721619206ef7debb05274a","family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126048,"semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":23,\"prime\":89}"},"fields":{"title":"A Complete Root Table Distinguishes Symbol Plus One from Minus One","stem":"Use odd prime p=89, original a=23. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).","answer":{"kind":"expression","value":"(89, 23, 23, [], -1, complete_square_table_has_no_root)","explanation":"Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (89, 23, 23, [], -1, complete_square_table_has_no_root)."},"skills":["A Complete Root Table Distinguishes Symbol Plus One from Minus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0049","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"becad5fb82edf8296080417c26b4f0a172b9b7295bef48ee07cdcca9cb83c8d0","family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126049,"semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":24,\"prime\":97}"},"fields":{"title":"A Complete Root Table Distinguishes Symbol Plus One from Minus One","stem":"Use odd prime p=97, original a=24. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).","answer":{"kind":"expression","value":"(97, 24, 24, [11, 86], 1, complete_nonzero_square_root_witness)","explanation":"Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (97, 24, 24, [11, 86], 1, complete_nonzero_square_root_witness)."},"skills":["A Complete Root Table Distinguishes Symbol Plus One from Minus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0050","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1d57a507b105b65dae058f73c019e81ecab825e98ff5ec912bf14665e20f3c8","family_id":"qf-mod43-legendre-residue-nonresidue-witness","seed":202611126050,"semantic_template":"qf-mod43-legendre-residue-nonresidue-witness/{\"a\":25,\"prime\":101}"},"fields":{"title":"A Complete Root Table Distinguishes Symbol Plus One from Minus One","stem":"Use odd prime p=101, original a=25. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).","answer":{"kind":"expression","value":"(101, 25, 25, [5, 96], 1, complete_nonzero_square_root_witness)","explanation":"Normalize a and require a nonzero class. List all nonzero square roots. Their presence gives symbol1; complete absence gives−1. The zero class has a separate branch, and no later Euler criterion is used here. The exact result, retaining the requested order, is (101, 25, 25, [5, 96], 1, complete_nonzero_square_root_witness)."},"skills":["A Complete Root Table Distinguishes Symbol Plus One from Minus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0051","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a1905d399040e114f0e327416072533a339c48925133eb5ceadea8d970d4255","family_id":"qf-mod43-legendre-zero-class","seed":202611126051,"semantic_template":"qf-mod43-legendre-zero-class/{\"a\":3,\"prime\":3}"},"fields":{"title":"Divisibility by p Gives Symbol Zero, Not Plus One","stem":"Use odd prime p=3, original a=3. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).","answer":{"kind":"expression","value":"(3, 3, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation":"Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (3, 3, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)."},"skills":["Divisibility by p Gives Symbol Zero, Not Plus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0052","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23e395b71f4d7357b436aea1fbbe79018aee96e83fe3d2f8270420d5f2e087bf","family_id":"qf-mod43-legendre-zero-class","seed":202611126052,"semantic_template":"qf-mod43-legendre-zero-class/{\"a\":10,\"prime\":5}"},"fields":{"title":"Divisibility by p Gives Symbol Zero, Not Plus One","stem":"Use odd prime p=5, original a=10. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).","answer":{"kind":"expression","value":"(5, 10, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation":"Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (5, 10, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)."},"skills":["Divisibility by p Gives Symbol Zero, Not Plus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0053","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c921b719fafe0540f9650e18af44d8a80c1af94d1e83a0dde02bbab6c6a788ee","family_id":"qf-mod43-legendre-zero-class","seed":202611126053,"semantic_template":"qf-mod43-legendre-zero-class/{\"a\":21,\"prime\":7}"},"fields":{"title":"Divisibility by p Gives Symbol Zero, Not Plus One","stem":"Use odd prime p=7, original a=21. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).","answer":{"kind":"expression","value":"(7, 21, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation":"Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (7, 21, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)."},"skills":["Divisibility by p Gives Symbol Zero, Not Plus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0054","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9fe97e8d9e9c66d7e8ded938a46c2e9d194ff441a9abb9e1984e0408f316883d","family_id":"qf-mod43-legendre-zero-class","seed":202611126054,"semantic_template":"qf-mod43-legendre-zero-class/{\"a\":44,\"prime\":11}"},"fields":{"title":"Divisibility by p Gives Symbol Zero, Not Plus One","stem":"Use odd prime p=11, original a=44. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).","answer":{"kind":"expression","value":"(11, 44, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation":"Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (11, 44, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)."},"skills":["Divisibility by p Gives Symbol Zero, Not Plus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0055","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2085b430e07d5686553fa5f4eaa9b530d1aa5cae98c9482d5bc71c23d7d93f84","family_id":"qf-mod43-legendre-zero-class","seed":202611126055,"semantic_template":"qf-mod43-legendre-zero-class/{\"a\":65,\"prime\":13}"},"fields":{"title":"Divisibility by p Gives Symbol Zero, Not Plus One","stem":"Use odd prime p=13, original a=65. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).","answer":{"kind":"expression","value":"(13, 65, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation":"Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (13, 65, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)."},"skills":["Divisibility by p Gives Symbol Zero, Not Plus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0056","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5228a14281380681d1a751803a21c35295a4b5d0097605a91a822c62c621412b","family_id":"qf-mod43-legendre-zero-class","seed":202611126056,"semantic_template":"qf-mod43-legendre-zero-class/{\"a\":102,\"prime\":17}"},"fields":{"title":"Divisibility by p Gives Symbol Zero, Not Plus One","stem":"Use odd prime p=17, original a=102. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).","answer":{"kind":"expression","value":"(17, 102, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation":"Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (17, 102, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)."},"skills":["Divisibility by p Gives Symbol Zero, Not Plus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0057","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f82e9bee58cadeb16fdc561118097d3e2e9e4a1b55e373e1bfa0a8153cf0cbd8","family_id":"qf-mod43-legendre-zero-class","seed":202611126057,"semantic_template":"qf-mod43-legendre-zero-class/{\"a\":133,\"prime\":19}"},"fields":{"title":"Divisibility by p Gives Symbol Zero, Not Plus One","stem":"Use odd prime p=19, original a=133. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).","answer":{"kind":"expression","value":"(19, 133, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation":"Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (19, 133, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)."},"skills":["Divisibility by p Gives Symbol Zero, Not Plus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0058","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d28edd2b285314ce42021864f1e4b830ef4648add3a167d82dbc4e07c2bb1eb","family_id":"qf-mod43-legendre-zero-class","seed":202611126058,"semantic_template":"qf-mod43-legendre-zero-class/{\"a\":184,\"prime\":23}"},"fields":{"title":"Divisibility by p Gives Symbol Zero, Not Plus One","stem":"Use odd prime p=23, original a=184. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).","answer":{"kind":"expression","value":"(23, 184, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation":"Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (23, 184, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)."},"skills":["Divisibility by p Gives Symbol Zero, Not Plus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0059","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b74cf56579fbb9736c24ac3d86ccebed49bfb604839e93c99d2b07e58abbe2f","family_id":"qf-mod43-legendre-zero-class","seed":202611126059,"semantic_template":"qf-mod43-legendre-zero-class/{\"a\":261,\"prime\":29}"},"fields":{"title":"Divisibility by p Gives Symbol Zero, Not Plus One","stem":"Use odd prime p=29, original a=261. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).","answer":{"kind":"expression","value":"(29, 261, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation":"Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (29, 261, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)."},"skills":["Divisibility by p Gives Symbol Zero, Not Plus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0060","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5dc4943bbbd7522e7b5350b1b9c469a8026a5cc915dc40a7261b25c84cd2a64","family_id":"qf-mod43-legendre-zero-class","seed":202611126060,"semantic_template":"qf-mod43-legendre-zero-class/{\"a\":310,\"prime\":31}"},"fields":{"title":"Divisibility by p Gives Symbol Zero, Not Plus One","stem":"Use odd prime p=31, original a=310. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).","answer":{"kind":"expression","value":"(31, 310, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation":"Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (31, 310, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)."},"skills":["Divisibility by p Gives Symbol Zero, Not Plus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0061","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43bba25cc667a3c06129a301ee773746a2296a0fd051dda50cb2d825b1ff7cf9","family_id":"qf-mod43-legendre-zero-class","seed":202611126061,"semantic_template":"qf-mod43-legendre-zero-class/{\"a\":407,\"prime\":37}"},"fields":{"title":"Divisibility by p Gives Symbol Zero, Not Plus One","stem":"Use odd prime p=37, original a=407. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).","answer":{"kind":"expression","value":"(37, 407, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation":"Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (37, 407, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)."},"skills":["Divisibility by p Gives Symbol Zero, Not Plus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0062","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"388b3a4a36622cf2b0fcfe871add6ab7443b46dbedb0359c89798668a27a5add","family_id":"qf-mod43-legendre-zero-class","seed":202611126062,"semantic_template":"qf-mod43-legendre-zero-class/{\"a\":492,\"prime\":41}"},"fields":{"title":"Divisibility by p Gives Symbol Zero, Not Plus One","stem":"Use odd prime p=41, original a=492. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).","answer":{"kind":"expression","value":"(41, 492, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation":"Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (41, 492, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)."},"skills":["Divisibility by p Gives Symbol Zero, Not Plus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0063","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"770ac14d25b4de2b516c3a93622a466e812229329cdb9765ba4f9ae3d20b90f8","family_id":"qf-mod43-legendre-zero-class","seed":202611126063,"semantic_template":"qf-mod43-legendre-zero-class/{\"a\":559,\"prime\":43}"},"fields":{"title":"Divisibility by p Gives Symbol Zero, Not Plus One","stem":"Use odd prime p=43, original a=559. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).","answer":{"kind":"expression","value":"(43, 559, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation":"Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (43, 559, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)."},"skills":["Divisibility by p Gives Symbol Zero, Not Plus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0064","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8099cf413de353bd2267ce872539ff406ba37d1cdadcf63aea6f2c751fba838e","family_id":"qf-mod43-legendre-zero-class","seed":202611126064,"semantic_template":"qf-mod43-legendre-zero-class/{\"a\":658,\"prime\":47}"},"fields":{"title":"Divisibility by p Gives Symbol Zero, Not Plus One","stem":"Use odd prime p=47, original a=658. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).","answer":{"kind":"expression","value":"(47, 658, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation":"Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (47, 658, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)."},"skills":["Divisibility by p Gives Symbol Zero, Not Plus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0065","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4979ef28a5a2070fd0298ad896eb514fd3d781b97f9ff78281444039f0242a61","family_id":"qf-mod43-legendre-zero-class","seed":202611126065,"semantic_template":"qf-mod43-legendre-zero-class/{\"a\":795,\"prime\":53}"},"fields":{"title":"Divisibility by p Gives Symbol Zero, Not Plus One","stem":"Use odd prime p=53, original a=795. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).","answer":{"kind":"expression","value":"(53, 795, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation":"Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (53, 795, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)."},"skills":["Divisibility by p Gives Symbol Zero, Not Plus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0066","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"938d92d347fd86808082d237044abad27887d9aa99912c4700be249359398c51","family_id":"qf-mod43-legendre-zero-class","seed":202611126066,"semantic_template":"qf-mod43-legendre-zero-class/{\"a\":944,\"prime\":59}"},"fields":{"title":"Divisibility by p Gives Symbol Zero, Not Plus One","stem":"Use odd prime p=59, original a=944. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).","answer":{"kind":"expression","value":"(59, 944, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation":"Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (59, 944, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)."},"skills":["Divisibility by p Gives Symbol Zero, Not Plus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0067","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90f85041250ee3eb0687051e62a490a9da4d499a29e0560bf8ec76403e89810e","family_id":"qf-mod43-legendre-zero-class","seed":202611126067,"semantic_template":"qf-mod43-legendre-zero-class/{\"a\":1037,\"prime\":61}"},"fields":{"title":"Divisibility by p Gives Symbol Zero, Not Plus One","stem":"Use odd prime p=61, original a=1037. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).","answer":{"kind":"expression","value":"(61, 1037, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation":"Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (61, 1037, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)."},"skills":["Divisibility by p Gives Symbol Zero, Not Plus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0068","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"628391cbec089a6986cf9cf05433339bcc6e76c84f631ac5f478f742809e3f3e","family_id":"qf-mod43-legendre-zero-class","seed":202611126068,"semantic_template":"qf-mod43-legendre-zero-class/{\"a\":1206,\"prime\":67}"},"fields":{"title":"Divisibility by p Gives Symbol Zero, Not Plus One","stem":"Use odd prime p=67, original a=1206. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).","answer":{"kind":"expression","value":"(67, 1206, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation":"Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (67, 1206, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)."},"skills":["Divisibility by p Gives Symbol Zero, Not Plus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0069","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"791338f6d8536eb382c684ae6ac1107463da752984f032b1686e14a5d7461e7f","family_id":"qf-mod43-legendre-zero-class","seed":202611126069,"semantic_template":"qf-mod43-legendre-zero-class/{\"a\":1349,\"prime\":71}"},"fields":{"title":"Divisibility by p Gives Symbol Zero, Not Plus One","stem":"Use odd prime p=71, original a=1349. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).","answer":{"kind":"expression","value":"(71, 1349, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation":"Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (71, 1349, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)."},"skills":["Divisibility by p Gives Symbol Zero, Not Plus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0070","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3e013b7fe2d8591a16a2fcdf947e9e2b14b50d4cdb0e086480a6c42898d439a","family_id":"qf-mod43-legendre-zero-class","seed":202611126070,"semantic_template":"qf-mod43-legendre-zero-class/{\"a\":1460,\"prime\":73}"},"fields":{"title":"Divisibility by p Gives Symbol Zero, Not Plus One","stem":"Use odd prime p=73, original a=1460. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).","answer":{"kind":"expression","value":"(73, 1460, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation":"Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (73, 1460, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)."},"skills":["Divisibility by p Gives Symbol Zero, Not Plus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0071","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf58ec0724492a10c73f922ef4aa90943def8c003302eb72bcc5bdc1ab320fe8","family_id":"qf-mod43-legendre-zero-class","seed":202611126071,"semantic_template":"qf-mod43-legendre-zero-class/{\"a\":1659,\"prime\":79}"},"fields":{"title":"Divisibility by p Gives Symbol Zero, Not Plus One","stem":"Use odd prime p=79, original a=1659. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).","answer":{"kind":"expression","value":"(79, 1659, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation":"Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (79, 1659, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)."},"skills":["Divisibility by p Gives Symbol Zero, Not Plus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0072","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4d92d9537961193989759379bf94431c01aaba114dc5d88a51478d7b1f66419","family_id":"qf-mod43-legendre-zero-class","seed":202611126072,"semantic_template":"qf-mod43-legendre-zero-class/{\"a\":1826,\"prime\":83}"},"fields":{"title":"Divisibility by p Gives Symbol Zero, Not Plus One","stem":"Use odd prime p=83, original a=1826. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).","answer":{"kind":"expression","value":"(83, 1826, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation":"Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (83, 1826, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)."},"skills":["Divisibility by p Gives Symbol Zero, Not Plus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0073","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2921ff2969ee650d4b703d843227fa7e1cabbaa247ab4fb45ffe9555f7841bd4","family_id":"qf-mod43-legendre-zero-class","seed":202611126073,"semantic_template":"qf-mod43-legendre-zero-class/{\"a\":2047,\"prime\":89}"},"fields":{"title":"Divisibility by p Gives Symbol Zero, Not Plus One","stem":"Use odd prime p=89, original a=2047. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).","answer":{"kind":"expression","value":"(89, 2047, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation":"Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (89, 2047, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)."},"skills":["Divisibility by p Gives Symbol Zero, Not Plus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0074","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1891178506ad6197268c3dfcbe7076cca8efdda81e48c515134063bcec7ad370","family_id":"qf-mod43-legendre-zero-class","seed":202611126074,"semantic_template":"qf-mod43-legendre-zero-class/{\"a\":2328,\"prime\":97}"},"fields":{"title":"Divisibility by p Gives Symbol Zero, Not Plus One","stem":"Use odd prime p=97, original a=2328. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).","answer":{"kind":"expression","value":"(97, 2328, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation":"Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (97, 2328, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)."},"skills":["Divisibility by p Gives Symbol Zero, Not Plus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0075","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4aaa96a982d76f0290a3430ce4076822f173b523c603f99655f7aa539cfd63d6","family_id":"qf-mod43-legendre-zero-class","seed":202611126075,"semantic_template":"qf-mod43-legendre-zero-class/{\"a\":2525,\"prime\":101}"},"fields":{"title":"Divisibility by p Gives Symbol Zero, Not Plus One","stem":"Use odd prime p=101, original a=2525. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).","answer":{"kind":"expression","value":"(101, 2525, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)","explanation":"Require a mod p=0 and report Legendre symbol0 before residue/nonresidue classification. The nonzero-root list is empty; this does not deny that0 itself squares to0. The exact result, retaining the requested order, is (101, 2525, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)."},"skills":["Divisibility by p Gives Symbol Zero, Not Plus One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0076","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1cfad8383d60aa22d52632185fa4ac2e4fc207ffc98de6a32147f2e39e40740c","family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126076,"semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":1,\"mode\":\"even_prime\",\"modulus\":2}"},"fields":{"title":"Domain Extensions and Premature Methods Are Different Legendre Boundaries","stem":"Use mode=even_prime, modulus=2, original a=1. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).","answer":{"kind":"expression","value":"(even_prime, 2, 1, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)","explanation":"Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (even_prime, 2, 1, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)."},"skills":["Domain Extensions and Premature Methods Are Different Legendre Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0077","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"871bdb1bee54b2212fb50d96ee6d364cc4e439f058b39f848419b8a6b4152c89","family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126077,"semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":2,\"mode\":\"composite_modulus\",\"modulus\":9}"},"fields":{"title":"Domain Extensions and Premature Methods Are Different Legendre Boundaries","stem":"Use mode=composite_modulus, modulus=9, original a=2. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).","answer":{"kind":"expression","value":"(composite_modulus, 9, 2, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)","explanation":"Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (composite_modulus, 9, 2, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)."},"skills":["Domain Extensions and Premature Methods Are Different Legendre Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0078","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"370ff6d5bb4c4cdc46ee039f6b3a77b1346a18beba0de9ecabbc7b3f78499dab","family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126078,"semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":1,\"mode\":\"premature_euler\",\"modulus\":7}"},"fields":{"title":"Domain Extensions and Premature Methods Are Different Legendre Boundaries","stem":"Use mode=premature_euler, modulus=7, original a=1. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).","answer":{"kind":"expression","value":"(premature_euler, 7, 1, No, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion)","explanation":"Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (premature_euler, 7, 1, No, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion)."},"skills":["Domain Extensions and Premature Methods Are Different Legendre Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0079","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c429183f2b29aad2370374d60d59e1321e8dbbc1ee2cea974a78d344897023a","family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126079,"semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":4,\"mode\":\"even_prime\",\"modulus\":2}"},"fields":{"title":"Domain Extensions and Premature Methods Are Different Legendre Boundaries","stem":"Use mode=even_prime, modulus=2, original a=4. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).","answer":{"kind":"expression","value":"(even_prime, 2, 0, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)","explanation":"Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (even_prime, 2, 0, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)."},"skills":["Domain Extensions and Premature Methods Are Different Legendre Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0080","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"261d8a91adb94ea8e1f54641b027cbc3d0b17b3a1495ce2e071d4c9a3e94e24c","family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126080,"semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":5,\"mode\":\"composite_modulus\",\"modulus\":15}"},"fields":{"title":"Domain Extensions and Premature Methods Are Different Legendre Boundaries","stem":"Use mode=composite_modulus, modulus=15, original a=5. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).","answer":{"kind":"expression","value":"(composite_modulus, 15, 5, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)","explanation":"Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (composite_modulus, 15, 5, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)."},"skills":["Domain Extensions and Premature Methods Are Different Legendre Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0081","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b40f1c41e0f25a4c4f52e041c5e7658b2dfafaeb58228e079d378bd35a5f7b7b","family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126081,"semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":1,\"mode\":\"premature_euler\",\"modulus\":17}"},"fields":{"title":"Domain Extensions and Premature Methods Are Different Legendre Boundaries","stem":"Use mode=premature_euler, modulus=17, original a=1. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).","answer":{"kind":"expression","value":"(premature_euler, 17, 1, No, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion)","explanation":"Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (premature_euler, 17, 1, No, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion)."},"skills":["Domain Extensions and Premature Methods Are Different Legendre Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0082","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d22ef9d61b2ac8e6dc22ebaee4ecf1eb7a4258d3629972da7379cd7362eba571","family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126082,"semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":7,\"mode\":\"even_prime\",\"modulus\":2}"},"fields":{"title":"Domain Extensions and Premature Methods Are Different Legendre Boundaries","stem":"Use mode=even_prime, modulus=2, original a=7. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).","answer":{"kind":"expression","value":"(even_prime, 2, 1, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)","explanation":"Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (even_prime, 2, 1, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)."},"skills":["Domain Extensions and Premature Methods Are Different Legendre Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0083","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f68578d413c867d942ec7668cb858360abb5349498285c498c6805faa42f072","family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126083,"semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":8,\"mode\":\"composite_modulus\",\"modulus\":21}"},"fields":{"title":"Domain Extensions and Premature Methods Are Different Legendre Boundaries","stem":"Use mode=composite_modulus, modulus=21, original a=8. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).","answer":{"kind":"expression","value":"(composite_modulus, 21, 8, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)","explanation":"Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (composite_modulus, 21, 8, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)."},"skills":["Domain Extensions and Premature Methods Are Different Legendre Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0084","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f47f38a68dcf5a2093b90ae68bd5ba06e6fb20caa128e345c20dfbd3cc40d23d","family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126084,"semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":1,\"mode\":\"premature_euler\",\"modulus\":29}"},"fields":{"title":"Domain Extensions and Premature Methods Are Different Legendre Boundaries","stem":"Use mode=premature_euler, modulus=29, original a=1. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).","answer":{"kind":"expression","value":"(premature_euler, 29, 1, No, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion)","explanation":"Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (premature_euler, 29, 1, No, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion)."},"skills":["Domain Extensions and Premature Methods Are Different Legendre Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0085","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e1fb50f8bc80cbfdcdc634d0ee41e44a23d72ca3d83a7b7c1b6a7055858e29b","family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126085,"semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":10,\"mode\":\"even_prime\",\"modulus\":2}"},"fields":{"title":"Domain Extensions and Premature Methods Are Different Legendre Boundaries","stem":"Use mode=even_prime, modulus=2, original a=10. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).","answer":{"kind":"expression","value":"(even_prime, 2, 0, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)","explanation":"Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (even_prime, 2, 0, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)."},"skills":["Domain Extensions and Premature Methods Are Different Legendre Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0086","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cbb0036a28f0c642a44104ba7ab7eead57c6b5ff5348f9e6c7a77510cc6a9de8","family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126086,"semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":11,\"mode\":\"composite_modulus\",\"modulus\":27}"},"fields":{"title":"Domain Extensions and Premature Methods Are Different Legendre Boundaries","stem":"Use mode=composite_modulus, modulus=27, original a=11. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).","answer":{"kind":"expression","value":"(composite_modulus, 27, 11, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)","explanation":"Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (composite_modulus, 27, 11, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)."},"skills":["Domain Extensions and Premature Methods Are Different Legendre Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0087","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b68aa287baed8821020704281d01c446a9e71f3cd55e08449f94a822f9feabc9","family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126087,"semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":1,\"mode\":\"premature_euler\",\"modulus\":41}"},"fields":{"title":"Domain Extensions and Premature Methods Are Different Legendre Boundaries","stem":"Use mode=premature_euler, modulus=41, original a=1. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).","answer":{"kind":"expression","value":"(premature_euler, 41, 1, No, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion)","explanation":"Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (premature_euler, 41, 1, No, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion)."},"skills":["Domain Extensions and Premature Methods Are Different Legendre Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0088","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe6cc1fbe76dcda5f221024bda21d1f0d481ddb062ad8c3600f590425bc515c1","family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126088,"semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":13,\"mode\":\"even_prime\",\"modulus\":2}"},"fields":{"title":"Domain Extensions and Premature Methods Are Different Legendre Boundaries","stem":"Use mode=even_prime, modulus=2, original a=13. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).","answer":{"kind":"expression","value":"(even_prime, 2, 1, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)","explanation":"Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (even_prime, 2, 1, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)."},"skills":["Domain Extensions and Premature Methods Are Different Legendre Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0089","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a37a861696d29767f752f788ca625a2c931103270ba26d38b8a8a778d5048136","family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126089,"semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":14,\"mode\":\"composite_modulus\",\"modulus\":33}"},"fields":{"title":"Domain Extensions and Premature Methods Are Different Legendre Boundaries","stem":"Use mode=composite_modulus, modulus=33, original a=14. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).","answer":{"kind":"expression","value":"(composite_modulus, 33, 14, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)","explanation":"Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (composite_modulus, 33, 14, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)."},"skills":["Domain Extensions and Premature Methods Are Different Legendre Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0090","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c9476b96f63dd769a7ca387e93f6d18fbb5b2fb534119464a40b5997d226d54","family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126090,"semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":1,\"mode\":\"premature_euler\",\"modulus\":53}"},"fields":{"title":"Domain Extensions and Premature Methods Are Different Legendre Boundaries","stem":"Use mode=premature_euler, modulus=53, original a=1. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).","answer":{"kind":"expression","value":"(premature_euler, 53, 1, No, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion)","explanation":"Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (premature_euler, 53, 1, No, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion)."},"skills":["Domain Extensions and Premature Methods Are Different Legendre Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0091","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c61681b6ce17b49238c745778f5809091d3daa17c893d30499c1d0d3e5fd010","family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126091,"semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":16,\"mode\":\"even_prime\",\"modulus\":2}"},"fields":{"title":"Domain Extensions and Premature Methods Are Different Legendre Boundaries","stem":"Use mode=even_prime, modulus=2, original a=16. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).","answer":{"kind":"expression","value":"(even_prime, 2, 0, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)","explanation":"Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (even_prime, 2, 0, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)."},"skills":["Domain Extensions and Premature Methods Are Different Legendre Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0092","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dcac3d0b8f516155a851b117b569c11bf9b86fd40fb377f907e208cfb00257a6","family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126092,"semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":17,\"mode\":\"composite_modulus\",\"modulus\":39}"},"fields":{"title":"Domain Extensions and Premature Methods Are Different Legendre Boundaries","stem":"Use mode=composite_modulus, modulus=39, original a=17. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).","answer":{"kind":"expression","value":"(composite_modulus, 39, 17, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)","explanation":"Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (composite_modulus, 39, 17, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)."},"skills":["Domain Extensions and Premature Methods Are Different Legendre Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0093","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd00ce04c65da636268263537da286459771f03ac4ac4e8df67b7335ca5b1be8","family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126093,"semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":1,\"mode\":\"premature_euler\",\"modulus\":67}"},"fields":{"title":"Domain Extensions and Premature Methods Are Different Legendre Boundaries","stem":"Use mode=premature_euler, modulus=67, original a=1. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).","answer":{"kind":"expression","value":"(premature_euler, 67, 1, No, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion)","explanation":"Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (premature_euler, 67, 1, No, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion)."},"skills":["Domain Extensions and Premature Methods Are Different Legendre Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0094","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55ca95fb593db03721679e465b3c7e2da3a4516bd2ab77f11a3cfdf90b61e11b","family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126094,"semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":19,\"mode\":\"even_prime\",\"modulus\":2}"},"fields":{"title":"Domain Extensions and Premature Methods Are Different Legendre Boundaries","stem":"Use mode=even_prime, modulus=2, original a=19. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).","answer":{"kind":"expression","value":"(even_prime, 2, 1, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)","explanation":"Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (even_prime, 2, 1, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)."},"skills":["Domain Extensions and Premature Methods Are Different Legendre Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0095","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0933eac609229880ec88fa612c06c4e74768526d7055d96bbc808eb83adb2d2","family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126095,"semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":20,\"mode\":\"composite_modulus\",\"modulus\":45}"},"fields":{"title":"Domain Extensions and Premature Methods Are Different Legendre Boundaries","stem":"Use mode=composite_modulus, modulus=45, original a=20. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).","answer":{"kind":"expression","value":"(composite_modulus, 45, 20, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)","explanation":"Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (composite_modulus, 45, 20, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)."},"skills":["Domain Extensions and Premature Methods Are Different Legendre Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0096","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ac26b5a483aaf521b57ebc187bfa64118ac3e2d3a14078f22c851e46e834218","family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126096,"semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":1,\"mode\":\"premature_euler\",\"modulus\":79}"},"fields":{"title":"Domain Extensions and Premature Methods Are Different Legendre Boundaries","stem":"Use mode=premature_euler, modulus=79, original a=1. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).","answer":{"kind":"expression","value":"(premature_euler, 79, 1, No, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion)","explanation":"Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (premature_euler, 79, 1, No, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion)."},"skills":["Domain Extensions and Premature Methods Are Different Legendre Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0097","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2109b0a0a08214661f207e18a2ca0c7f5e37d2e70bf2628e9ca6f1baa830cb6c","family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126097,"semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":22,\"mode\":\"even_prime\",\"modulus\":2}"},"fields":{"title":"Domain Extensions and Premature Methods Are Different Legendre Boundaries","stem":"Use mode=even_prime, modulus=2, original a=22. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).","answer":{"kind":"expression","value":"(even_prime, 2, 0, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)","explanation":"Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (even_prime, 2, 0, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)."},"skills":["Domain Extensions and Premature Methods Are Different Legendre Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0098","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8360e6b76d19b174ab4a026514ed97028f6c5a172b06e3e5e9d64359aa0e86f3","family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126098,"semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":23,\"mode\":\"composite_modulus\",\"modulus\":51}"},"fields":{"title":"Domain Extensions and Premature Methods Are Different Legendre Boundaries","stem":"Use mode=composite_modulus, modulus=51, original a=23. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).","answer":{"kind":"expression","value":"(composite_modulus, 51, 23, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)","explanation":"Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (composite_modulus, 51, 23, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)."},"skills":["Domain Extensions and Premature Methods Are Different Legendre Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0099","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7b42b55abda72167dc852545e6c3fbfd0cdd94cbdf088b7595055310e7dc0cf","family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126099,"semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":1,\"mode\":\"premature_euler\",\"modulus\":97}"},"fields":{"title":"Domain Extensions and Premature Methods Are Different Legendre Boundaries","stem":"Use mode=premature_euler, modulus=97, original a=1. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).","answer":{"kind":"expression","value":"(premature_euler, 97, 1, No, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion)","explanation":"Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (premature_euler, 97, 1, No, classification_must_come_from_the_complete_square_table_not_the_later_Euler_criterion)."},"skills":["Domain Extensions and Premature Methods Are Different Legendre Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r06-legendre-symbol-0100","batch_id":"batch-20261014-advance-20261112-r06-legendre-symbol-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61bdb26d0fdc27d5877b6f32b578d3daa6c27c6e9cd636b97d288006af78949c","family_id":"qf-mod43-legendre-domain-method-boundary","seed":202611126100,"semantic_template":"qf-mod43-legendre-domain-method-boundary/{\"a\":25,\"mode\":\"even_prime\",\"modulus\":2}"},"fields":{"title":"Domain Extensions and Premature Methods Are Different Legendre Boundaries","stem":"Use mode=even_prime, modulus=2, original a=25. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).","answer":{"kind":"expression","value":"(even_prime, 2, 1, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)","explanation":"Modulus2 or a composite modulus does not meet this Legendre domain, and no Jacobi extension is silently adopted. premature_euler instead has a valid odd-prime nonzero input but withholds a certificate from a not-yet-established method. Its No neither refutes Euler criterion nor calls that valid input out of domain. The exact result, retaining the requested order, is (even_prime, 2, 1, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)."},"skills":["Domain Extensions and Premature Methods Are Different Legendre Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}