{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0001","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21c41f444f135aabe4578d1fd3e5a29fd399fe6b6af0dcfa08ab49c638e7b6a0","family_id":"qf-mod23-complexmod-conjugate","seed":202610231001,"semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[-12,-3]}"},"fields":{"title":"Conjugation of an Ordered Complex Pair","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-12, -3]. Give (value,conjugate).","answer":{"kind":"expression","value":"([-12, -3], [-12, 3])","explanation":"Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-12, -3], [-12, 3])."},"skills":["Conjugation of an Ordered Complex Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0002","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7c7ba38b116eb681faa2e710b1d8fa79196dc1d77b3f6c3792cf934f2db03024","family_id":"qf-mod23-complexmod-conjugate","seed":202610231002,"semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[-11,-2]}"},"fields":{"title":"Conjugation of an Ordered Complex Pair","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-11, -2]. Give (value,conjugate).","answer":{"kind":"expression","value":"([-11, -2], [-11, 2])","explanation":"Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-11, -2], [-11, 2])."},"skills":["Conjugation of an Ordered Complex Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0003","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3197b8b879336c62b3af4428e3dabe08274b5d9e93a6ca3db753cdb560d30a80","family_id":"qf-mod23-complexmod-conjugate","seed":202610231003,"semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[-10,-1]}"},"fields":{"title":"Conjugation of an Ordered Complex Pair","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-10, -1]. Give (value,conjugate).","answer":{"kind":"expression","value":"([-10, -1], [-10, 1])","explanation":"Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-10, -1], [-10, 1])."},"skills":["Conjugation of an Ordered Complex Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0004","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c376cbd9872f0570f21529309a4347f039bb809a09f076986d980b52ec156f7e","family_id":"qf-mod23-complexmod-conjugate","seed":202610231004,"semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[-9,0]}"},"fields":{"title":"Conjugation of an Ordered Complex Pair","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-9, 0]. Give (value,conjugate).","answer":{"kind":"expression","value":"([-9, 0], [-9, 0])","explanation":"Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 0], [-9, 0])."},"skills":["Conjugation of an Ordered Complex Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0005","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51809c3fd9986f184b8cf1e0da9de21d9d13d14c4e9c8972819e90f484bfb784","family_id":"qf-mod23-complexmod-conjugate","seed":202610231005,"semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[-8,1]}"},"fields":{"title":"Conjugation of an Ordered Complex Pair","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-8, 1]. Give (value,conjugate).","answer":{"kind":"expression","value":"([-8, 1], [-8, -1])","explanation":"Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 1], [-8, -1])."},"skills":["Conjugation of an Ordered Complex Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0006","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48f53b56770555946bfe8e1117fe3800e821b43c806c17ca49f24f977643511e","family_id":"qf-mod23-complexmod-conjugate","seed":202610231006,"semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[-7,2]}"},"fields":{"title":"Conjugation of an Ordered Complex Pair","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-7, 2]. Give (value,conjugate).","answer":{"kind":"expression","value":"([-7, 2], [-7, -2])","explanation":"Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-7, 2], [-7, -2])."},"skills":["Conjugation of an Ordered Complex Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0007","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8292879cf44f30e3881ceefcb830351901815115f45ecb6d124c4d6f78365bf","family_id":"qf-mod23-complexmod-conjugate","seed":202610231007,"semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[-6,3]}"},"fields":{"title":"Conjugation of an Ordered Complex Pair","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-6, 3]. Give (value,conjugate).","answer":{"kind":"expression","value":"([-6, 3], [-6, -3])","explanation":"Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-6, 3], [-6, -3])."},"skills":["Conjugation of an Ordered Complex Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0008","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6986767af4b6cf5ed32ee7ab11fd98fbf4856f6e6b5d076691e5b22536a9bdbb","family_id":"qf-mod23-complexmod-conjugate","seed":202610231008,"semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[-5,-3]}"},"fields":{"title":"Conjugation of an Ordered Complex Pair","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-5, -3]. Give (value,conjugate).","answer":{"kind":"expression","value":"([-5, -3], [-5, 3])","explanation":"Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-5, -3], [-5, 3])."},"skills":["Conjugation of an Ordered Complex Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0009","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8ac33845977cfcd2ead94a617c505c4390a79012dd37dc9dae9542198bf3f8e","family_id":"qf-mod23-complexmod-conjugate","seed":202610231009,"semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[-4,-2]}"},"fields":{"title":"Conjugation of an Ordered Complex Pair","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-4, -2]. Give (value,conjugate).","answer":{"kind":"expression","value":"([-4, -2], [-4, 2])","explanation":"Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-4, -2], [-4, 2])."},"skills":["Conjugation of an Ordered Complex Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0010","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b710f47a9e2a85bb2609196af234ad9a290a1e7c184d9d39fb5030684465b601","family_id":"qf-mod23-complexmod-conjugate","seed":202610231010,"semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[-3,-1]}"},"fields":{"title":"Conjugation of an Ordered Complex Pair","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-3, -1]. Give (value,conjugate).","answer":{"kind":"expression","value":"([-3, -1], [-3, 1])","explanation":"Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-3, -1], [-3, 1])."},"skills":["Conjugation of an Ordered Complex Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0011","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a7ea2f00c5a3bf86da3dd9c75d26aceb6682a15b8061ce8f8c37e0c0de500ed","family_id":"qf-mod23-complexmod-conjugate","seed":202610231011,"semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[-2,0]}"},"fields":{"title":"Conjugation of an Ordered Complex Pair","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-2, 0]. Give (value,conjugate).","answer":{"kind":"expression","value":"([-2, 0], [-2, 0])","explanation":"Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-2, 0], [-2, 0])."},"skills":["Conjugation of an Ordered Complex Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0012","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d76b235741370a861e7a9a0f56c4ef8162d99469ed1e441b3fe39764fab7608","family_id":"qf-mod23-complexmod-conjugate","seed":202610231012,"semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[-1,1]}"},"fields":{"title":"Conjugation of an Ordered Complex Pair","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-1, 1]. Give (value,conjugate).","answer":{"kind":"expression","value":"([-1, 1], [-1, -1])","explanation":"Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-1, 1], [-1, -1])."},"skills":["Conjugation of an Ordered Complex Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0013","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3548a8945cad5b6d814d3516c793f19e01d331735abd2a961cd868806c0cfa27","family_id":"qf-mod23-complexmod-conjugate","seed":202610231013,"semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[0,2]}"},"fields":{"title":"Conjugation of an Ordered Complex Pair","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[0, 2]. Give (value,conjugate).","answer":{"kind":"expression","value":"([0, 2], [0, -2])","explanation":"Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 2], [0, -2])."},"skills":["Conjugation of an Ordered Complex Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0014","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7e214533025fd5d00b38ee9ba95725e40b878ccac0e03e58f425a6ace3a7aca","family_id":"qf-mod23-complexmod-conjugate","seed":202610231014,"semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[1,3]}"},"fields":{"title":"Conjugation of an Ordered Complex Pair","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[1, 3]. Give (value,conjugate).","answer":{"kind":"expression","value":"([1, 3], [1, -3])","explanation":"Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 3], [1, -3])."},"skills":["Conjugation of an Ordered Complex Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0015","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"811c031d7931597befe23f0617f50a31632729c5ba84b3bdf071ee0c6e974b7b","family_id":"qf-mod23-complexmod-conjugate","seed":202610231015,"semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[2,-3]}"},"fields":{"title":"Conjugation of an Ordered Complex Pair","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[2, -3]. Give (value,conjugate).","answer":{"kind":"expression","value":"([2, -3], [2, 3])","explanation":"Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, -3], [2, 3])."},"skills":["Conjugation of an Ordered Complex Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0016","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08ba7fd4d3af689a8ebc52122e8de93b8415b670835d19132f541bfe3683996a","family_id":"qf-mod23-complexmod-conjugate","seed":202610231016,"semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[3,-2]}"},"fields":{"title":"Conjugation of an Ordered Complex Pair","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[3, -2]. Give (value,conjugate).","answer":{"kind":"expression","value":"([3, -2], [3, 2])","explanation":"Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, -2], [3, 2])."},"skills":["Conjugation of an Ordered Complex Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0017","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1468d5cddb3a2b1a343b313297a83464284094b71474d00aa963933f61ac385c","family_id":"qf-mod23-complexmod-conjugate","seed":202610231017,"semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[4,-1]}"},"fields":{"title":"Conjugation of an Ordered Complex Pair","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[4, -1]. Give (value,conjugate).","answer":{"kind":"expression","value":"([4, -1], [4, 1])","explanation":"Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1], [4, 1])."},"skills":["Conjugation of an Ordered Complex Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0018","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"56cb6cd66584eba20810b5a9b9b5e1fd64035fe7690cfa81d2135ca2cae999d1","family_id":"qf-mod23-complexmod-conjugate","seed":202610231018,"semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[5,0]}"},"fields":{"title":"Conjugation of an Ordered Complex Pair","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[5, 0]. Give (value,conjugate).","answer":{"kind":"expression","value":"([5, 0], [5, 0])","explanation":"Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0], [5, 0])."},"skills":["Conjugation of an Ordered Complex Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0019","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5ebc3f09ac13c60693ac7e7d9174ea452fe1bf4f3f3a896535cbb9a80f3b283","family_id":"qf-mod23-complexmod-conjugate","seed":202610231019,"semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[6,1]}"},"fields":{"title":"Conjugation of an Ordered Complex Pair","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[6, 1]. Give (value,conjugate).","answer":{"kind":"expression","value":"([6, 1], [6, -1])","explanation":"Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([6, 1], [6, -1])."},"skills":["Conjugation of an Ordered Complex Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0020","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"377ee926e1a41450df6627796eef2beb019502a7b801a7a614bb1fe1a5f586fd","family_id":"qf-mod23-complexmod-conjugate","seed":202610231020,"semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[7,2]}"},"fields":{"title":"Conjugation of an Ordered Complex Pair","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[7, 2]. Give (value,conjugate).","answer":{"kind":"expression","value":"([7, 2], [7, -2])","explanation":"Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([7, 2], [7, -2])."},"skills":["Conjugation of an Ordered Complex Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0021","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a35965f16bc3696bd71cd78eec9337b7d91867b12765364a5ed6b3b25eb43f54","family_id":"qf-mod23-complexmod-conjugate","seed":202610231021,"semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[8,3]}"},"fields":{"title":"Conjugation of an Ordered Complex Pair","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[8, 3]. Give (value,conjugate).","answer":{"kind":"expression","value":"([8, 3], [8, -3])","explanation":"Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([8, 3], [8, -3])."},"skills":["Conjugation of an Ordered Complex Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0022","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"608f13a09c6be38bfce4d62b7aaaa055d4f7e3f1f13f6bba16e61b35a6eb3945","family_id":"qf-mod23-complexmod-conjugate","seed":202610231022,"semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[9,-3]}"},"fields":{"title":"Conjugation of an Ordered Complex Pair","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[9, -3]. Give (value,conjugate).","answer":{"kind":"expression","value":"([9, -3], [9, 3])","explanation":"Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([9, -3], [9, 3])."},"skills":["Conjugation of an Ordered Complex Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0023","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99996a852cdb86d3f9f0e4192f342a5cf3341d9cf9a66fce5a063231953fc575","family_id":"qf-mod23-complexmod-conjugate","seed":202610231023,"semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[10,-2]}"},"fields":{"title":"Conjugation of an Ordered Complex Pair","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[10, -2]. Give (value,conjugate).","answer":{"kind":"expression","value":"([10, -2], [10, 2])","explanation":"Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([10, -2], [10, 2])."},"skills":["Conjugation of an Ordered Complex Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0024","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f69b4e270dec266b60bcc4ef66fc6257db0529bd86914d8efd9af08ce3f3bdd8","family_id":"qf-mod23-complexmod-conjugate","seed":202610231024,"semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[11,-1]}"},"fields":{"title":"Conjugation of an Ordered Complex Pair","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[11, -1]. Give (value,conjugate).","answer":{"kind":"expression","value":"([11, -1], [11, 1])","explanation":"Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([11, -1], [11, 1])."},"skills":["Conjugation of an Ordered Complex Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0025","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb99e4297c8dae854fb74a6add5f65da637f1439585ebfd2d13cd4f7875b655d","family_id":"qf-mod23-complexmod-conjugate","seed":202610231025,"semantic_template":"qf-mod23-complexmod-conjugate/{\"value\":[12,0]}"},"fields":{"title":"Conjugation of an Ordered Complex Pair","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[12, 0]. Give (value,conjugate).","answer":{"kind":"expression","value":"([12, 0], [12, 0])","explanation":"Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([12, 0], [12, 0])."},"skills":["Conjugation of an Ordered Complex Pair"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0026","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3920cc81a9523235a5290c901f96268cc9560718e70484470e579373efb1752b","family_id":"qf-mod23-complexmod-modulus-square","seed":202610231026,"semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[-12,-3]}"},"fields":{"title":"Exact Modulus Squared without an Unrequested Root","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-12, -3]. Give (value,modulus squared).","answer":{"kind":"expression","value":"([-12, -3], 153)","explanation":"Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-12, -3], 153)."},"skills":["Exact Modulus Squared without an Unrequested Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0027","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5887848a1ce95bacaa43a7807b459eec5053fa2ee2cecbf76f810d06b40d060f","family_id":"qf-mod23-complexmod-modulus-square","seed":202610231027,"semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[-11,-2]}"},"fields":{"title":"Exact Modulus Squared without an Unrequested Root","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-11, -2]. Give (value,modulus squared).","answer":{"kind":"expression","value":"([-11, -2], 125)","explanation":"Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-11, -2], 125)."},"skills":["Exact Modulus Squared without an Unrequested Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0028","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae22f002177d2c1451d2bf47013a399a7a4dd0dfe849b0264c2c0307446ad6e2","family_id":"qf-mod23-complexmod-modulus-square","seed":202610231028,"semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[-10,-1]}"},"fields":{"title":"Exact Modulus Squared without an Unrequested Root","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-10, -1]. Give (value,modulus squared).","answer":{"kind":"expression","value":"([-10, -1], 101)","explanation":"Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-10, -1], 101)."},"skills":["Exact Modulus Squared without an Unrequested Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0029","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dadbe2cdc6404ab2ed1e006db79f5f1c0ff599b55407ae8c8e647487b2b0573e","family_id":"qf-mod23-complexmod-modulus-square","seed":202610231029,"semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[-9,0]}"},"fields":{"title":"Exact Modulus Squared without an Unrequested Root","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-9, 0]. Give (value,modulus squared).","answer":{"kind":"expression","value":"([-9, 0], 81)","explanation":"Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 0], 81)."},"skills":["Exact Modulus Squared without an Unrequested Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0030","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ee839a09e5ecae31e29a93d6865cd9267b58a532147705f1a57045fcce68167","family_id":"qf-mod23-complexmod-modulus-square","seed":202610231030,"semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[-8,1]}"},"fields":{"title":"Exact Modulus Squared without an Unrequested Root","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-8, 1]. Give (value,modulus squared).","answer":{"kind":"expression","value":"([-8, 1], 65)","explanation":"Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 1], 65)."},"skills":["Exact Modulus Squared without an Unrequested Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0031","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78d854e8980609a035c7be7bf31695d77b6899e43822fc5b53f0a172b82fa61a","family_id":"qf-mod23-complexmod-modulus-square","seed":202610231031,"semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[-7,2]}"},"fields":{"title":"Exact Modulus Squared without an Unrequested Root","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-7, 2]. Give (value,modulus squared).","answer":{"kind":"expression","value":"([-7, 2], 53)","explanation":"Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-7, 2], 53)."},"skills":["Exact Modulus Squared without an Unrequested Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0032","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"09e6a4870d218983d96e43aed651c7d150077a129c107044b322676792564c1e","family_id":"qf-mod23-complexmod-modulus-square","seed":202610231032,"semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[-6,3]}"},"fields":{"title":"Exact Modulus Squared without an Unrequested Root","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-6, 3]. Give (value,modulus squared).","answer":{"kind":"expression","value":"([-6, 3], 45)","explanation":"Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-6, 3], 45)."},"skills":["Exact Modulus Squared without an Unrequested Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0033","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35488480ec7d52b98016f52b7db8342df1885e99f12a48358fca73faae298ffe","family_id":"qf-mod23-complexmod-modulus-square","seed":202610231033,"semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[-5,-3]}"},"fields":{"title":"Exact Modulus Squared without an Unrequested Root","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-5, -3]. Give (value,modulus squared).","answer":{"kind":"expression","value":"([-5, -3], 34)","explanation":"Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-5, -3], 34)."},"skills":["Exact Modulus Squared without an Unrequested Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0034","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad6a32607bf2be7e327dc3465ee5609861d56e43eefbcd37345d5aa702a87b47","family_id":"qf-mod23-complexmod-modulus-square","seed":202610231034,"semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[-4,-2]}"},"fields":{"title":"Exact Modulus Squared without an Unrequested Root","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-4, -2]. Give (value,modulus squared).","answer":{"kind":"expression","value":"([-4, -2], 20)","explanation":"Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-4, -2], 20)."},"skills":["Exact Modulus Squared without an Unrequested Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0035","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cba42e397096bc247d62091b1591367128d088e7b6f8171d59431d40efb95d13","family_id":"qf-mod23-complexmod-modulus-square","seed":202610231035,"semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[-3,-1]}"},"fields":{"title":"Exact Modulus Squared without an Unrequested Root","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-3, -1]. Give (value,modulus squared).","answer":{"kind":"expression","value":"([-3, -1], 10)","explanation":"Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-3, -1], 10)."},"skills":["Exact Modulus Squared without an Unrequested Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0036","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1e508b58e56935281ee72f829dd370d092a3bc98b68c5dd1f479493a7c7ad90","family_id":"qf-mod23-complexmod-modulus-square","seed":202610231036,"semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[-2,0]}"},"fields":{"title":"Exact Modulus Squared without an Unrequested Root","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-2, 0]. Give (value,modulus squared).","answer":{"kind":"expression","value":"([-2, 0], 4)","explanation":"Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-2, 0], 4)."},"skills":["Exact Modulus Squared without an Unrequested Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0037","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2ba6e76249e2accf1015ade85c4d0894037cbc06405bc9e1266acfb3a3d3e61","family_id":"qf-mod23-complexmod-modulus-square","seed":202610231037,"semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[-1,1]}"},"fields":{"title":"Exact Modulus Squared without an Unrequested Root","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-1, 1]. Give (value,modulus squared).","answer":{"kind":"expression","value":"([-1, 1], 2)","explanation":"Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-1, 1], 2)."},"skills":["Exact Modulus Squared without an Unrequested Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0038","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"09bc7fd2db04b4edc4ad333305e7121f8675caf264bafb74ab5e3b914791d569","family_id":"qf-mod23-complexmod-modulus-square","seed":202610231038,"semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[0,2]}"},"fields":{"title":"Exact Modulus Squared without an Unrequested Root","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[0, 2]. Give (value,modulus squared).","answer":{"kind":"expression","value":"([0, 2], 4)","explanation":"Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 2], 4)."},"skills":["Exact Modulus Squared without an Unrequested Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0039","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1aec684b3a0c7d0e690a9a28363085743fb81201d50329d64f908ce7bfa17d1","family_id":"qf-mod23-complexmod-modulus-square","seed":202610231039,"semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[1,3]}"},"fields":{"title":"Exact Modulus Squared without an Unrequested Root","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[1, 3]. Give (value,modulus squared).","answer":{"kind":"expression","value":"([1, 3], 10)","explanation":"Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 3], 10)."},"skills":["Exact Modulus Squared without an Unrequested Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0040","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b5104ce7cbc754f75944e399b5cf6c2ff5381b58cac7c1809914c34b136b550","family_id":"qf-mod23-complexmod-modulus-square","seed":202610231040,"semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[2,-3]}"},"fields":{"title":"Exact Modulus Squared without an Unrequested Root","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[2, -3]. Give (value,modulus squared).","answer":{"kind":"expression","value":"([2, -3], 13)","explanation":"Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, -3], 13)."},"skills":["Exact Modulus Squared without an Unrequested Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0041","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"edd7836ac097e79940b31d45ee125142157082bf14e3ce2e6e39a737dabdbda5","family_id":"qf-mod23-complexmod-modulus-square","seed":202610231041,"semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[3,-2]}"},"fields":{"title":"Exact Modulus Squared without an Unrequested Root","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[3, -2]. Give (value,modulus squared).","answer":{"kind":"expression","value":"([3, -2], 13)","explanation":"Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, -2], 13)."},"skills":["Exact Modulus Squared without an Unrequested Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0042","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74557b6e63d37c9d3a12033f594de3bccdce454658b66cd88dd7cd0103bf179d","family_id":"qf-mod23-complexmod-modulus-square","seed":202610231042,"semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[4,-1]}"},"fields":{"title":"Exact Modulus Squared without an Unrequested Root","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[4, -1]. Give (value,modulus squared).","answer":{"kind":"expression","value":"([4, -1], 17)","explanation":"Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1], 17)."},"skills":["Exact Modulus Squared without an Unrequested Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0043","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"560567566d9b209089b46721a852ed61874e91316d44701683ae33c2d3758042","family_id":"qf-mod23-complexmod-modulus-square","seed":202610231043,"semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[5,0]}"},"fields":{"title":"Exact Modulus Squared without an Unrequested Root","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[5, 0]. Give (value,modulus squared).","answer":{"kind":"expression","value":"([5, 0], 25)","explanation":"Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0], 25)."},"skills":["Exact Modulus Squared without an Unrequested Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0044","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ab9436da6b22ec3733dafd2c13b13d91edbd9f430adc1a4db0167c9811b005b","family_id":"qf-mod23-complexmod-modulus-square","seed":202610231044,"semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[6,1]}"},"fields":{"title":"Exact Modulus Squared without an Unrequested Root","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[6, 1]. Give (value,modulus squared).","answer":{"kind":"expression","value":"([6, 1], 37)","explanation":"Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([6, 1], 37)."},"skills":["Exact Modulus Squared without an Unrequested Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0045","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0cc9fff18effaf91b33db1fccb64866e987ea1705b7c75265d4173c947e47800","family_id":"qf-mod23-complexmod-modulus-square","seed":202610231045,"semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[7,2]}"},"fields":{"title":"Exact Modulus Squared without an Unrequested Root","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[7, 2]. Give (value,modulus squared).","answer":{"kind":"expression","value":"([7, 2], 53)","explanation":"Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([7, 2], 53)."},"skills":["Exact Modulus Squared without an Unrequested Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0046","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed4740ae39ee5de7accf75bd3bef813f2a964c83ce4700f41dc5988c25c7f481","family_id":"qf-mod23-complexmod-modulus-square","seed":202610231046,"semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[8,3]}"},"fields":{"title":"Exact Modulus Squared without an Unrequested Root","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[8, 3]. Give (value,modulus squared).","answer":{"kind":"expression","value":"([8, 3], 73)","explanation":"Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([8, 3], 73)."},"skills":["Exact Modulus Squared without an Unrequested Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0047","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"475af15bdf4f7ceb7a0dba5ce1ceca7a3e5b956f1889b90266c95c3fb551e2fe","family_id":"qf-mod23-complexmod-modulus-square","seed":202610231047,"semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[9,-3]}"},"fields":{"title":"Exact Modulus Squared without an Unrequested Root","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[9, -3]. Give (value,modulus squared).","answer":{"kind":"expression","value":"([9, -3], 90)","explanation":"Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([9, -3], 90)."},"skills":["Exact Modulus Squared without an Unrequested Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0048","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e3b2d22559b253ca9f1d93747f5b722f069460079a88e854abe35323a098370e","family_id":"qf-mod23-complexmod-modulus-square","seed":202610231048,"semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[10,-2]}"},"fields":{"title":"Exact Modulus Squared without an Unrequested Root","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[10, -2]. Give (value,modulus squared).","answer":{"kind":"expression","value":"([10, -2], 104)","explanation":"Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([10, -2], 104)."},"skills":["Exact Modulus Squared without an Unrequested Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0049","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"769bdf6bb95215d5b9c49378d7a94a523b6965ac23c3146484d4ad2bdaa36a04","family_id":"qf-mod23-complexmod-modulus-square","seed":202610231049,"semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[11,-1]}"},"fields":{"title":"Exact Modulus Squared without an Unrequested Root","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[11, -1]. Give (value,modulus squared).","answer":{"kind":"expression","value":"([11, -1], 122)","explanation":"Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([11, -1], 122)."},"skills":["Exact Modulus Squared without an Unrequested Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0050","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d7390cb9f3a712810a06f5f97b045fbb0f9d4abaa6eabed0b3eb58bb297049e","family_id":"qf-mod23-complexmod-modulus-square","seed":202610231050,"semantic_template":"qf-mod23-complexmod-modulus-square/{\"value\":[12,0]}"},"fields":{"title":"Exact Modulus Squared without an Unrequested Root","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[12, 0]. Give (value,modulus squared).","answer":{"kind":"expression","value":"([12, 0], 144)","explanation":"Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([12, 0], 144)."},"skills":["Exact Modulus Squared without an Unrequested Root"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0051","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4cb37f98fb34813ca391d1afbf6b71953d5bda78c49258f6024a6ee743354a49","family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231051,"semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[-12,-3]}"},"fields":{"title":"The Product with the Conjugate Is a Real Embedding","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-12, -3]. Give (value,conjugate,product as a complex pair,modulus squared).","answer":{"kind":"expression","value":"([-12, -3], [-12, 3], [153, 0], 153)","explanation":"Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-12, -3], [-12, 3], [153, 0], 153)."},"skills":["The Product with the Conjugate Is a Real Embedding"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0052","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba979e654717b52a6aee49c60decf056ba9e19eaa150d822b31cb7a3be3d8d30","family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231052,"semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[-11,-2]}"},"fields":{"title":"The Product with the Conjugate Is a Real Embedding","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-11, -2]. Give (value,conjugate,product as a complex pair,modulus squared).","answer":{"kind":"expression","value":"([-11, -2], [-11, 2], [125, 0], 125)","explanation":"Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-11, -2], [-11, 2], [125, 0], 125)."},"skills":["The Product with the Conjugate Is a Real Embedding"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0053","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14e9ec0c90fa9431e37329bd1757f7b675f3ddb635fc5c3f0cb04ca209a46d0f","family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231053,"semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[-10,-1]}"},"fields":{"title":"The Product with the Conjugate Is a Real Embedding","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-10, -1]. Give (value,conjugate,product as a complex pair,modulus squared).","answer":{"kind":"expression","value":"([-10, -1], [-10, 1], [101, 0], 101)","explanation":"Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-10, -1], [-10, 1], [101, 0], 101)."},"skills":["The Product with the Conjugate Is a Real Embedding"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0054","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ffb1bf50ceb5b566cb6247aadb7ee3f03093c5ff98b0764d8222209f6c9b3fd","family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231054,"semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[-9,0]}"},"fields":{"title":"The Product with the Conjugate Is a Real Embedding","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-9, 0]. Give (value,conjugate,product as a complex pair,modulus squared).","answer":{"kind":"expression","value":"([-9, 0], [-9, 0], [81, 0], 81)","explanation":"Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 0], [-9, 0], [81, 0], 81)."},"skills":["The Product with the Conjugate Is a Real Embedding"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0055","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1d991be0008ea80efbe4593f45c62acfd71c362dd5f70d7796dfc929d556f58","family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231055,"semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[-8,1]}"},"fields":{"title":"The Product with the Conjugate Is a Real Embedding","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-8, 1]. Give (value,conjugate,product as a complex pair,modulus squared).","answer":{"kind":"expression","value":"([-8, 1], [-8, -1], [65, 0], 65)","explanation":"Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 1], [-8, -1], [65, 0], 65)."},"skills":["The Product with the Conjugate Is a Real Embedding"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0056","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a4e9d15883bba50e84e4dde33df515d2a98f439665266ebdc3b707887684fdc","family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231056,"semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[-7,2]}"},"fields":{"title":"The Product with the Conjugate Is a Real Embedding","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-7, 2]. Give (value,conjugate,product as a complex pair,modulus squared).","answer":{"kind":"expression","value":"([-7, 2], [-7, -2], [53, 0], 53)","explanation":"Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-7, 2], [-7, -2], [53, 0], 53)."},"skills":["The Product with the Conjugate Is a Real Embedding"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0057","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"918178874cb8989558b75765995a45fea6708ee575eb76327527ed7c8b65ec8a","family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231057,"semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[-6,3]}"},"fields":{"title":"The Product with the Conjugate Is a Real Embedding","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-6, 3]. Give (value,conjugate,product as a complex pair,modulus squared).","answer":{"kind":"expression","value":"([-6, 3], [-6, -3], [45, 0], 45)","explanation":"Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-6, 3], [-6, -3], [45, 0], 45)."},"skills":["The Product with the Conjugate Is a Real Embedding"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0058","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3815ac63c1542455adfb902bed60a29adc64e97173cf7514616d4ebf3eb8686a","family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231058,"semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[-5,-3]}"},"fields":{"title":"The Product with the Conjugate Is a Real Embedding","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-5, -3]. Give (value,conjugate,product as a complex pair,modulus squared).","answer":{"kind":"expression","value":"([-5, -3], [-5, 3], [34, 0], 34)","explanation":"Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-5, -3], [-5, 3], [34, 0], 34)."},"skills":["The Product with the Conjugate Is a Real Embedding"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0059","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cbe612ad653600d763ce45230eb110a08e32fa7dab68466177caf4257b90b788","family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231059,"semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[-4,-2]}"},"fields":{"title":"The Product with the Conjugate Is a Real Embedding","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-4, -2]. Give (value,conjugate,product as a complex pair,modulus squared).","answer":{"kind":"expression","value":"([-4, -2], [-4, 2], [20, 0], 20)","explanation":"Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-4, -2], [-4, 2], [20, 0], 20)."},"skills":["The Product with the Conjugate Is a Real Embedding"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0060","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e0f5e9675271b409f66f59366ca65126faf97f9d93f369e2f60f79c7bc5a81e","family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231060,"semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[-3,-1]}"},"fields":{"title":"The Product with the Conjugate Is a Real Embedding","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-3, -1]. Give (value,conjugate,product as a complex pair,modulus squared).","answer":{"kind":"expression","value":"([-3, -1], [-3, 1], [10, 0], 10)","explanation":"Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-3, -1], [-3, 1], [10, 0], 10)."},"skills":["The Product with the Conjugate Is a Real Embedding"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0061","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2cec670ba82cc1bcef59a5b85182e74a0e7a5d07792bc9afe63ca34a9f48f227","family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231061,"semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[-2,0]}"},"fields":{"title":"The Product with the Conjugate Is a Real Embedding","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-2, 0]. Give (value,conjugate,product as a complex pair,modulus squared).","answer":{"kind":"expression","value":"([-2, 0], [-2, 0], [4, 0], 4)","explanation":"Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-2, 0], [-2, 0], [4, 0], 4)."},"skills":["The Product with the Conjugate Is a Real Embedding"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0062","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"409b6e68467841d4e2aac8ffbc434882f2a401fe02a79310018b77a981c09144","family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231062,"semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[-1,1]}"},"fields":{"title":"The Product with the Conjugate Is a Real Embedding","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-1, 1]. Give (value,conjugate,product as a complex pair,modulus squared).","answer":{"kind":"expression","value":"([-1, 1], [-1, -1], [2, 0], 2)","explanation":"Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-1, 1], [-1, -1], [2, 0], 2)."},"skills":["The Product with the Conjugate Is a Real Embedding"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0063","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"092e25750761bfad16a81536db8e5dd3b4ee9764aa0ef5379d16a29a41e2427e","family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231063,"semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[0,2]}"},"fields":{"title":"The Product with the Conjugate Is a Real Embedding","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[0, 2]. Give (value,conjugate,product as a complex pair,modulus squared).","answer":{"kind":"expression","value":"([0, 2], [0, -2], [4, 0], 4)","explanation":"Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 2], [0, -2], [4, 0], 4)."},"skills":["The Product with the Conjugate Is a Real Embedding"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0064","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69ad48edc3a925f1eb517ae208ffc5bfdc3972acfbab799a33089b7bf7b94741","family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231064,"semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[1,3]}"},"fields":{"title":"The Product with the Conjugate Is a Real Embedding","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[1, 3]. Give (value,conjugate,product as a complex pair,modulus squared).","answer":{"kind":"expression","value":"([1, 3], [1, -3], [10, 0], 10)","explanation":"Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 3], [1, -3], [10, 0], 10)."},"skills":["The Product with the Conjugate Is a Real Embedding"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0065","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"748fbd102cf1df5e1965ced640fa4143b6ebdeafa7f72c0ff6dea5fbcaf2168f","family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231065,"semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[2,-3]}"},"fields":{"title":"The Product with the Conjugate Is a Real Embedding","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[2, -3]. Give (value,conjugate,product as a complex pair,modulus squared).","answer":{"kind":"expression","value":"([2, -3], [2, 3], [13, 0], 13)","explanation":"Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, -3], [2, 3], [13, 0], 13)."},"skills":["The Product with the Conjugate Is a Real Embedding"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0066","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f83fe5cf51bb873d5cb53127fe41f45efb4579b2833541d9bdf7efce46817034","family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231066,"semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[3,-2]}"},"fields":{"title":"The Product with the Conjugate Is a Real Embedding","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[3, -2]. Give (value,conjugate,product as a complex pair,modulus squared).","answer":{"kind":"expression","value":"([3, -2], [3, 2], [13, 0], 13)","explanation":"Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, -2], [3, 2], [13, 0], 13)."},"skills":["The Product with the Conjugate Is a Real Embedding"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0067","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e59ba13865c5a0b7be2982d896ec6f12f3fbb83a65741a0008b5b9d8c60e04a","family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231067,"semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[4,-1]}"},"fields":{"title":"The Product with the Conjugate Is a Real Embedding","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[4, -1]. Give (value,conjugate,product as a complex pair,modulus squared).","answer":{"kind":"expression","value":"([4, -1], [4, 1], [17, 0], 17)","explanation":"Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1], [4, 1], [17, 0], 17)."},"skills":["The Product with the Conjugate Is a Real Embedding"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0068","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c3090e3976facec05a4d3854b62217307d396d6e2eb4062bcb66528eae06373","family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231068,"semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[5,0]}"},"fields":{"title":"The Product with the Conjugate Is a Real Embedding","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[5, 0]. Give (value,conjugate,product as a complex pair,modulus squared).","answer":{"kind":"expression","value":"([5, 0], [5, 0], [25, 0], 25)","explanation":"Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0], [5, 0], [25, 0], 25)."},"skills":["The Product with the Conjugate Is a Real Embedding"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0069","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa93ff6180e8c7f7af9c89f06859a3d873551cfc49b25d6273bca978286285d9","family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231069,"semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[6,1]}"},"fields":{"title":"The Product with the Conjugate Is a Real Embedding","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[6, 1]. Give (value,conjugate,product as a complex pair,modulus squared).","answer":{"kind":"expression","value":"([6, 1], [6, -1], [37, 0], 37)","explanation":"Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([6, 1], [6, -1], [37, 0], 37)."},"skills":["The Product with the Conjugate Is a Real Embedding"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0070","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba4c279d3e1143223adf3137ce1b211778d79586ec7bbdce5d17608f26cec0ef","family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231070,"semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[7,2]}"},"fields":{"title":"The Product with the Conjugate Is a Real Embedding","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[7, 2]. Give (value,conjugate,product as a complex pair,modulus squared).","answer":{"kind":"expression","value":"([7, 2], [7, -2], [53, 0], 53)","explanation":"Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([7, 2], [7, -2], [53, 0], 53)."},"skills":["The Product with the Conjugate Is a Real Embedding"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0071","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a5d39762e074109d8bb3fc0f11b078c38176d8a9ae750d77b4460dc15211f3e","family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231071,"semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[8,3]}"},"fields":{"title":"The Product with the Conjugate Is a Real Embedding","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[8, 3]. Give (value,conjugate,product as a complex pair,modulus squared).","answer":{"kind":"expression","value":"([8, 3], [8, -3], [73, 0], 73)","explanation":"Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([8, 3], [8, -3], [73, 0], 73)."},"skills":["The Product with the Conjugate Is a Real Embedding"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0072","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da5fe6c0a19afb1d3f1c7381fbb320422bcd11247ad8b33b52dd2465e344d140","family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231072,"semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[9,-3]}"},"fields":{"title":"The Product with the Conjugate Is a Real Embedding","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[9, -3]. Give (value,conjugate,product as a complex pair,modulus squared).","answer":{"kind":"expression","value":"([9, -3], [9, 3], [90, 0], 90)","explanation":"Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([9, -3], [9, 3], [90, 0], 90)."},"skills":["The Product with the Conjugate Is a Real Embedding"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0073","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7a54a9109b36c57aec9111dfadfe0de1a082e9997e31dfc300ff36fe3fdca8e","family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231073,"semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[10,-2]}"},"fields":{"title":"The Product with the Conjugate Is a Real Embedding","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[10, -2]. Give (value,conjugate,product as a complex pair,modulus squared).","answer":{"kind":"expression","value":"([10, -2], [10, 2], [104, 0], 104)","explanation":"Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([10, -2], [10, 2], [104, 0], 104)."},"skills":["The Product with the Conjugate Is a Real Embedding"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0074","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97a548f8d60ea6a34c6cc9f83e61175db59c69bb46f9e715bff7c0c81936c03a","family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231074,"semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[11,-1]}"},"fields":{"title":"The Product with the Conjugate Is a Real Embedding","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[11, -1]. Give (value,conjugate,product as a complex pair,modulus squared).","answer":{"kind":"expression","value":"([11, -1], [11, 1], [122, 0], 122)","explanation":"Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([11, -1], [11, 1], [122, 0], 122)."},"skills":["The Product with the Conjugate Is a Real Embedding"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0075","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dcce65d9d655cbe499e5a0e1a3db8c4874a45d55bbdeea64cfc595220662716c","family_id":"qf-mod23-complexmod-product-conjugate","seed":202610231075,"semantic_template":"qf-mod23-complexmod-product-conjugate/{\"value\":[12,0]}"},"fields":{"title":"The Product with the Conjugate Is a Real Embedding","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[12, 0]. Give (value,conjugate,product as a complex pair,modulus squared).","answer":{"kind":"expression","value":"([12, 0], [12, 0], [144, 0], 144)","explanation":"Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([12, 0], [12, 0], [144, 0], 144)."},"skills":["The Product with the Conjugate Is a Real Embedding"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0076","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43a4e76f030a7183cf1b560b9b226419a1697181ac2828c5373103be523f27bc","family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231076,"semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[-12,-3],\"right\":[-2,-9]}"},"fields":{"title":"Conjugating a Product versus Multiplying the Conjugates","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[-12, -3], right=[-2, -9]. Give (left,right,product,conjugate of product,product of conjugates,equal?).","answer":{"kind":"expression","value":"([-12, -3], [-2, -9], [-3, 114], [-3, -114], [-3, -114], Yes)","explanation":"Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-12, -3], [-2, -9], [-3, 114], [-3, -114], [-3, -114], Yes)."},"skills":["Conjugating a Product versus Multiplying the Conjugates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0077","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91ef1e8988cc7d28ff3549b6e38fd1def0d1edd95442e63a2f14a37d2b7b8618","family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231077,"semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[-11,-2],\"right\":[-1,-7]}"},"fields":{"title":"Conjugating a Product versus Multiplying the Conjugates","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[-11, -2], right=[-1, -7]. Give (left,right,product,conjugate of product,product of conjugates,equal?).","answer":{"kind":"expression","value":"([-11, -2], [-1, -7], [-3, 79], [-3, -79], [-3, -79], Yes)","explanation":"Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-11, -2], [-1, -7], [-3, 79], [-3, -79], [-3, -79], Yes)."},"skills":["Conjugating a Product versus Multiplying the Conjugates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0078","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0cd34332f6f9f9acce22f95be820baec04cbcaf300ab0f915342cf9fab34f13e","family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231078,"semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[-10,-1],\"right\":[0,-5]}"},"fields":{"title":"Conjugating a Product versus Multiplying the Conjugates","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[-10, -1], right=[0, -5]. Give (left,right,product,conjugate of product,product of conjugates,equal?).","answer":{"kind":"expression","value":"([-10, -1], [0, -5], [-5, 50], [-5, -50], [-5, -50], Yes)","explanation":"Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-10, -1], [0, -5], [-5, 50], [-5, -50], [-5, -50], Yes)."},"skills":["Conjugating a Product versus Multiplying the Conjugates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0079","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1309707512251387a9fae1f11b5247a3f85a7fb90fb09a660f65645cd6963503","family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231079,"semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[-9,0],\"right\":[1,-3]}"},"fields":{"title":"Conjugating a Product versus Multiplying the Conjugates","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[-9, 0], right=[1, -3]. Give (left,right,product,conjugate of product,product of conjugates,equal?).","answer":{"kind":"expression","value":"([-9, 0], [1, -3], [-9, 27], [-9, -27], [-9, -27], Yes)","explanation":"Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 0], [1, -3], [-9, 27], [-9, -27], [-9, -27], Yes)."},"skills":["Conjugating a Product versus Multiplying the Conjugates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0080","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"10b3cdd81e0cba2434c3c0f6ee8ca9e76fe4b5a3b4154b21515410b18b2e4aaf","family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231080,"semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[-8,1],\"right\":[2,-1]}"},"fields":{"title":"Conjugating a Product versus Multiplying the Conjugates","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[-8, 1], right=[2, -1]. Give (left,right,product,conjugate of product,product of conjugates,equal?).","answer":{"kind":"expression","value":"([-8, 1], [2, -1], [-15, 10], [-15, -10], [-15, -10], Yes)","explanation":"Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 1], [2, -1], [-15, 10], [-15, -10], [-15, -10], Yes)."},"skills":["Conjugating a Product versus Multiplying the Conjugates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0081","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"98e677b4b83ee13fda42fc9dc6298114bbd8c6ce30e4bbfd4c835749fb633fb4","family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231081,"semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[-7,2],\"right\":[-2,1]}"},"fields":{"title":"Conjugating a Product versus Multiplying the Conjugates","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[-7, 2], right=[-2, 1]. Give (left,right,product,conjugate of product,product of conjugates,equal?).","answer":{"kind":"expression","value":"([-7, 2], [-2, 1], [12, -11], [12, 11], [12, 11], Yes)","explanation":"Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-7, 2], [-2, 1], [12, -11], [12, 11], [12, 11], Yes)."},"skills":["Conjugating a Product versus Multiplying the Conjugates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0082","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ec04c9aa346b093ef1062795ea49ddc89f4dd692f0b257ebe80df5b949e05cf","family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231082,"semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[-6,3],\"right\":[-1,3]}"},"fields":{"title":"Conjugating a Product versus Multiplying the Conjugates","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[-6, 3], right=[-1, 3]. Give (left,right,product,conjugate of product,product of conjugates,equal?).","answer":{"kind":"expression","value":"([-6, 3], [-1, 3], [-3, -21], [-3, 21], [-3, 21], Yes)","explanation":"Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-6, 3], [-1, 3], [-3, -21], [-3, 21], [-3, 21], Yes)."},"skills":["Conjugating a Product versus Multiplying the Conjugates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0083","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"09d976dfa8645aee89eb44252715119ce7c8d0428da22951d7a83fa9729223c8","family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231083,"semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[-5,-3],\"right\":[0,5]}"},"fields":{"title":"Conjugating a Product versus Multiplying the Conjugates","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[-5, -3], right=[0, 5]. Give (left,right,product,conjugate of product,product of conjugates,equal?).","answer":{"kind":"expression","value":"([-5, -3], [0, 5], [15, -25], [15, 25], [15, 25], Yes)","explanation":"Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-5, -3], [0, 5], [15, -25], [15, 25], [15, 25], Yes)."},"skills":["Conjugating a Product versus Multiplying the Conjugates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0084","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"820bbffd9dff5347d067712362ac8bd613c9ca9aabf8aaea0ac7c2516803d986","family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231084,"semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[-4,-2],\"right\":[1,7]}"},"fields":{"title":"Conjugating a Product versus Multiplying the Conjugates","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[-4, -2], right=[1, 7]. Give (left,right,product,conjugate of product,product of conjugates,equal?).","answer":{"kind":"expression","value":"([-4, -2], [1, 7], [10, -30], [10, 30], [10, 30], Yes)","explanation":"Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-4, -2], [1, 7], [10, -30], [10, 30], [10, 30], Yes)."},"skills":["Conjugating a Product versus Multiplying the Conjugates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0085","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9ca3325522101c7b97df774459626a376ed58d9246c14f2f4ff2475b533336e","family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231085,"semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[-3,-1],\"right\":[2,9]}"},"fields":{"title":"Conjugating a Product versus Multiplying the Conjugates","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[-3, -1], right=[2, 9]. Give (left,right,product,conjugate of product,product of conjugates,equal?).","answer":{"kind":"expression","value":"([-3, -1], [2, 9], [3, -29], [3, 29], [3, 29], Yes)","explanation":"Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-3, -1], [2, 9], [3, -29], [3, 29], [3, 29], Yes)."},"skills":["Conjugating a Product versus Multiplying the Conjugates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0086","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14a5d47f397be677ce1ccc9152956abd0d975338768bd88ac6d8272e3d5f5573","family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231086,"semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[-2,0],\"right\":[-2,11]}"},"fields":{"title":"Conjugating a Product versus Multiplying the Conjugates","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[-2, 0], right=[-2, 11]. Give (left,right,product,conjugate of product,product of conjugates,equal?).","answer":{"kind":"expression","value":"([-2, 0], [-2, 11], [4, -22], [4, 22], [4, 22], Yes)","explanation":"Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-2, 0], [-2, 11], [4, -22], [4, 22], [4, 22], Yes)."},"skills":["Conjugating a Product versus Multiplying the Conjugates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0087","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17577824e0447283057c50d4b5ffbbbbb316bbd8272f9107331a7d8e0313dc4f","family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231087,"semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[-1,1],\"right\":[-1,13]}"},"fields":{"title":"Conjugating a Product versus Multiplying the Conjugates","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[-1, 1], right=[-1, 13]. Give (left,right,product,conjugate of product,product of conjugates,equal?).","answer":{"kind":"expression","value":"([-1, 1], [-1, 13], [-12, -14], [-12, 14], [-12, 14], Yes)","explanation":"Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-1, 1], [-1, 13], [-12, -14], [-12, 14], [-12, 14], Yes)."},"skills":["Conjugating a Product versus Multiplying the Conjugates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0088","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e87f3a7c60109964a11a696bfe402ade56d868467695a9c50bc8e26fab0992f","family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231088,"semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[0,2],\"right\":[0,15]}"},"fields":{"title":"Conjugating a Product versus Multiplying the Conjugates","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[0, 2], right=[0, 15]. Give (left,right,product,conjugate of product,product of conjugates,equal?).","answer":{"kind":"expression","value":"([0, 2], [0, 15], [-30, 0], [-30, 0], [-30, 0], Yes)","explanation":"Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 2], [0, 15], [-30, 0], [-30, 0], [-30, 0], Yes)."},"skills":["Conjugating a Product versus Multiplying the Conjugates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0089","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6375e7e9d1bbde00485c0ad55d365a594b6ce5f7ed5806374f28ca48da9ad7df","family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231089,"semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[1,3],\"right\":[1,17]}"},"fields":{"title":"Conjugating a Product versus Multiplying the Conjugates","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[1, 3], right=[1, 17]. Give (left,right,product,conjugate of product,product of conjugates,equal?).","answer":{"kind":"expression","value":"([1, 3], [1, 17], [-50, 20], [-50, -20], [-50, -20], Yes)","explanation":"Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 3], [1, 17], [-50, 20], [-50, -20], [-50, -20], Yes)."},"skills":["Conjugating a Product versus Multiplying the Conjugates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0090","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f6c6e70be79812b61ed000217373df3ecbcd24a6372d152b6ef7e7795785e8d","family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231090,"semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[2,-3],\"right\":[2,19]}"},"fields":{"title":"Conjugating a Product versus Multiplying the Conjugates","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[2, -3], right=[2, 19]. Give (left,right,product,conjugate of product,product of conjugates,equal?).","answer":{"kind":"expression","value":"([2, -3], [2, 19], [61, 32], [61, -32], [61, -32], Yes)","explanation":"Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, -3], [2, 19], [61, 32], [61, -32], [61, -32], Yes)."},"skills":["Conjugating a Product versus Multiplying the Conjugates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0091","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21efa076035c198ad3e46812b3413714b0c3f8e6ce63930f679158385a855e6d","family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231091,"semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[3,-2],\"right\":[-2,21]}"},"fields":{"title":"Conjugating a Product versus Multiplying the Conjugates","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[3, -2], right=[-2, 21]. Give (left,right,product,conjugate of product,product of conjugates,equal?).","answer":{"kind":"expression","value":"([3, -2], [-2, 21], [36, 67], [36, -67], [36, -67], Yes)","explanation":"Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, -2], [-2, 21], [36, 67], [36, -67], [36, -67], Yes)."},"skills":["Conjugating a Product versus Multiplying the Conjugates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0092","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"710dd19b671f03a7845293aaeda5b0678696d33c06402996885160302c3f02d4","family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231092,"semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[4,-1],\"right\":[-1,23]}"},"fields":{"title":"Conjugating a Product versus Multiplying the Conjugates","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[4, -1], right=[-1, 23]. Give (left,right,product,conjugate of product,product of conjugates,equal?).","answer":{"kind":"expression","value":"([4, -1], [-1, 23], [19, 93], [19, -93], [19, -93], Yes)","explanation":"Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1], [-1, 23], [19, 93], [19, -93], [19, -93], Yes)."},"skills":["Conjugating a Product versus Multiplying the Conjugates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0093","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b2d609ad2dfc9fde4c92e2ba99a411ecf669abb47836638874c917e951e292c","family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231093,"semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[5,0],\"right\":[0,25]}"},"fields":{"title":"Conjugating a Product versus Multiplying the Conjugates","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[5, 0], right=[0, 25]. Give (left,right,product,conjugate of product,product of conjugates,equal?).","answer":{"kind":"expression","value":"([5, 0], [0, 25], [0, 125], [0, -125], [0, -125], Yes)","explanation":"Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0], [0, 25], [0, 125], [0, -125], [0, -125], Yes)."},"skills":["Conjugating a Product versus Multiplying the Conjugates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0094","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7375d4e8601a32b0c867c5dd6b098489e31450e8c36177d50135ed043beecae4","family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231094,"semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[6,1],\"right\":[1,27]}"},"fields":{"title":"Conjugating a Product versus Multiplying the Conjugates","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[6, 1], right=[1, 27]. Give (left,right,product,conjugate of product,product of conjugates,equal?).","answer":{"kind":"expression","value":"([6, 1], [1, 27], [-21, 163], [-21, -163], [-21, -163], Yes)","explanation":"Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([6, 1], [1, 27], [-21, 163], [-21, -163], [-21, -163], Yes)."},"skills":["Conjugating a Product versus Multiplying the Conjugates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0095","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"25167ded5c8efdcf1cb04788cdbd1c37dbcfbb8e5406488dd2136f9692b31ab4","family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231095,"semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[7,2],\"right\":[2,29]}"},"fields":{"title":"Conjugating a Product versus Multiplying the Conjugates","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[7, 2], right=[2, 29]. Give (left,right,product,conjugate of product,product of conjugates,equal?).","answer":{"kind":"expression","value":"([7, 2], [2, 29], [-44, 207], [-44, -207], [-44, -207], Yes)","explanation":"Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([7, 2], [2, 29], [-44, 207], [-44, -207], [-44, -207], Yes)."},"skills":["Conjugating a Product versus Multiplying the Conjugates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0096","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9df54c33772621fe0545a3f8f8c22e4b2c4ba477a37b7aa15d8f7c42b52d9dd7","family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231096,"semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[8,3],\"right\":[-2,31]}"},"fields":{"title":"Conjugating a Product versus Multiplying the Conjugates","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[8, 3], right=[-2, 31]. Give (left,right,product,conjugate of product,product of conjugates,equal?).","answer":{"kind":"expression","value":"([8, 3], [-2, 31], [-109, 242], [-109, -242], [-109, -242], Yes)","explanation":"Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([8, 3], [-2, 31], [-109, 242], [-109, -242], [-109, -242], Yes)."},"skills":["Conjugating a Product versus Multiplying the Conjugates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0097","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2814b582e532b3026bac076e4e38af82a807092052a50e8b5ef9b728aee5da66","family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231097,"semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[9,-3],\"right\":[-1,33]}"},"fields":{"title":"Conjugating a Product versus Multiplying the Conjugates","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[9, -3], right=[-1, 33]. Give (left,right,product,conjugate of product,product of conjugates,equal?).","answer":{"kind":"expression","value":"([9, -3], [-1, 33], [90, 300], [90, -300], [90, -300], Yes)","explanation":"Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([9, -3], [-1, 33], [90, 300], [90, -300], [90, -300], Yes)."},"skills":["Conjugating a Product versus Multiplying the Conjugates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0098","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd0603f0660c5590b6763b229c657f47ca76f95ddf53fd6a3f8e22f1cc58ab68","family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231098,"semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[10,-2],\"right\":[0,35]}"},"fields":{"title":"Conjugating a Product versus Multiplying the Conjugates","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[10, -2], right=[0, 35]. Give (left,right,product,conjugate of product,product of conjugates,equal?).","answer":{"kind":"expression","value":"([10, -2], [0, 35], [70, 350], [70, -350], [70, -350], Yes)","explanation":"Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([10, -2], [0, 35], [70, 350], [70, -350], [70, -350], Yes)."},"skills":["Conjugating a Product versus Multiplying the Conjugates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0099","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dde6b3ca771bb6ede0f96d85ef05bbeabc383b1e81c2c2ec27baa8ab14d815e3","family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231099,"semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[11,-1],\"right\":[1,37]}"},"fields":{"title":"Conjugating a Product versus Multiplying the Conjugates","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[11, -1], right=[1, 37]. Give (left,right,product,conjugate of product,product of conjugates,equal?).","answer":{"kind":"expression","value":"([11, -1], [1, 37], [48, 406], [48, -406], [48, -406], Yes)","explanation":"Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([11, -1], [1, 37], [48, 406], [48, -406], [48, -406], Yes)."},"skills":["Conjugating a Product versus Multiplying the Conjugates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r01-complex-modulus-conjugate-0100","batch_id":"batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2c442b0322dc036b3729e5800ed50ff4bcc86044e1c72ca61a0127a9eea82a6","family_id":"qf-mod23-complexmod-conjugation-product","seed":202610231100,"semantic_template":"qf-mod23-complexmod-conjugation-product/{\"left\":[12,0],\"right\":[2,39]}"},"fields":{"title":"Conjugating a Product versus Multiplying the Conjugates","stem":"Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[12, 0], right=[2, 39]. Give (left,right,product,conjugate of product,product of conjugates,equal?).","answer":{"kind":"expression","value":"([12, 0], [2, 39], [24, 468], [24, -468], [24, -468], Yes)","explanation":"Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([12, 0], [2, 39], [24, 468], [24, -468], [24, -468], Yes)."},"skills":["Conjugating a Product versus Multiplying the Conjugates"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}