{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0001","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92316fe7fbad413a34d4226761369c73f88f84f74a1f9abe03ab4982d3ca973c","family_id":"qf-eigenpair-candidate","seed":202610074001,"semantic_template":"qf-eigenpair-candidate/{\"inputs\":[\"zero_vector\"],\"operation\":\"eigenpair\"}"},"fields":{"title":"Eigenpair Candidate with a Nonzero Vector","stem":"Over R, the 2×2 matrix is M=[[2, 1], [0, 3]]. Is λ=99, v=[0, 0] a real eigenpair under the nonzero-eigenvector convention?","answer":{"kind":"short_text","value":"No","explanation":"Mv=[0, 0] and λv=[0, 0]. Both equality and v≠0 are required, giving No. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded."},"skills":["Eigenpair Candidate with a Nonzero Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0002","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43908e689a54de821fdf76f8795545c4833023e3d3b92b69e01e816e2076ed60","family_id":"qf-eigenpair-candidate","seed":202610074002,"semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"-11\",\"0\"],[\"0\",\"3\"]],\"3\",[\"0\",\"1\"]],\"operation\":\"eigenpair\"}"},"fields":{"title":"Eigenpair Candidate with a Nonzero Vector","stem":"Over R, the 2×2 matrix is M=[[-11, 0], [0, 3]]. Is λ=3, v=[0, 1] a real eigenpair under the nonzero-eigenvector convention?","answer":{"kind":"short_text","value":"Yes","explanation":"Mv=[0, 3] and λv=[0, 3]. Both equality and v≠0 are required, giving Yes. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded."},"skills":["Eigenpair Candidate with a Nonzero Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0003","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"017b65cc46ce57b2bdc40647323a9797409ca3ee81fc75e8be8cdf9059cd2c15","family_id":"qf-eigenpair-candidate","seed":202610074003,"semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"-10\",\"0\"],[\"0\",\"4\"]],\"-10\",[\"1\",\"1\"]],\"operation\":\"eigenpair\"}"},"fields":{"title":"Eigenpair Candidate with a Nonzero Vector","stem":"Over R, the 2×2 matrix is M=[[-10, 0], [0, 4]]. Is λ=-10, v=[1, 1] a real eigenpair under the nonzero-eigenvector convention?","answer":{"kind":"short_text","value":"No","explanation":"Mv=[-10, 4] and λv=[-10, -10]. Both equality and v≠0 are required, giving No. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded."},"skills":["Eigenpair Candidate with a Nonzero Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0004","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"794d7ff603dec19fc0f2a75b2e43c915e424f1dcbe3cc862069ec8ac9e6d12b8","family_id":"qf-eigenpair-candidate","seed":202610074004,"semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"-9\",\"0\"],[\"0\",\"5\"]],\"-8\",[\"1\",\"-1\"]],\"operation\":\"eigenpair\"}"},"fields":{"title":"Eigenpair Candidate with a Nonzero Vector","stem":"Over R, the 2×2 matrix is M=[[-9, 0], [0, 5]]. Is λ=-8, v=[1, -1] a real eigenpair under the nonzero-eigenvector convention?","answer":{"kind":"short_text","value":"No","explanation":"Mv=[-9, -5] and λv=[-8, 8]. Both equality and v≠0 are required, giving No. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded."},"skills":["Eigenpair Candidate with a Nonzero Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0005","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c95f25b93ec0db42c7a03ec12412f5a6155defb448c42d95c440cf767a5c5e39","family_id":"qf-eigenpair-candidate","seed":202610074005,"semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"4\",\"1\"],[\"1\",\"4\"]],\"5\",[\"1\",\"1\"]],\"operation\":\"eigenpair\"}"},"fields":{"title":"Eigenpair Candidate with a Nonzero Vector","stem":"Over R, the 2×2 matrix is M=[[4, 1], [1, 4]]. Is λ=5, v=[1, 1] a real eigenpair under the nonzero-eigenvector convention?","answer":{"kind":"short_text","value":"Yes","explanation":"Mv=[5, 5] and λv=[5, 5]. Both equality and v≠0 are required, giving Yes. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded."},"skills":["Eigenpair Candidate with a Nonzero Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0006","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d91edd702b0833a5424dc2954eb5bf3c3d99e24126ed8684c4dc8d19dbb6103b","family_id":"qf-eigenpair-candidate","seed":202610074006,"semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"5\",\"0\"],[\"1\",\"6\"]],\"5\",[\"1\",\"-1\"]],\"operation\":\"eigenpair\"}"},"fields":{"title":"Eigenpair Candidate with a Nonzero Vector","stem":"Over R, the 2×2 matrix is M=[[5, 0], [1, 6]]. Is λ=5, v=[1, -1] a real eigenpair under the nonzero-eigenvector convention?","answer":{"kind":"short_text","value":"Yes","explanation":"Mv=[5, -5] and λv=[5, -5]. Both equality and v≠0 are required, giving Yes. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded."},"skills":["Eigenpair Candidate with a Nonzero Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0007","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5db639f674b4e4c0e71eef8516e0a73b41b77789ec8dd5da88162c2c19bc75a","family_id":"qf-eigenpair-candidate","seed":202610074007,"semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"-6\",\"0\"],[\"0\",\"8\"]],\"-6\",[\"1\",\"1\"]],\"operation\":\"eigenpair\"}"},"fields":{"title":"Eigenpair Candidate with a Nonzero Vector","stem":"Over R, the 2×2 matrix is M=[[-6, 0], [0, 8]]. Is λ=-6, v=[1, 1] a real eigenpair under the nonzero-eigenvector convention?","answer":{"kind":"short_text","value":"No","explanation":"Mv=[-6, 8] and λv=[-6, -6]. Both equality and v≠0 are required, giving No. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded."},"skills":["Eigenpair Candidate with a Nonzero Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0008","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"820668c382aa8170c25747686fb1b19f648d62c1a1f88a98005cdfc3a71fc123","family_id":"qf-eigenpair-candidate","seed":202610074008,"semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"-5\",\"0\"],[\"0\",\"9\"]],\"-4\",[\"1\",\"-1\"]],\"operation\":\"eigenpair\"}"},"fields":{"title":"Eigenpair Candidate with a Nonzero Vector","stem":"Over R, the 2×2 matrix is M=[[-5, 0], [0, 9]]. Is λ=-4, v=[1, -1] a real eigenpair under the nonzero-eigenvector convention?","answer":{"kind":"short_text","value":"No","explanation":"Mv=[-5, -9] and λv=[-4, 4]. Both equality and v≠0 are required, giving No. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded."},"skills":["Eigenpair Candidate with a Nonzero Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0009","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e97204591a8ff3ce18ea7ff95ceaba78d2b8ea2a5a199b5cd69e587b2b611284","family_id":"qf-eigenpair-candidate","seed":202610074009,"semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"-4\",\"0\"],[\"0\",\"10\"]],\"-4\",[\"1\",\"0\"]],\"operation\":\"eigenpair\"}"},"fields":{"title":"Eigenpair Candidate with a Nonzero Vector","stem":"Over R, the 2×2 matrix is M=[[-4, 0], [0, 10]]. Is λ=-4, v=[1, 0] a real eigenpair under the nonzero-eigenvector convention?","answer":{"kind":"short_text","value":"Yes","explanation":"Mv=[-4, 0] and λv=[-4, 0]. Both equality and v≠0 are required, giving Yes. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded."},"skills":["Eigenpair Candidate with a Nonzero Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0010","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e72334d204d3c25610d9a27528db087cbcdbe966bf371bda3188e310ab34cea8","family_id":"qf-eigenpair-candidate","seed":202610074010,"semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"-3\",\"0\"],[\"0\",\"11\"]],\"11\",[\"0\",\"1\"]],\"operation\":\"eigenpair\"}"},"fields":{"title":"Eigenpair Candidate with a Nonzero Vector","stem":"Over R, the 2×2 matrix is M=[[-3, 0], [0, 11]]. Is λ=11, v=[0, 1] a real eigenpair under the nonzero-eigenvector convention?","answer":{"kind":"short_text","value":"Yes","explanation":"Mv=[0, 11] and λv=[0, 11]. Both equality and v≠0 are required, giving Yes. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded."},"skills":["Eigenpair Candidate with a Nonzero Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0011","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36dc927eaabadd501de91fc3317b7f94dbd838d2d07c0023b30a0e8eaaa66465","family_id":"qf-eigenpair-candidate","seed":202610074011,"semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"-2\",\"0\"],[\"0\",\"12\"]],\"-2\",[\"1\",\"1\"]],\"operation\":\"eigenpair\"}"},"fields":{"title":"Eigenpair Candidate with a Nonzero Vector","stem":"Over R, the 2×2 matrix is M=[[-2, 0], [0, 12]]. Is λ=-2, v=[1, 1] a real eigenpair under the nonzero-eigenvector convention?","answer":{"kind":"short_text","value":"No","explanation":"Mv=[-2, 12] and λv=[-2, -2]. Both equality and v≠0 are required, giving No. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded."},"skills":["Eigenpair Candidate with a Nonzero Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0012","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a75ad4b83f065e5ccdc76af9ea9fb5ca11bf668b5897dbd37eaa3a8d8a2e4e74","family_id":"qf-eigenpair-candidate","seed":202610074012,"semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"-1\",\"0\"],[\"0\",\"13\"]],\"0\",[\"1\",\"-1\"]],\"operation\":\"eigenpair\"}"},"fields":{"title":"Eigenpair Candidate with a Nonzero Vector","stem":"Over R, the 2×2 matrix is M=[[-1, 0], [0, 13]]. Is λ=0, v=[1, -1] a real eigenpair under the nonzero-eigenvector convention?","answer":{"kind":"short_text","value":"No","explanation":"Mv=[-1, -13] and λv=[0, 0]. Both equality and v≠0 are required, giving No. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded."},"skills":["Eigenpair Candidate with a Nonzero Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0013","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34122e76d6c9622856f1464c19b33ceaa2de45f37013b5f665cd70a6446585e4","family_id":"qf-eigenpair-candidate","seed":202610074013,"semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"0\",\"0\"],[\"0\",\"14\"]],\"0\",[\"1\",\"0\"]],\"operation\":\"eigenpair\"}"},"fields":{"title":"Eigenpair Candidate with a Nonzero Vector","stem":"Over R, the 2×2 matrix is M=[[0, 0], [0, 14]]. Is λ=0, v=[1, 0] a real eigenpair under the nonzero-eigenvector convention?","answer":{"kind":"short_text","value":"Yes","explanation":"Mv=[0, 0] and λv=[0, 0]. Both equality and v≠0 are required, giving Yes. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded."},"skills":["Eigenpair Candidate with a Nonzero Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0014","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e4f321109b9095b35afcc8fa87c386cd0320c878175bafabd8b247fd4d4fd58","family_id":"qf-eigenpair-candidate","seed":202610074014,"semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"1\",\"0\"],[\"0\",\"15\"]],\"15\",[\"0\",\"1\"]],\"operation\":\"eigenpair\"}"},"fields":{"title":"Eigenpair Candidate with a Nonzero Vector","stem":"Over R, the 2×2 matrix is M=[[1, 0], [0, 15]]. Is λ=15, v=[0, 1] a real eigenpair under the nonzero-eigenvector convention?","answer":{"kind":"short_text","value":"Yes","explanation":"Mv=[0, 15] and λv=[0, 15]. Both equality and v≠0 are required, giving Yes. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded."},"skills":["Eigenpair Candidate with a Nonzero Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0015","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4513c1f1a94b4fc9859c34d3af497f8895f32e587e6a7a147ddae047bb7bde1b","family_id":"qf-eigenpair-candidate","seed":202610074015,"semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"2\",\"0\"],[\"0\",\"16\"]],\"2\",[\"1\",\"1\"]],\"operation\":\"eigenpair\"}"},"fields":{"title":"Eigenpair Candidate with a Nonzero Vector","stem":"Over R, the 2×2 matrix is M=[[2, 0], [0, 16]]. Is λ=2, v=[1, 1] a real eigenpair under the nonzero-eigenvector convention?","answer":{"kind":"short_text","value":"No","explanation":"Mv=[2, 16] and λv=[2, 2]. Both equality and v≠0 are required, giving No. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded."},"skills":["Eigenpair Candidate with a Nonzero Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0016","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75ddb27bf335598632dab508cb976d64435a91ae4a0f1c5d7f238b0eb3786c03","family_id":"qf-eigenpair-candidate","seed":202610074016,"semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"3\",\"0\"],[\"0\",\"17\"]],\"4\",[\"1\",\"-1\"]],\"operation\":\"eigenpair\"}"},"fields":{"title":"Eigenpair Candidate with a Nonzero Vector","stem":"Over R, the 2×2 matrix is M=[[3, 0], [0, 17]]. Is λ=4, v=[1, -1] a real eigenpair under the nonzero-eigenvector convention?","answer":{"kind":"short_text","value":"No","explanation":"Mv=[3, -17] and λv=[4, -4]. Both equality and v≠0 are required, giving No. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded."},"skills":["Eigenpair Candidate with a Nonzero Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0017","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"09a2a44e589eb5d8aaa30c7363412c3fc5decc2b95cbd0dbde6bcf114c5c5e5d","family_id":"qf-eigenpair-candidate","seed":202610074017,"semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"4\",\"0\"],[\"0\",\"18\"]],\"4\",[\"1\",\"0\"]],\"operation\":\"eigenpair\"}"},"fields":{"title":"Eigenpair Candidate with a Nonzero Vector","stem":"Over R, the 2×2 matrix is M=[[4, 0], [0, 18]]. Is λ=4, v=[1, 0] a real eigenpair under the nonzero-eigenvector convention?","answer":{"kind":"short_text","value":"Yes","explanation":"Mv=[4, 0] and λv=[4, 0]. Both equality and v≠0 are required, giving Yes. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded."},"skills":["Eigenpair Candidate with a Nonzero Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0018","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c044777e9c4ab826d2d5dc6ed20594b37042c21616b6c9dde153977c8106bc65","family_id":"qf-eigenpair-candidate","seed":202610074018,"semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"5\",\"0\"],[\"0\",\"19\"]],\"19\",[\"0\",\"1\"]],\"operation\":\"eigenpair\"}"},"fields":{"title":"Eigenpair Candidate with a Nonzero Vector","stem":"Over R, the 2×2 matrix is M=[[5, 0], [0, 19]]. Is λ=19, v=[0, 1] a real eigenpair under the nonzero-eigenvector convention?","answer":{"kind":"short_text","value":"Yes","explanation":"Mv=[0, 19] and λv=[0, 19]. Both equality and v≠0 are required, giving Yes. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded."},"skills":["Eigenpair Candidate with a Nonzero Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0019","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97614e2b337eada16232d32b96de3175aa2b0437e96d02f84e6cb6d67ea39f77","family_id":"qf-eigenpair-candidate","seed":202610074019,"semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"6\",\"0\"],[\"0\",\"20\"]],\"6\",[\"1\",\"1\"]],\"operation\":\"eigenpair\"}"},"fields":{"title":"Eigenpair Candidate with a Nonzero Vector","stem":"Over R, the 2×2 matrix is M=[[6, 0], [0, 20]]. Is λ=6, v=[1, 1] a real eigenpair under the nonzero-eigenvector convention?","answer":{"kind":"short_text","value":"No","explanation":"Mv=[6, 20] and λv=[6, 6]. Both equality and v≠0 are required, giving No. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded."},"skills":["Eigenpair Candidate with a Nonzero Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0020","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8276471f8fcc9ddab07aaadbd9957d277e230e9fb923ed324f49ed2eee6fd16a","family_id":"qf-eigenpair-candidate","seed":202610074020,"semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"7\",\"0\"],[\"0\",\"21\"]],\"8\",[\"1\",\"-1\"]],\"operation\":\"eigenpair\"}"},"fields":{"title":"Eigenpair Candidate with a Nonzero Vector","stem":"Over R, the 2×2 matrix is M=[[7, 0], [0, 21]]. Is λ=8, v=[1, -1] a real eigenpair under the nonzero-eigenvector convention?","answer":{"kind":"short_text","value":"No","explanation":"Mv=[7, -21] and λv=[8, -8]. Both equality and v≠0 are required, giving No. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded."},"skills":["Eigenpair Candidate with a Nonzero Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0021","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca2699027089dd3bcf0b149d2ed248f628db7bb973465537ca6e44d358b55fcc","family_id":"qf-eigenpair-candidate","seed":202610074021,"semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"8\",\"0\"],[\"0\",\"22\"]],\"8\",[\"1\",\"0\"]],\"operation\":\"eigenpair\"}"},"fields":{"title":"Eigenpair Candidate with a Nonzero Vector","stem":"Over R, the 2×2 matrix is M=[[8, 0], [0, 22]]. Is λ=8, v=[1, 0] a real eigenpair under the nonzero-eigenvector convention?","answer":{"kind":"short_text","value":"Yes","explanation":"Mv=[8, 0] and λv=[8, 0]. Both equality and v≠0 are required, giving Yes. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded."},"skills":["Eigenpair Candidate with a Nonzero Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0022","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a71007505e6a83e106e4cb56bf552ebf8bb3ac956bd3c49266274f95b7fd53f8","family_id":"qf-eigenpair-candidate","seed":202610074022,"semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"9\",\"0\"],[\"0\",\"23\"]],\"23\",[\"0\",\"1\"]],\"operation\":\"eigenpair\"}"},"fields":{"title":"Eigenpair Candidate with a Nonzero Vector","stem":"Over R, the 2×2 matrix is M=[[9, 0], [0, 23]]. Is λ=23, v=[0, 1] a real eigenpair under the nonzero-eigenvector convention?","answer":{"kind":"short_text","value":"Yes","explanation":"Mv=[0, 23] and λv=[0, 23]. Both equality and v≠0 are required, giving Yes. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded."},"skills":["Eigenpair Candidate with a Nonzero Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0023","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d807c0019d932127a086f638494619a07a6ca8d74230746c904f37c6f034f243","family_id":"qf-eigenpair-candidate","seed":202610074023,"semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"10\",\"0\"],[\"0\",\"24\"]],\"10\",[\"1\",\"1\"]],\"operation\":\"eigenpair\"}"},"fields":{"title":"Eigenpair Candidate with a Nonzero Vector","stem":"Over R, the 2×2 matrix is M=[[10, 0], [0, 24]]. Is λ=10, v=[1, 1] a real eigenpair under the nonzero-eigenvector convention?","answer":{"kind":"short_text","value":"No","explanation":"Mv=[10, 24] and λv=[10, 10]. Both equality and v≠0 are required, giving No. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded."},"skills":["Eigenpair Candidate with a Nonzero Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0024","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9281ebfd76d5549916c95ca846c63bd86009733dca73bed8e6eef8cc3083c94a","family_id":"qf-eigenpair-candidate","seed":202610074024,"semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"11\",\"0\"],[\"0\",\"25\"]],\"12\",[\"1\",\"-1\"]],\"operation\":\"eigenpair\"}"},"fields":{"title":"Eigenpair Candidate with a Nonzero Vector","stem":"Over R, the 2×2 matrix is M=[[11, 0], [0, 25]]. Is λ=12, v=[1, -1] a real eigenpair under the nonzero-eigenvector convention?","answer":{"kind":"short_text","value":"No","explanation":"Mv=[11, -25] and λv=[12, -12]. Both equality and v≠0 are required, giving No. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded."},"skills":["Eigenpair Candidate with a Nonzero Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0025","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"def8c13f09ff887b718fb5e997b1830bfef73be714e4f20f0d848d91daeb4455","family_id":"qf-eigenpair-candidate","seed":202610074025,"semantic_template":"qf-eigenpair-candidate/{\"inputs\":[[[\"12\",\"0\"],[\"0\",\"26\"]],\"12\",[\"1\",\"0\"]],\"operation\":\"eigenpair\"}"},"fields":{"title":"Eigenpair Candidate with a Nonzero Vector","stem":"Over R, the 2×2 matrix is M=[[12, 0], [0, 26]]. Is λ=12, v=[1, 0] a real eigenpair under the nonzero-eigenvector convention?","answer":{"kind":"short_text","value":"Yes","explanation":"Mv=[12, 0] and λv=[12, 0]. Both equality and v≠0 are required, giving Yes. The zero vector is not an eigenvector, even though the equality then holds; zero or negative eigenvalues are not automatically excluded."},"skills":["Eigenpair Candidate with a Nonzero Vector"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0026","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2638a1ad1279549d1753270df25c8d4d7c23d22f7549c24e69b7a24fd08b31dd","family_id":"qf-eigenvalue-diagonal-set","seed":202610074026,"semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"0\",\"0\"],\"operation\":\"diagonal_eigenvalues\"}"},"fields":{"title":"Distinct Real Eigenvalues of a Diagonal Matrix","stem":"Over R, the 2×2 matrix is diag(0,0). Give its distinct real eigenvalues as an increasing set with each value once.","answer":{"kind":"expression","value":"{0}","explanation":"The two diagonal values give {0}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count."},"skills":["Distinct Real Eigenvalues of a Diagonal Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0027","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c59a25e9bf96261d58d8b1790497de40262770b87988dc22673ad52ceb7eedfe","family_id":"qf-eigenvalue-diagonal-set","seed":202610074027,"semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"-2\",\"-2\"],\"operation\":\"diagonal_eigenvalues\"}"},"fields":{"title":"Distinct Real Eigenvalues of a Diagonal Matrix","stem":"Over R, the 2×2 matrix is diag(-2,-2). Give its distinct real eigenvalues as an increasing set with each value once.","answer":{"kind":"expression","value":"{-2}","explanation":"The two diagonal values give {-2}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count."},"skills":["Distinct Real Eigenvalues of a Diagonal Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0028","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16dfe67725dbdcd94e0d11eac1eee580f6217f6f1b355eb82f87b636714345b7","family_id":"qf-eigenvalue-diagonal-set","seed":202610074028,"semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"-10\",\"3\"],\"operation\":\"diagonal_eigenvalues\"}"},"fields":{"title":"Distinct Real Eigenvalues of a Diagonal Matrix","stem":"Over R, the 2×2 matrix is diag(-10,3). Give its distinct real eigenvalues as an increasing set with each value once.","answer":{"kind":"expression","value":"{-10, 3}","explanation":"The two diagonal values give {-10, 3}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count."},"skills":["Distinct Real Eigenvalues of a Diagonal Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0029","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f77d90fc2d3bd731e89d73db03d08e66c0c9b2acd18291b8595adf98fc913189","family_id":"qf-eigenvalue-diagonal-set","seed":202610074029,"semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"-9\",\"4\"],\"operation\":\"diagonal_eigenvalues\"}"},"fields":{"title":"Distinct Real Eigenvalues of a Diagonal Matrix","stem":"Over R, the 2×2 matrix is diag(-9,4). Give its distinct real eigenvalues as an increasing set with each value once.","answer":{"kind":"expression","value":"{-9, 4}","explanation":"The two diagonal values give {-9, 4}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count."},"skills":["Distinct Real Eigenvalues of a Diagonal Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0030","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2cfd8d68d13bc31f4c4ed9d71f47c88c35c3fdf5eddc38db3f5ed3018095c0d2","family_id":"qf-eigenvalue-diagonal-set","seed":202610074030,"semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"-8\",\"5\"],\"operation\":\"diagonal_eigenvalues\"}"},"fields":{"title":"Distinct Real Eigenvalues of a Diagonal Matrix","stem":"Over R, the 2×2 matrix is diag(-8,5). Give its distinct real eigenvalues as an increasing set with each value once.","answer":{"kind":"expression","value":"{-8, 5}","explanation":"The two diagonal values give {-8, 5}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count."},"skills":["Distinct Real Eigenvalues of a Diagonal Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0031","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0cae43af9140f753792d6d6db5b39c0e0d3053a190baacb80521e053cbd2fb25","family_id":"qf-eigenvalue-diagonal-set","seed":202610074031,"semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"-7\",\"6\"],\"operation\":\"diagonal_eigenvalues\"}"},"fields":{"title":"Distinct Real Eigenvalues of a Diagonal Matrix","stem":"Over R, the 2×2 matrix is diag(-7,6). Give its distinct real eigenvalues as an increasing set with each value once.","answer":{"kind":"expression","value":"{-7, 6}","explanation":"The two diagonal values give {-7, 6}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count."},"skills":["Distinct Real Eigenvalues of a Diagonal Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0032","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba289ebaebed0b5de7991951103ca3b41d3466eeb7af81dab1ea990e8afd5fad","family_id":"qf-eigenvalue-diagonal-set","seed":202610074032,"semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"-6\",\"7\"],\"operation\":\"diagonal_eigenvalues\"}"},"fields":{"title":"Distinct Real Eigenvalues of a Diagonal Matrix","stem":"Over R, the 2×2 matrix is diag(-6,7). Give its distinct real eigenvalues as an increasing set with each value once.","answer":{"kind":"expression","value":"{-6, 7}","explanation":"The two diagonal values give {-6, 7}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count."},"skills":["Distinct Real Eigenvalues of a Diagonal Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0033","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6029be192ff3f7e135f69e2b26a24bb4acf8e0a344608ada62c9fee7baa3dc17","family_id":"qf-eigenvalue-diagonal-set","seed":202610074033,"semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"-5\",\"8\"],\"operation\":\"diagonal_eigenvalues\"}"},"fields":{"title":"Distinct Real Eigenvalues of a Diagonal Matrix","stem":"Over R, the 2×2 matrix is diag(-5,8). Give its distinct real eigenvalues as an increasing set with each value once.","answer":{"kind":"expression","value":"{-5, 8}","explanation":"The two diagonal values give {-5, 8}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count."},"skills":["Distinct Real Eigenvalues of a Diagonal Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0034","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2cfc214c4336ab486c92c33aa1ac0ff4eb505c42b004ca0634b1bc4438f55393","family_id":"qf-eigenvalue-diagonal-set","seed":202610074034,"semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"-4\",\"9\"],\"operation\":\"diagonal_eigenvalues\"}"},"fields":{"title":"Distinct Real Eigenvalues of a Diagonal Matrix","stem":"Over R, the 2×2 matrix is diag(-4,9). Give its distinct real eigenvalues as an increasing set with each value once.","answer":{"kind":"expression","value":"{-4, 9}","explanation":"The two diagonal values give {-4, 9}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count."},"skills":["Distinct Real Eigenvalues of a Diagonal Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0035","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1afa72952785c2bd1a5d445fee3ed33f803cfe9981c01c5f447a7811e6fe3bf1","family_id":"qf-eigenvalue-diagonal-set","seed":202610074035,"semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"-3\",\"10\"],\"operation\":\"diagonal_eigenvalues\"}"},"fields":{"title":"Distinct Real Eigenvalues of a Diagonal Matrix","stem":"Over R, the 2×2 matrix is diag(-3,10). Give its distinct real eigenvalues as an increasing set with each value once.","answer":{"kind":"expression","value":"{-3, 10}","explanation":"The two diagonal values give {-3, 10}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count."},"skills":["Distinct Real Eigenvalues of a Diagonal Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0036","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd09b9d367e4391b88aaac952e88004a6195473c22f787b0b3ebcfb5ad6d6cb5","family_id":"qf-eigenvalue-diagonal-set","seed":202610074036,"semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"-2\",\"11\"],\"operation\":\"diagonal_eigenvalues\"}"},"fields":{"title":"Distinct Real Eigenvalues of a Diagonal Matrix","stem":"Over R, the 2×2 matrix is diag(-2,11). Give its distinct real eigenvalues as an increasing set with each value once.","answer":{"kind":"expression","value":"{-2, 11}","explanation":"The two diagonal values give {-2, 11}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count."},"skills":["Distinct Real Eigenvalues of a Diagonal Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0037","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b04c02262c8f8c6f913a2a4ede8f9f63290ae4e0231a8a12026be82862900fea","family_id":"qf-eigenvalue-diagonal-set","seed":202610074037,"semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"-1\",\"12\"],\"operation\":\"diagonal_eigenvalues\"}"},"fields":{"title":"Distinct Real Eigenvalues of a Diagonal Matrix","stem":"Over R, the 2×2 matrix is diag(-1,12). Give its distinct real eigenvalues as an increasing set with each value once.","answer":{"kind":"expression","value":"{-1, 12}","explanation":"The two diagonal values give {-1, 12}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count."},"skills":["Distinct Real Eigenvalues of a Diagonal Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0038","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc8c6a3a8466167b3e60832be03a7f3893521d20de959bb0833d9e0a8622b290","family_id":"qf-eigenvalue-diagonal-set","seed":202610074038,"semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"0\",\"13\"],\"operation\":\"diagonal_eigenvalues\"}"},"fields":{"title":"Distinct Real Eigenvalues of a Diagonal Matrix","stem":"Over R, the 2×2 matrix is diag(0,13). Give its distinct real eigenvalues as an increasing set with each value once.","answer":{"kind":"expression","value":"{0, 13}","explanation":"The two diagonal values give {0, 13}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count."},"skills":["Distinct Real Eigenvalues of a Diagonal Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0039","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"973acad67bc675515a1fb0afa4a8b8327a76a43bca552fbad66473e28133e5f1","family_id":"qf-eigenvalue-diagonal-set","seed":202610074039,"semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"1\",\"14\"],\"operation\":\"diagonal_eigenvalues\"}"},"fields":{"title":"Distinct Real Eigenvalues of a Diagonal Matrix","stem":"Over R, the 2×2 matrix is diag(1,14). Give its distinct real eigenvalues as an increasing set with each value once.","answer":{"kind":"expression","value":"{1, 14}","explanation":"The two diagonal values give {1, 14}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count."},"skills":["Distinct Real Eigenvalues of a Diagonal Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0040","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e9041d9886afaad7819646093cd2b283209cfe751de0638fc7ee61b0d69f971","family_id":"qf-eigenvalue-diagonal-set","seed":202610074040,"semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"2\",\"15\"],\"operation\":\"diagonal_eigenvalues\"}"},"fields":{"title":"Distinct Real Eigenvalues of a Diagonal Matrix","stem":"Over R, the 2×2 matrix is diag(2,15). Give its distinct real eigenvalues as an increasing set with each value once.","answer":{"kind":"expression","value":"{2, 15}","explanation":"The two diagonal values give {2, 15}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count."},"skills":["Distinct Real Eigenvalues of a Diagonal Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0041","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79612bb546995ad81b776ac55e9d342ec728b050b8eada271752d1e0ecba4744","family_id":"qf-eigenvalue-diagonal-set","seed":202610074041,"semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"3\",\"16\"],\"operation\":\"diagonal_eigenvalues\"}"},"fields":{"title":"Distinct Real Eigenvalues of a Diagonal Matrix","stem":"Over R, the 2×2 matrix is diag(3,16). Give its distinct real eigenvalues as an increasing set with each value once.","answer":{"kind":"expression","value":"{3, 16}","explanation":"The two diagonal values give {3, 16}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count."},"skills":["Distinct Real Eigenvalues of a Diagonal Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0042","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec669e7ae1fb7603887de984d5d9f1e4a689224ea6aa2a8be39e5ad0e17dc584","family_id":"qf-eigenvalue-diagonal-set","seed":202610074042,"semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"4\",\"17\"],\"operation\":\"diagonal_eigenvalues\"}"},"fields":{"title":"Distinct Real Eigenvalues of a Diagonal Matrix","stem":"Over R, the 2×2 matrix is diag(4,17). Give its distinct real eigenvalues as an increasing set with each value once.","answer":{"kind":"expression","value":"{4, 17}","explanation":"The two diagonal values give {4, 17}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count."},"skills":["Distinct Real Eigenvalues of a Diagonal Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0043","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63aac503560d6a7ef02a4e04d3af7159f87fa3d0297710e395e53bdf1f19ffdd","family_id":"qf-eigenvalue-diagonal-set","seed":202610074043,"semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"5\",\"18\"],\"operation\":\"diagonal_eigenvalues\"}"},"fields":{"title":"Distinct Real Eigenvalues of a Diagonal Matrix","stem":"Over R, the 2×2 matrix is diag(5,18). Give its distinct real eigenvalues as an increasing set with each value once.","answer":{"kind":"expression","value":"{5, 18}","explanation":"The two diagonal values give {5, 18}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count."},"skills":["Distinct Real Eigenvalues of a Diagonal Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0044","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f25db7429c7556f6af29b9a20f78d07dd018585baccc59ee86af402d78b4171","family_id":"qf-eigenvalue-diagonal-set","seed":202610074044,"semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"6\",\"19\"],\"operation\":\"diagonal_eigenvalues\"}"},"fields":{"title":"Distinct Real Eigenvalues of a Diagonal Matrix","stem":"Over R, the 2×2 matrix is diag(6,19). Give its distinct real eigenvalues as an increasing set with each value once.","answer":{"kind":"expression","value":"{6, 19}","explanation":"The two diagonal values give {6, 19}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count."},"skills":["Distinct Real Eigenvalues of a Diagonal Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0045","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1507005f50509aea90b9ce25e4d1eff9e75e29f2164b7e846d3faa74c3cfb32d","family_id":"qf-eigenvalue-diagonal-set","seed":202610074045,"semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"7\",\"20\"],\"operation\":\"diagonal_eigenvalues\"}"},"fields":{"title":"Distinct Real Eigenvalues of a Diagonal Matrix","stem":"Over R, the 2×2 matrix is diag(7,20). Give its distinct real eigenvalues as an increasing set with each value once.","answer":{"kind":"expression","value":"{7, 20}","explanation":"The two diagonal values give {7, 20}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count."},"skills":["Distinct Real Eigenvalues of a Diagonal Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0046","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a08f3dca1877aa100e0d6fc19a96b640bf00cd0120e838deaf765252a706ba8f","family_id":"qf-eigenvalue-diagonal-set","seed":202610074046,"semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"8\",\"21\"],\"operation\":\"diagonal_eigenvalues\"}"},"fields":{"title":"Distinct Real Eigenvalues of a Diagonal Matrix","stem":"Over R, the 2×2 matrix is diag(8,21). Give its distinct real eigenvalues as an increasing set with each value once.","answer":{"kind":"expression","value":"{8, 21}","explanation":"The two diagonal values give {8, 21}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count."},"skills":["Distinct Real Eigenvalues of a Diagonal Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0047","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa6d1e9abc694e360b15ff7be3b73f904a55f7ae20f5f8a057336b4ba0cc540a","family_id":"qf-eigenvalue-diagonal-set","seed":202610074047,"semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"9\",\"22\"],\"operation\":\"diagonal_eigenvalues\"}"},"fields":{"title":"Distinct Real Eigenvalues of a Diagonal Matrix","stem":"Over R, the 2×2 matrix is diag(9,22). Give its distinct real eigenvalues as an increasing set with each value once.","answer":{"kind":"expression","value":"{9, 22}","explanation":"The two diagonal values give {9, 22}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count."},"skills":["Distinct Real Eigenvalues of a Diagonal Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0048","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d2b59d2ab126f3049ef25a0acc3b60c24b902491d05e70fcb95d70d50bb531e","family_id":"qf-eigenvalue-diagonal-set","seed":202610074048,"semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"10\",\"23\"],\"operation\":\"diagonal_eigenvalues\"}"},"fields":{"title":"Distinct Real Eigenvalues of a Diagonal Matrix","stem":"Over R, the 2×2 matrix is diag(10,23). Give its distinct real eigenvalues as an increasing set with each value once.","answer":{"kind":"expression","value":"{10, 23}","explanation":"The two diagonal values give {10, 23}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count."},"skills":["Distinct Real Eigenvalues of a Diagonal Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0049","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31db21b971cb67b10ce0e1d85607e00131f30f4133fe0a6d88ab49d7848e4f13","family_id":"qf-eigenvalue-diagonal-set","seed":202610074049,"semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"11\",\"24\"],\"operation\":\"diagonal_eigenvalues\"}"},"fields":{"title":"Distinct Real Eigenvalues of a Diagonal Matrix","stem":"Over R, the 2×2 matrix is diag(11,24). Give its distinct real eigenvalues as an increasing set with each value once.","answer":{"kind":"expression","value":"{11, 24}","explanation":"The two diagonal values give {11, 24}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count."},"skills":["Distinct Real Eigenvalues of a Diagonal Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0050","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27bed51a5f761ed82522fb5ec1efca0e6ae3a745a227476f7946c0d781a35cd1","family_id":"qf-eigenvalue-diagonal-set","seed":202610074050,"semantic_template":"qf-eigenvalue-diagonal-set/{\"inputs\":[\"12\",\"25\"],\"operation\":\"diagonal_eigenvalues\"}"},"fields":{"title":"Distinct Real Eigenvalues of a Diagonal Matrix","stem":"Over R, the 2×2 matrix is diag(12,25). Give its distinct real eigenvalues as an increasing set with each value once.","answer":{"kind":"expression","value":"{12, 25}","explanation":"The two diagonal values give {12, 25}. Equal values are listed once; this is a set of distinct eigenvalues, not an algebraic-multiplicity count."},"skills":["Distinct Real Eigenvalues of a Diagonal Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0051","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"301e16af27d7e9d42410e299ec89dcddb551bff5b5d33279e9ec579b3f00aa13","family_id":"qf-eigenvalue-kernel-membership","seed":202610074051,"semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[\"zero_vector\"],\"operation\":\"kernel_membership\"}"},"fields":{"title":"Kernel Membership for a Candidate Scalar","stem":"Over R, the 2×2 matrix is M=[[2, 1], [0, 3]]. For the candidate scalar λ=99 and v=[0, 0], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.","answer":{"kind":"short_text","value":"Yes","explanation":"The residual (M−λI)v is [0, 0], giving Yes. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue."},"skills":["Kernel Membership for a Candidate Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0052","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c8dc86be72c89ed065e81bf74e9b1d17c4e99485b99f51bcb548019b055e0518","family_id":"qf-eigenvalue-kernel-membership","seed":202610074052,"semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"-11\",\"0\"],[\"0\",\"3\"]],\"3\",[\"0\",\"1\"]],\"operation\":\"kernel_membership\"}"},"fields":{"title":"Kernel Membership for a Candidate Scalar","stem":"Over R, the 2×2 matrix is M=[[-11, 0], [0, 3]]. For the candidate scalar λ=3 and v=[0, 1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.","answer":{"kind":"short_text","value":"Yes","explanation":"The residual (M−λI)v is [0, 0], giving Yes. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue."},"skills":["Kernel Membership for a Candidate Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0053","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c5652d3838a9d92602d86185d039912d60c38a74c2998772f673c11f1717891","family_id":"qf-eigenvalue-kernel-membership","seed":202610074053,"semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"-10\",\"0\"],[\"0\",\"4\"]],\"-10\",[\"1\",\"1\"]],\"operation\":\"kernel_membership\"}"},"fields":{"title":"Kernel Membership for a Candidate Scalar","stem":"Over R, the 2×2 matrix is M=[[-10, 0], [0, 4]]. For the candidate scalar λ=-10 and v=[1, 1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.","answer":{"kind":"short_text","value":"No","explanation":"The residual (M−λI)v is [0, 14], giving No. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue."},"skills":["Kernel Membership for a Candidate Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0054","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47ee3c7cd812b82f95a9eb146fc88c04278659c32739df39111a33079367c6a0","family_id":"qf-eigenvalue-kernel-membership","seed":202610074054,"semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"-9\",\"0\"],[\"0\",\"5\"]],\"-8\",[\"1\",\"-1\"]],\"operation\":\"kernel_membership\"}"},"fields":{"title":"Kernel Membership for a Candidate Scalar","stem":"Over R, the 2×2 matrix is M=[[-9, 0], [0, 5]]. For the candidate scalar λ=-8 and v=[1, -1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.","answer":{"kind":"short_text","value":"No","explanation":"The residual (M−λI)v is [-1, -13], giving No. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue."},"skills":["Kernel Membership for a Candidate Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0055","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"922a79a98710cda4b1717df504aea37a3b4ab276790ebece84e47a00561f9ca2","family_id":"qf-eigenvalue-kernel-membership","seed":202610074055,"semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"4\",\"1\"],[\"1\",\"4\"]],\"5\",[\"1\",\"1\"]],\"operation\":\"kernel_membership\"}"},"fields":{"title":"Kernel Membership for a Candidate Scalar","stem":"Over R, the 2×2 matrix is M=[[4, 1], [1, 4]]. For the candidate scalar λ=5 and v=[1, 1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.","answer":{"kind":"short_text","value":"Yes","explanation":"The residual (M−λI)v is [0, 0], giving Yes. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue."},"skills":["Kernel Membership for a Candidate Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0056","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2beca5e54ad0cd56ccd998cfe8d047677e02e91045d6ccde2ee841ef5ef7b1fd","family_id":"qf-eigenvalue-kernel-membership","seed":202610074056,"semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"5\",\"0\"],[\"1\",\"6\"]],\"5\",[\"1\",\"-1\"]],\"operation\":\"kernel_membership\"}"},"fields":{"title":"Kernel Membership for a Candidate Scalar","stem":"Over R, the 2×2 matrix is M=[[5, 0], [1, 6]]. For the candidate scalar λ=5 and v=[1, -1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.","answer":{"kind":"short_text","value":"Yes","explanation":"The residual (M−λI)v is [0, 0], giving Yes. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue."},"skills":["Kernel Membership for a Candidate Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0057","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5362fddb21fa0b7689c66cbc7cc4ee308729fb285e242af15ac3f0d71cd61493","family_id":"qf-eigenvalue-kernel-membership","seed":202610074057,"semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"-6\",\"0\"],[\"0\",\"8\"]],\"-6\",[\"1\",\"1\"]],\"operation\":\"kernel_membership\"}"},"fields":{"title":"Kernel Membership for a Candidate Scalar","stem":"Over R, the 2×2 matrix is M=[[-6, 0], [0, 8]]. For the candidate scalar λ=-6 and v=[1, 1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.","answer":{"kind":"short_text","value":"No","explanation":"The residual (M−λI)v is [0, 14], giving No. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue."},"skills":["Kernel Membership for a Candidate Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0058","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"730887417ada7c14fe0468a11bca836118b3e47356bbbc804840434591a27567","family_id":"qf-eigenvalue-kernel-membership","seed":202610074058,"semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"-5\",\"0\"],[\"0\",\"9\"]],\"-4\",[\"1\",\"-1\"]],\"operation\":\"kernel_membership\"}"},"fields":{"title":"Kernel Membership for a Candidate Scalar","stem":"Over R, the 2×2 matrix is M=[[-5, 0], [0, 9]]. For the candidate scalar λ=-4 and v=[1, -1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.","answer":{"kind":"short_text","value":"No","explanation":"The residual (M−λI)v is [-1, -13], giving No. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue."},"skills":["Kernel Membership for a Candidate Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0059","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e161030428ae2fceec6c08e3e802f1c7166052865675b86aac91e4d3138e92f0","family_id":"qf-eigenvalue-kernel-membership","seed":202610074059,"semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"-4\",\"0\"],[\"0\",\"10\"]],\"-4\",[\"1\",\"0\"]],\"operation\":\"kernel_membership\"}"},"fields":{"title":"Kernel Membership for a Candidate Scalar","stem":"Over R, the 2×2 matrix is M=[[-4, 0], [0, 10]]. For the candidate scalar λ=-4 and v=[1, 0], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.","answer":{"kind":"short_text","value":"Yes","explanation":"The residual (M−λI)v is [0, 0], giving Yes. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue."},"skills":["Kernel Membership for a Candidate Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0060","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0c5d74ad818b5f50915b50fa23f40096e3d48a879d947d22fbee1b9e653dc0f","family_id":"qf-eigenvalue-kernel-membership","seed":202610074060,"semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"-3\",\"0\"],[\"0\",\"11\"]],\"11\",[\"0\",\"1\"]],\"operation\":\"kernel_membership\"}"},"fields":{"title":"Kernel Membership for a Candidate Scalar","stem":"Over R, the 2×2 matrix is M=[[-3, 0], [0, 11]]. For the candidate scalar λ=11 and v=[0, 1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.","answer":{"kind":"short_text","value":"Yes","explanation":"The residual (M−λI)v is [0, 0], giving Yes. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue."},"skills":["Kernel Membership for a Candidate Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0061","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"faf1c2301581e3dd89faeb4630a127f9f59c3b1cf33cbba0a0b49631c8a44c97","family_id":"qf-eigenvalue-kernel-membership","seed":202610074061,"semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"-2\",\"0\"],[\"0\",\"12\"]],\"-2\",[\"1\",\"1\"]],\"operation\":\"kernel_membership\"}"},"fields":{"title":"Kernel Membership for a Candidate Scalar","stem":"Over R, the 2×2 matrix is M=[[-2, 0], [0, 12]]. For the candidate scalar λ=-2 and v=[1, 1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.","answer":{"kind":"short_text","value":"No","explanation":"The residual (M−λI)v is [0, 14], giving No. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue."},"skills":["Kernel Membership for a Candidate Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0062","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3743a10a567f210c35e3e884369aaea9fef1d5d657802e43d67dee687960c490","family_id":"qf-eigenvalue-kernel-membership","seed":202610074062,"semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"-1\",\"0\"],[\"0\",\"13\"]],\"0\",[\"1\",\"-1\"]],\"operation\":\"kernel_membership\"}"},"fields":{"title":"Kernel Membership for a Candidate Scalar","stem":"Over R, the 2×2 matrix is M=[[-1, 0], [0, 13]]. For the candidate scalar λ=0 and v=[1, -1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.","answer":{"kind":"short_text","value":"No","explanation":"The residual (M−λI)v is [-1, -13], giving No. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue."},"skills":["Kernel Membership for a Candidate Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0063","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"968bbe53f8e8b07d5871b6ea13800880b09236628e7be52d35e1f41ef4103904","family_id":"qf-eigenvalue-kernel-membership","seed":202610074063,"semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"0\",\"0\"],[\"0\",\"14\"]],\"0\",[\"1\",\"0\"]],\"operation\":\"kernel_membership\"}"},"fields":{"title":"Kernel Membership for a Candidate Scalar","stem":"Over R, the 2×2 matrix is M=[[0, 0], [0, 14]]. For the candidate scalar λ=0 and v=[1, 0], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.","answer":{"kind":"short_text","value":"Yes","explanation":"The residual (M−λI)v is [0, 0], giving Yes. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue."},"skills":["Kernel Membership for a Candidate Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0064","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff267d08959e30705c9db2c73bfeabca1a074f35ff5da3749d16ea34d4ac6054","family_id":"qf-eigenvalue-kernel-membership","seed":202610074064,"semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"1\",\"0\"],[\"0\",\"15\"]],\"15\",[\"0\",\"1\"]],\"operation\":\"kernel_membership\"}"},"fields":{"title":"Kernel Membership for a Candidate Scalar","stem":"Over R, the 2×2 matrix is M=[[1, 0], [0, 15]]. For the candidate scalar λ=15 and v=[0, 1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.","answer":{"kind":"short_text","value":"Yes","explanation":"The residual (M−λI)v is [0, 0], giving Yes. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue."},"skills":["Kernel Membership for a Candidate Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0065","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9df1df75fa0374e46e72b2a2d99aeea758e6fc9f6114e5ca7b1ed27e2ab1dbdb","family_id":"qf-eigenvalue-kernel-membership","seed":202610074065,"semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"2\",\"0\"],[\"0\",\"16\"]],\"2\",[\"1\",\"1\"]],\"operation\":\"kernel_membership\"}"},"fields":{"title":"Kernel Membership for a Candidate Scalar","stem":"Over R, the 2×2 matrix is M=[[2, 0], [0, 16]]. For the candidate scalar λ=2 and v=[1, 1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.","answer":{"kind":"short_text","value":"No","explanation":"The residual (M−λI)v is [0, 14], giving No. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue."},"skills":["Kernel Membership for a Candidate Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0066","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9287d8ec96e5351c759553b903c573a612fad4a7e271c01ac687a1d01ca2aa13","family_id":"qf-eigenvalue-kernel-membership","seed":202610074066,"semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"3\",\"0\"],[\"0\",\"17\"]],\"4\",[\"1\",\"-1\"]],\"operation\":\"kernel_membership\"}"},"fields":{"title":"Kernel Membership for a Candidate Scalar","stem":"Over R, the 2×2 matrix is M=[[3, 0], [0, 17]]. For the candidate scalar λ=4 and v=[1, -1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.","answer":{"kind":"short_text","value":"No","explanation":"The residual (M−λI)v is [-1, -13], giving No. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue."},"skills":["Kernel Membership for a Candidate Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0067","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71b3c2c484a04c2e653f29b1bb70ce1758ed4f07b7e64ad53bc740f99a475e88","family_id":"qf-eigenvalue-kernel-membership","seed":202610074067,"semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"4\",\"0\"],[\"0\",\"18\"]],\"4\",[\"1\",\"0\"]],\"operation\":\"kernel_membership\"}"},"fields":{"title":"Kernel Membership for a Candidate Scalar","stem":"Over R, the 2×2 matrix is M=[[4, 0], [0, 18]]. For the candidate scalar λ=4 and v=[1, 0], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.","answer":{"kind":"short_text","value":"Yes","explanation":"The residual (M−λI)v is [0, 0], giving Yes. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue."},"skills":["Kernel Membership for a Candidate Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0068","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23ed7f96b0f38c03215dc3ae36a38c6235fa2163d4fbe34e93a14b6724c0da82","family_id":"qf-eigenvalue-kernel-membership","seed":202610074068,"semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"5\",\"0\"],[\"0\",\"19\"]],\"19\",[\"0\",\"1\"]],\"operation\":\"kernel_membership\"}"},"fields":{"title":"Kernel Membership for a Candidate Scalar","stem":"Over R, the 2×2 matrix is M=[[5, 0], [0, 19]]. For the candidate scalar λ=19 and v=[0, 1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.","answer":{"kind":"short_text","value":"Yes","explanation":"The residual (M−λI)v is [0, 0], giving Yes. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue."},"skills":["Kernel Membership for a Candidate Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0069","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ce1d56e8a5e5f59af1c975555bcf38718f4bf5afa056f659d92f63b777e52a7","family_id":"qf-eigenvalue-kernel-membership","seed":202610074069,"semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"6\",\"0\"],[\"0\",\"20\"]],\"6\",[\"1\",\"1\"]],\"operation\":\"kernel_membership\"}"},"fields":{"title":"Kernel Membership for a Candidate Scalar","stem":"Over R, the 2×2 matrix is M=[[6, 0], [0, 20]]. For the candidate scalar λ=6 and v=[1, 1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.","answer":{"kind":"short_text","value":"No","explanation":"The residual (M−λI)v is [0, 14], giving No. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue."},"skills":["Kernel Membership for a Candidate Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0070","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6268b6d6707538e0771f058c6e560ff7d88fce2cb1f2c7f7c2e62a18f0ef8d7","family_id":"qf-eigenvalue-kernel-membership","seed":202610074070,"semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"7\",\"0\"],[\"0\",\"21\"]],\"8\",[\"1\",\"-1\"]],\"operation\":\"kernel_membership\"}"},"fields":{"title":"Kernel Membership for a Candidate Scalar","stem":"Over R, the 2×2 matrix is M=[[7, 0], [0, 21]]. For the candidate scalar λ=8 and v=[1, -1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.","answer":{"kind":"short_text","value":"No","explanation":"The residual (M−λI)v is [-1, -13], giving No. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue."},"skills":["Kernel Membership for a Candidate Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0071","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b86d2d807f1ec58a86eb8a367b2dd8c17fd7221e7036cdd3d190e7336874df5","family_id":"qf-eigenvalue-kernel-membership","seed":202610074071,"semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"8\",\"0\"],[\"0\",\"22\"]],\"8\",[\"1\",\"0\"]],\"operation\":\"kernel_membership\"}"},"fields":{"title":"Kernel Membership for a Candidate Scalar","stem":"Over R, the 2×2 matrix is M=[[8, 0], [0, 22]]. For the candidate scalar λ=8 and v=[1, 0], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.","answer":{"kind":"short_text","value":"Yes","explanation":"The residual (M−λI)v is [0, 0], giving Yes. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue."},"skills":["Kernel Membership for a Candidate Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0072","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"831ec9e11e9a2ade45365f4ebd3e119fb44c66be5ec8e000aaef0f68ef49f39d","family_id":"qf-eigenvalue-kernel-membership","seed":202610074072,"semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"9\",\"0\"],[\"0\",\"23\"]],\"23\",[\"0\",\"1\"]],\"operation\":\"kernel_membership\"}"},"fields":{"title":"Kernel Membership for a Candidate Scalar","stem":"Over R, the 2×2 matrix is M=[[9, 0], [0, 23]]. For the candidate scalar λ=23 and v=[0, 1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.","answer":{"kind":"short_text","value":"Yes","explanation":"The residual (M−λI)v is [0, 0], giving Yes. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue."},"skills":["Kernel Membership for a Candidate Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0073","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"defe2fad9ee51459e3634f4f304043432d51399e3d38845409dfcc33b60677df","family_id":"qf-eigenvalue-kernel-membership","seed":202610074073,"semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"10\",\"0\"],[\"0\",\"24\"]],\"10\",[\"1\",\"1\"]],\"operation\":\"kernel_membership\"}"},"fields":{"title":"Kernel Membership for a Candidate Scalar","stem":"Over R, the 2×2 matrix is M=[[10, 0], [0, 24]]. For the candidate scalar λ=10 and v=[1, 1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.","answer":{"kind":"short_text","value":"No","explanation":"The residual (M−λI)v is [0, 14], giving No. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue."},"skills":["Kernel Membership for a Candidate Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0074","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d1b59d1d4585785308c4e77e1c3d26f1d208423e47c0f5d726931a611f7ab16","family_id":"qf-eigenvalue-kernel-membership","seed":202610074074,"semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"11\",\"0\"],[\"0\",\"25\"]],\"12\",[\"1\",\"-1\"]],\"operation\":\"kernel_membership\"}"},"fields":{"title":"Kernel Membership for a Candidate Scalar","stem":"Over R, the 2×2 matrix is M=[[11, 0], [0, 25]]. For the candidate scalar λ=12 and v=[1, -1], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.","answer":{"kind":"short_text","value":"No","explanation":"The residual (M−λI)v is [-1, -13], giving No. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue."},"skills":["Kernel Membership for a Candidate Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0075","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"449c70e672e77292a69c4d1ad3c75f5b8ccbcb376f019f318e2cdd3afb0f3955","family_id":"qf-eigenvalue-kernel-membership","seed":202610074075,"semantic_template":"qf-eigenvalue-kernel-membership/{\"inputs\":[[[\"12\",\"0\"],[\"0\",\"26\"]],\"12\",[\"1\",\"0\"]],\"operation\":\"kernel_membership\"}"},"fields":{"title":"Kernel Membership for a Candidate Scalar","stem":"Over R, the 2×2 matrix is M=[[12, 0], [0, 26]]. For the candidate scalar λ=12 and v=[1, 0], does v belong to ker(M−λI)? Do not assume λ is an eigenvalue.","answer":{"kind":"short_text","value":"Yes","explanation":"The residual (M−λI)v is [0, 0], giving Yes. Every such kernel contains zero, and zero-vector membership does not establish that λ is an eigenvalue."},"skills":["Kernel Membership for a Candidate Scalar"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0076","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0574943a30e6338dc8fb7b596a7f1381f32fd70df15848f6cc03ce079c5ad9af","family_id":"qf-eigenvalue-real-count","seed":202610074076,"semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"0\",\"0\"],\"operation\":\"real_eigenvalue_count\"}"},"fields":{"title":"Number of Distinct Real Eigenvalues of a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[0, 0], [0, 0]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.","answer":{"kind":"short_text","value":"1","explanation":"The characteristic polynomial coefficients are (1,0,0), with discriminant Δ=0. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 1. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here."},"skills":["Number of Distinct Real Eigenvalues of a 2×2 Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0077","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b0e874edfe1a465421990cc1dff248962a2bca7b1aacc7a1919d2da077256cd","family_id":"qf-eigenvalue-real-count","seed":202610074077,"semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"2\",\"3\"],\"operation\":\"real_eigenvalue_count\"}"},"fields":{"title":"Number of Distinct Real Eigenvalues of a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[1, -1], [2, 1]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.","answer":{"kind":"short_text","value":"0","explanation":"The characteristic polynomial coefficients are (1,-2,3), with discriminant Δ=-8. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 0. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here."},"skills":["Number of Distinct Real Eigenvalues of a 2×2 Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0078","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6aebcaf2f931fb0b922ebc1ed5ee473fd9843c8d0b4bd354a7558a7cc2193c36","family_id":"qf-eigenvalue-real-count","seed":202610074078,"semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"2\",\"1\"],\"operation\":\"real_eigenvalue_count\"}"},"fields":{"title":"Number of Distinct Real Eigenvalues of a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[1, 1], [0, 1]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.","answer":{"kind":"short_text","value":"1","explanation":"The characteristic polynomial coefficients are (1,-2,1), with discriminant Δ=0. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 1. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here."},"skills":["Number of Distinct Real Eigenvalues of a 2×2 Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0079","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71dfbc5f5a2967843983cdc340a8472beccd883f29cfe58ad8b377db0ffb9143","family_id":"qf-eigenvalue-real-count","seed":202610074079,"semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"0\",\"-2\"],\"operation\":\"real_eigenvalue_count\"}"},"fields":{"title":"Number of Distinct Real Eigenvalues of a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[-1, 1], [1, 1]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.","answer":{"kind":"short_text","value":"2","explanation":"The characteristic polynomial coefficients are (1,0,-2), with discriminant Δ=8. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 2. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here."},"skills":["Number of Distinct Real Eigenvalues of a 2×2 Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0080","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"285151c1d13460368fd2002f8f31843e0de0e1d84d6e48bcd4a9408a7d835de6","family_id":"qf-eigenvalue-real-count","seed":202610074080,"semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"4\",\"7\"],\"operation\":\"real_eigenvalue_count\"}"},"fields":{"title":"Number of Distinct Real Eigenvalues of a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[2, -1], [3, 2]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.","answer":{"kind":"short_text","value":"0","explanation":"The characteristic polynomial coefficients are (1,-4,7), with discriminant Δ=-12. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 0. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here."},"skills":["Number of Distinct Real Eigenvalues of a 2×2 Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0081","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9031942e3f504a87dbd9e6746baf260dafc7f0980d8e7e43fec4ef66a5f0a0bf","family_id":"qf-eigenvalue-real-count","seed":202610074081,"semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"4\",\"4\"],\"operation\":\"real_eigenvalue_count\"}"},"fields":{"title":"Number of Distinct Real Eigenvalues of a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[2, 1], [0, 2]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.","answer":{"kind":"short_text","value":"1","explanation":"The characteristic polynomial coefficients are (1,-4,4), with discriminant Δ=0. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 1. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here."},"skills":["Number of Distinct Real Eigenvalues of a 2×2 Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0082","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"245d1a6efb135b30df514a13ef9f5d3111f4e08b1f1aa0c590ca9db459477a31","family_id":"qf-eigenvalue-real-count","seed":202610074082,"semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"0\",\"-5\"],\"operation\":\"real_eigenvalue_count\"}"},"fields":{"title":"Number of Distinct Real Eigenvalues of a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[-2, 1], [1, 2]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.","answer":{"kind":"short_text","value":"2","explanation":"The characteristic polynomial coefficients are (1,0,-5), with discriminant Δ=20. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 2. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here."},"skills":["Number of Distinct Real Eigenvalues of a 2×2 Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0083","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e24875cf745d2bbe5ce3b3c1449fecfe9b422f1ed243f5e2d9168a2ec0ce997","family_id":"qf-eigenvalue-real-count","seed":202610074083,"semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"6\",\"13\"],\"operation\":\"real_eigenvalue_count\"}"},"fields":{"title":"Number of Distinct Real Eigenvalues of a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[3, -1], [4, 3]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.","answer":{"kind":"short_text","value":"0","explanation":"The characteristic polynomial coefficients are (1,-6,13), with discriminant Δ=-16. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 0. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here."},"skills":["Number of Distinct Real Eigenvalues of a 2×2 Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0084","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d67bb411999dfb695860fe0c5c4e5ee4093fb33922c2ec76a5ccdddd17f9ccc8","family_id":"qf-eigenvalue-real-count","seed":202610074084,"semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"6\",\"9\"],\"operation\":\"real_eigenvalue_count\"}"},"fields":{"title":"Number of Distinct Real Eigenvalues of a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[3, 1], [0, 3]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.","answer":{"kind":"short_text","value":"1","explanation":"The characteristic polynomial coefficients are (1,-6,9), with discriminant Δ=0. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 1. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here."},"skills":["Number of Distinct Real Eigenvalues of a 2×2 Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0085","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"440c9363f04da2e0d63350faba3904fb04c516ffdd52c2ce8684dbd48ba9d105","family_id":"qf-eigenvalue-real-count","seed":202610074085,"semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"0\",\"-10\"],\"operation\":\"real_eigenvalue_count\"}"},"fields":{"title":"Number of Distinct Real Eigenvalues of a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[-3, 1], [1, 3]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.","answer":{"kind":"short_text","value":"2","explanation":"The characteristic polynomial coefficients are (1,0,-10), with discriminant Δ=40. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 2. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here."},"skills":["Number of Distinct Real Eigenvalues of a 2×2 Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0086","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3226bd59e230c282f5c55d873895bc296792a91753e3fd2be7ae1c252d80ad82","family_id":"qf-eigenvalue-real-count","seed":202610074086,"semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"8\",\"21\"],\"operation\":\"real_eigenvalue_count\"}"},"fields":{"title":"Number of Distinct Real Eigenvalues of a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[4, -1], [5, 4]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.","answer":{"kind":"short_text","value":"0","explanation":"The characteristic polynomial coefficients are (1,-8,21), with discriminant Δ=-20. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 0. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here."},"skills":["Number of Distinct Real Eigenvalues of a 2×2 Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0087","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"043256caeb45009e9e80caaa5ef3aa5c2748a7859c6ee63e1b4df81a42ebb161","family_id":"qf-eigenvalue-real-count","seed":202610074087,"semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"8\",\"16\"],\"operation\":\"real_eigenvalue_count\"}"},"fields":{"title":"Number of Distinct Real Eigenvalues of a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[4, 1], [0, 4]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.","answer":{"kind":"short_text","value":"1","explanation":"The characteristic polynomial coefficients are (1,-8,16), with discriminant Δ=0. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 1. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here."},"skills":["Number of Distinct Real Eigenvalues of a 2×2 Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0088","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d0c175f0038ba0395373729714500b5f94c02eac2a994306abd5f403ce5dad0","family_id":"qf-eigenvalue-real-count","seed":202610074088,"semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"0\",\"-17\"],\"operation\":\"real_eigenvalue_count\"}"},"fields":{"title":"Number of Distinct Real Eigenvalues of a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[-4, 1], [1, 4]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.","answer":{"kind":"short_text","value":"2","explanation":"The characteristic polynomial coefficients are (1,0,-17), with discriminant Δ=68. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 2. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here."},"skills":["Number of Distinct Real Eigenvalues of a 2×2 Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0089","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74848e41e1110a5917a1483fd4605cb4539e1e1f4808e1a38595dc463f0d8f2a","family_id":"qf-eigenvalue-real-count","seed":202610074089,"semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"10\",\"31\"],\"operation\":\"real_eigenvalue_count\"}"},"fields":{"title":"Number of Distinct Real Eigenvalues of a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[5, -1], [6, 5]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.","answer":{"kind":"short_text","value":"0","explanation":"The characteristic polynomial coefficients are (1,-10,31), with discriminant Δ=-24. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 0. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here."},"skills":["Number of Distinct Real Eigenvalues of a 2×2 Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0090","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97b37d783b176dde01a5fed55f198ae6da3a3635a8aed90358e3e43bebc71d44","family_id":"qf-eigenvalue-real-count","seed":202610074090,"semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"10\",\"25\"],\"operation\":\"real_eigenvalue_count\"}"},"fields":{"title":"Number of Distinct Real Eigenvalues of a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[5, 1], [0, 5]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.","answer":{"kind":"short_text","value":"1","explanation":"The characteristic polynomial coefficients are (1,-10,25), with discriminant Δ=0. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 1. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here."},"skills":["Number of Distinct Real Eigenvalues of a 2×2 Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0091","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"860a7b96107a011117d2c58d4ee8fdadeb30a1961c96be0ad160b94b08c237f7","family_id":"qf-eigenvalue-real-count","seed":202610074091,"semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"0\",\"-26\"],\"operation\":\"real_eigenvalue_count\"}"},"fields":{"title":"Number of Distinct Real Eigenvalues of a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[-5, 1], [1, 5]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.","answer":{"kind":"short_text","value":"2","explanation":"The characteristic polynomial coefficients are (1,0,-26), with discriminant Δ=104. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 2. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here."},"skills":["Number of Distinct Real Eigenvalues of a 2×2 Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0092","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0110e57863d26371cd2aa4450f907991d3c93398859c716aa74c6d8d0a396216","family_id":"qf-eigenvalue-real-count","seed":202610074092,"semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"12\",\"43\"],\"operation\":\"real_eigenvalue_count\"}"},"fields":{"title":"Number of Distinct Real Eigenvalues of a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[6, -1], [7, 6]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.","answer":{"kind":"short_text","value":"0","explanation":"The characteristic polynomial coefficients are (1,-12,43), with discriminant Δ=-28. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 0. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here."},"skills":["Number of Distinct Real Eigenvalues of a 2×2 Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0093","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27a13f755e97888b9448ac263ab39fdcd4c2596c5008855097a9ffb18ce175a9","family_id":"qf-eigenvalue-real-count","seed":202610074093,"semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"12\",\"36\"],\"operation\":\"real_eigenvalue_count\"}"},"fields":{"title":"Number of Distinct Real Eigenvalues of a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[6, 1], [0, 6]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.","answer":{"kind":"short_text","value":"1","explanation":"The characteristic polynomial coefficients are (1,-12,36), with discriminant Δ=0. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 1. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here."},"skills":["Number of Distinct Real Eigenvalues of a 2×2 Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0094","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ead80b661c6baf025339e4d623342b1a16922a2751c1858dc088492c9bb95f97","family_id":"qf-eigenvalue-real-count","seed":202610074094,"semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"0\",\"-37\"],\"operation\":\"real_eigenvalue_count\"}"},"fields":{"title":"Number of Distinct Real Eigenvalues of a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[-6, 1], [1, 6]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.","answer":{"kind":"short_text","value":"2","explanation":"The characteristic polynomial coefficients are (1,0,-37), with discriminant Δ=148. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 2. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here."},"skills":["Number of Distinct Real Eigenvalues of a 2×2 Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0095","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1221e5bcb5a1c77a08f9fc36f7d5e6d948075abf5bdc825a9acaa32721b406fc","family_id":"qf-eigenvalue-real-count","seed":202610074095,"semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"14\",\"57\"],\"operation\":\"real_eigenvalue_count\"}"},"fields":{"title":"Number of Distinct Real Eigenvalues of a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[7, -1], [8, 7]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.","answer":{"kind":"short_text","value":"0","explanation":"The characteristic polynomial coefficients are (1,-14,57), with discriminant Δ=-32. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 0. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here."},"skills":["Number of Distinct Real Eigenvalues of a 2×2 Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0096","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d5afc79ea204be72f44c4d53306ca55cede34dc7f9eebc124b5f6d461d03117","family_id":"qf-eigenvalue-real-count","seed":202610074096,"semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"14\",\"49\"],\"operation\":\"real_eigenvalue_count\"}"},"fields":{"title":"Number of Distinct Real Eigenvalues of a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[7, 1], [0, 7]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.","answer":{"kind":"short_text","value":"1","explanation":"The characteristic polynomial coefficients are (1,-14,49), with discriminant Δ=0. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 1. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here."},"skills":["Number of Distinct Real Eigenvalues of a 2×2 Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0097","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9da74a3a4a3654ee3b6f053847bf401d3e843c38060d410218160a59acc64838","family_id":"qf-eigenvalue-real-count","seed":202610074097,"semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"0\",\"-50\"],\"operation\":\"real_eigenvalue_count\"}"},"fields":{"title":"Number of Distinct Real Eigenvalues of a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[-7, 1], [1, 7]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.","answer":{"kind":"short_text","value":"2","explanation":"The characteristic polynomial coefficients are (1,0,-50), with discriminant Δ=200. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 2. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here."},"skills":["Number of Distinct Real Eigenvalues of a 2×2 Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0098","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b78d66898bf9b80be259b77852575189464738413f0fdeedaf61cfabd4fbec8","family_id":"qf-eigenvalue-real-count","seed":202610074098,"semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"16\",\"73\"],\"operation\":\"real_eigenvalue_count\"}"},"fields":{"title":"Number of Distinct Real Eigenvalues of a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[8, -1], [9, 8]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.","answer":{"kind":"short_text","value":"0","explanation":"The characteristic polynomial coefficients are (1,-16,73), with discriminant Δ=-36. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 0. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here."},"skills":["Number of Distinct Real Eigenvalues of a 2×2 Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0099","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b4af20ed3e966cb59b49cd99e22c04ad616a3554439226e11c2278c8f3a94ee","family_id":"qf-eigenvalue-real-count","seed":202610074099,"semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"16\",\"64\"],\"operation\":\"real_eigenvalue_count\"}"},"fields":{"title":"Number of Distinct Real Eigenvalues of a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[8, 1], [0, 8]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.","answer":{"kind":"short_text","value":"1","explanation":"The characteristic polynomial coefficients are (1,-16,64), with discriminant Δ=0. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 1. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here."},"skills":["Number of Distinct Real Eigenvalues of a 2×2 Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261006-advance-20261007-r04-eigenvalue-0100","batch_id":"batch-20261006-advance-20261007-r04-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27c2e80f5f01a46510467e364c8baa56473b19d24d1d5803f4eb5a80fcd0da87","family_id":"qf-eigenvalue-real-count","seed":202610074100,"semantic_template":"qf-eigenvalue-real-count/{\"inputs\":[\"0\",\"-65\"],\"operation\":\"real_eigenvalue_count\"}"},"fields":{"title":"Number of Distinct Real Eigenvalues of a 2×2 Matrix","stem":"Over R, the 2×2 matrix is M=[[-8, 1], [1, 8]]. Find only the number of distinct real eigenvalues: zero, one, or two. Do not output approximate roots or count algebraic multiplicity.","answer":{"kind":"short_text","value":"2","explanation":"The characteristic polynomial coefficients are (1,0,-65), with discriminant Δ=260. Negative, zero, and positive Δ give zero, one, and two distinct real roots, so the answer is 2. A positive nonsquare Δ still gives two real roots; no diagonalizability, complex-spectrum, or Jordan conclusion follows here."},"skills":["Number of Distinct Real Eigenvalues of a 2×2 Matrix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}