{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0001","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1e2b38f0b6405ab9770f202d93eed79dbe2557525cc96843d0b1d9c624f512c","family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097001,"semantic_template":"qf-density-eigen-characteristic-coefficients/[\"0\",\"0\"]"},"fields":{"title":"Descending Characteristic Coefficients of a Real 2×2 Matrix","stem":"Over R, M=[[0, 0], [0, 0]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).","answer":{"kind":"expression","value":"(1, 0, 0)","explanation":"Use (1,−tr(M),det(M)) to obtain (1, 0, 0), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added."},"skills":["Descending Characteristic Coefficients of a Real 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-20261007-advance-20261009-r07-eigenvalue-0002","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dfc5e0a693721b9100c64734df3e6300c1f15e188a1c2c662fb8842291f755af","family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097002,"semantic_template":"qf-density-eigen-characteristic-coefficients/[\"2\",\"1\"]"},"fields":{"title":"Descending Characteristic Coefficients of a Real 2×2 Matrix","stem":"Over R, M=[[1, 0], [0, 1]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).","answer":{"kind":"expression","value":"(1, -2, 1)","explanation":"Use (1,−tr(M),det(M)) to obtain (1, -2, 1), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added."},"skills":["Descending Characteristic Coefficients of a Real 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-20261007-advance-20261009-r07-eigenvalue-0003","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b13c57ef3c987c12dbeca78c5ee40c059818b867a981b07cc4b3a13e089e102","family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097003,"semantic_template":"qf-density-eigen-characteristic-coefficients/[\"-7\",\"-33\"]"},"fields":{"title":"Descending Characteristic Coefficients of a Real 2×2 Matrix","stem":"Over R, M=[[-10, 3], [1, 3]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).","answer":{"kind":"expression","value":"(1, 7, -33)","explanation":"Use (1,−tr(M),det(M)) to obtain (1, 7, -33), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added."},"skills":["Descending Characteristic Coefficients of a Real 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-20261007-advance-20261009-r07-eigenvalue-0004","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c3da3247b23a5ff4bb2eb4f3015c22cfcb065c40e96b39c5cd54d0601f508821","family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097004,"semantic_template":"qf-density-eigen-characteristic-coefficients/[\"-5\",\"-35\"]"},"fields":{"title":"Descending Characteristic Coefficients of a Real 2×2 Matrix","stem":"Over R, M=[[-9, 1], [-1, 4]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).","answer":{"kind":"expression","value":"(1, 5, -35)","explanation":"Use (1,−tr(M),det(M)) to obtain (1, 5, -35), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added."},"skills":["Descending Characteristic Coefficients of a Real 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-20261007-advance-20261009-r07-eigenvalue-0005","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eac0798c2fbe1c10aa96edd5b2ee3b25bae700b0e953100ed5d3df252de120be","family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097005,"semantic_template":"qf-density-eigen-characteristic-coefficients/[\"-3\",\"-42\"]"},"fields":{"title":"Descending Characteristic Coefficients of a Real 2×2 Matrix","stem":"Over R, M=[[-8, 2], [1, 5]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).","answer":{"kind":"expression","value":"(1, 3, -42)","explanation":"Use (1,−tr(M),det(M)) to obtain (1, 3, -42), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added."},"skills":["Descending Characteristic Coefficients of a Real 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-20261007-advance-20261009-r07-eigenvalue-0006","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cdeb54874b78a63d1bd433aace895bd612ab4247685dce71ca432943b13766d5","family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097006,"semantic_template":"qf-density-eigen-characteristic-coefficients/[\"-1\",\"-39\"]"},"fields":{"title":"Descending Characteristic Coefficients of a Real 2×2 Matrix","stem":"Over R, M=[[-7, 3], [-1, 6]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).","answer":{"kind":"expression","value":"(1, 1, -39)","explanation":"Use (1,−tr(M),det(M)) to obtain (1, 1, -39), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added."},"skills":["Descending Characteristic Coefficients of a Real 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-20261007-advance-20261009-r07-eigenvalue-0007","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40d51ebf97465acdd621f548cdfa24155eab26ea86b019142b2a05676d64a6bd","family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097007,"semantic_template":"qf-density-eigen-characteristic-coefficients/[\"1\",\"-43\"]"},"fields":{"title":"Descending Characteristic Coefficients of a Real 2×2 Matrix","stem":"Over R, M=[[-6, 1], [1, 7]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).","answer":{"kind":"expression","value":"(1, -1, -43)","explanation":"Use (1,−tr(M),det(M)) to obtain (1, -1, -43), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added."},"skills":["Descending Characteristic Coefficients of a Real 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-20261007-advance-20261009-r07-eigenvalue-0008","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b47162ab85c0fbfefaf35c79921a7a8e71a28a235a8d110d17b5258ecd36936","family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097008,"semantic_template":"qf-density-eigen-characteristic-coefficients/[\"3\",\"-38\"]"},"fields":{"title":"Descending Characteristic Coefficients of a Real 2×2 Matrix","stem":"Over R, M=[[-5, 2], [-1, 8]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).","answer":{"kind":"expression","value":"(1, -3, -38)","explanation":"Use (1,−tr(M),det(M)) to obtain (1, -3, -38), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added."},"skills":["Descending Characteristic Coefficients of a Real 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-20261007-advance-20261009-r07-eigenvalue-0009","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8faf333c1654c2a7f287c6ff3abaaf7315c1bbdc65880a0dc5a3532f14a1c16","family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097009,"semantic_template":"qf-density-eigen-characteristic-coefficients/[\"5\",\"-39\"]"},"fields":{"title":"Descending Characteristic Coefficients of a Real 2×2 Matrix","stem":"Over R, M=[[-4, 3], [1, 9]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).","answer":{"kind":"expression","value":"(1, -5, -39)","explanation":"Use (1,−tr(M),det(M)) to obtain (1, -5, -39), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added."},"skills":["Descending Characteristic Coefficients of a Real 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-20261007-advance-20261009-r07-eigenvalue-0010","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57c8b4e0bf17c8b5faa6e9ba1347b1697af2995582c573f2d5c14dbe4844230d","family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097010,"semantic_template":"qf-density-eigen-characteristic-coefficients/[\"7\",\"-29\"]"},"fields":{"title":"Descending Characteristic Coefficients of a Real 2×2 Matrix","stem":"Over R, M=[[-3, 1], [-1, 10]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).","answer":{"kind":"expression","value":"(1, -7, -29)","explanation":"Use (1,−tr(M),det(M)) to obtain (1, -7, -29), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added."},"skills":["Descending Characteristic Coefficients of a Real 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-20261007-advance-20261009-r07-eigenvalue-0011","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19e5b694d90e76dad9b181f76f95ef33de0a7d65b937b3722e22c6107e24dde2","family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097011,"semantic_template":"qf-density-eigen-characteristic-coefficients/[\"9\",\"-24\"]"},"fields":{"title":"Descending Characteristic Coefficients of a Real 2×2 Matrix","stem":"Over R, M=[[-2, 2], [1, 11]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).","answer":{"kind":"expression","value":"(1, -9, -24)","explanation":"Use (1,−tr(M),det(M)) to obtain (1, -9, -24), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added."},"skills":["Descending Characteristic Coefficients of a Real 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-20261007-advance-20261009-r07-eigenvalue-0012","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f98da6081baa3b28b624bf4ada9e96b5560b233bb04768535886783b03ba3eef","family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097012,"semantic_template":"qf-density-eigen-characteristic-coefficients/[\"11\",\"-9\"]"},"fields":{"title":"Descending Characteristic Coefficients of a Real 2×2 Matrix","stem":"Over R, M=[[-1, 3], [-1, 12]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).","answer":{"kind":"expression","value":"(1, -11, -9)","explanation":"Use (1,−tr(M),det(M)) to obtain (1, -11, -9), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added."},"skills":["Descending Characteristic Coefficients of a Real 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-20261007-advance-20261009-r07-eigenvalue-0013","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f279353d2083061dcfe8a3a96bddb5a8723e465b6e789de5917fe5ab5d297338","family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097013,"semantic_template":"qf-density-eigen-characteristic-coefficients/[\"13\",\"-1\"]"},"fields":{"title":"Descending Characteristic Coefficients of a Real 2×2 Matrix","stem":"Over R, M=[[0, 1], [1, 13]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).","answer":{"kind":"expression","value":"(1, -13, -1)","explanation":"Use (1,−tr(M),det(M)) to obtain (1, -13, -1), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added."},"skills":["Descending Characteristic Coefficients of a Real 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-20261007-advance-20261009-r07-eigenvalue-0014","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0aa73e18631440010411e27f7de3f7e9cc29ed88be99766809112ae74ad2f5a","family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097014,"semantic_template":"qf-density-eigen-characteristic-coefficients/[\"15\",\"16\"]"},"fields":{"title":"Descending Characteristic Coefficients of a Real 2×2 Matrix","stem":"Over R, M=[[1, 2], [-1, 14]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).","answer":{"kind":"expression","value":"(1, -15, 16)","explanation":"Use (1,−tr(M),det(M)) to obtain (1, -15, 16), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added."},"skills":["Descending Characteristic Coefficients of a Real 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-20261007-advance-20261009-r07-eigenvalue-0015","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6654b848881c1937650baf7d1671aad73c63c6e3dca1b94839e71b5bd43bfb7e","family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097015,"semantic_template":"qf-density-eigen-characteristic-coefficients/[\"17\",\"27\"]"},"fields":{"title":"Descending Characteristic Coefficients of a Real 2×2 Matrix","stem":"Over R, M=[[2, 3], [1, 15]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).","answer":{"kind":"expression","value":"(1, -17, 27)","explanation":"Use (1,−tr(M),det(M)) to obtain (1, -17, 27), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added."},"skills":["Descending Characteristic Coefficients of a Real 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-20261007-advance-20261009-r07-eigenvalue-0016","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1964930a835ef78fa57d737b135969debf7f51a213a9f5d3c640d3bf8c7a74b","family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097016,"semantic_template":"qf-density-eigen-characteristic-coefficients/[\"19\",\"49\"]"},"fields":{"title":"Descending Characteristic Coefficients of a Real 2×2 Matrix","stem":"Over R, M=[[3, 1], [-1, 16]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).","answer":{"kind":"expression","value":"(1, -19, 49)","explanation":"Use (1,−tr(M),det(M)) to obtain (1, -19, 49), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added."},"skills":["Descending Characteristic Coefficients of a Real 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-20261007-advance-20261009-r07-eigenvalue-0017","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27f99390c16e7e271e67bda6448f9a11848ba81df9934e7bfb89dd12fb36c9b6","family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097017,"semantic_template":"qf-density-eigen-characteristic-coefficients/[\"21\",\"66\"]"},"fields":{"title":"Descending Characteristic Coefficients of a Real 2×2 Matrix","stem":"Over R, M=[[4, 2], [1, 17]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).","answer":{"kind":"expression","value":"(1, -21, 66)","explanation":"Use (1,−tr(M),det(M)) to obtain (1, -21, 66), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added."},"skills":["Descending Characteristic Coefficients of a Real 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-20261007-advance-20261009-r07-eigenvalue-0018","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a04cc7fbb3186138b93ea9f6fbb024e35c4b9507560db917193d5ddafb419fa4","family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097018,"semantic_template":"qf-density-eigen-characteristic-coefficients/[\"23\",\"93\"]"},"fields":{"title":"Descending Characteristic Coefficients of a Real 2×2 Matrix","stem":"Over R, M=[[5, 3], [-1, 18]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).","answer":{"kind":"expression","value":"(1, -23, 93)","explanation":"Use (1,−tr(M),det(M)) to obtain (1, -23, 93), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added."},"skills":["Descending Characteristic Coefficients of a Real 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-20261007-advance-20261009-r07-eigenvalue-0019","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab806b4e6910897a365b2649755a2a627f3a359ea3ce4f3e6e4e498b802f757b","family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097019,"semantic_template":"qf-density-eigen-characteristic-coefficients/[\"25\",\"113\"]"},"fields":{"title":"Descending Characteristic Coefficients of a Real 2×2 Matrix","stem":"Over R, M=[[6, 1], [1, 19]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).","answer":{"kind":"expression","value":"(1, -25, 113)","explanation":"Use (1,−tr(M),det(M)) to obtain (1, -25, 113), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added."},"skills":["Descending Characteristic Coefficients of a Real 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-20261007-advance-20261009-r07-eigenvalue-0020","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e3577995c1506541926b8471b1b6f1af296e991146ec64476948bb36aa394d7c","family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097020,"semantic_template":"qf-density-eigen-characteristic-coefficients/[\"27\",\"142\"]"},"fields":{"title":"Descending Characteristic Coefficients of a Real 2×2 Matrix","stem":"Over R, M=[[7, 2], [-1, 20]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).","answer":{"kind":"expression","value":"(1, -27, 142)","explanation":"Use (1,−tr(M),det(M)) to obtain (1, -27, 142), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added."},"skills":["Descending Characteristic Coefficients of a Real 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-20261007-advance-20261009-r07-eigenvalue-0021","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"235b4a48d88e0755cb4c45e02ce20f37d76926d6f38deca9f48472932756ef7c","family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097021,"semantic_template":"qf-density-eigen-characteristic-coefficients/[\"29\",\"165\"]"},"fields":{"title":"Descending Characteristic Coefficients of a Real 2×2 Matrix","stem":"Over R, M=[[8, 3], [1, 21]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).","answer":{"kind":"expression","value":"(1, -29, 165)","explanation":"Use (1,−tr(M),det(M)) to obtain (1, -29, 165), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added."},"skills":["Descending Characteristic Coefficients of a Real 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-20261007-advance-20261009-r07-eigenvalue-0022","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2ef2245ba11c3a249acc14c062d76ed09024090bcd00148f0425dd748285810","family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097022,"semantic_template":"qf-density-eigen-characteristic-coefficients/[\"31\",\"199\"]"},"fields":{"title":"Descending Characteristic Coefficients of a Real 2×2 Matrix","stem":"Over R, M=[[9, 1], [-1, 22]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).","answer":{"kind":"expression","value":"(1, -31, 199)","explanation":"Use (1,−tr(M),det(M)) to obtain (1, -31, 199), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added."},"skills":["Descending Characteristic Coefficients of a Real 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-20261007-advance-20261009-r07-eigenvalue-0023","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1fa7fdc0d9a6bc6d1512c95fa3515d896be9ccaf00b9da71526c5622a4d6410c","family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097023,"semantic_template":"qf-density-eigen-characteristic-coefficients/[\"33\",\"228\"]"},"fields":{"title":"Descending Characteristic Coefficients of a Real 2×2 Matrix","stem":"Over R, M=[[10, 2], [1, 23]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).","answer":{"kind":"expression","value":"(1, -33, 228)","explanation":"Use (1,−tr(M),det(M)) to obtain (1, -33, 228), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added."},"skills":["Descending Characteristic Coefficients of a Real 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-20261007-advance-20261009-r07-eigenvalue-0024","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a45deb500445a518a43a0c91e6f6b7eac92c29832c17fd92c3d6dec2fd9816c","family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097024,"semantic_template":"qf-density-eigen-characteristic-coefficients/[\"35\",\"267\"]"},"fields":{"title":"Descending Characteristic Coefficients of a Real 2×2 Matrix","stem":"Over R, M=[[11, 3], [-1, 24]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).","answer":{"kind":"expression","value":"(1, -35, 267)","explanation":"Use (1,−tr(M),det(M)) to obtain (1, -35, 267), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added."},"skills":["Descending Characteristic Coefficients of a Real 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-20261007-advance-20261009-r07-eigenvalue-0025","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d65ae1b311382b30f160dd76b2c49083c5af2f441d42e1943b6f8335bff0871","family_id":"qf-density-eigen-characteristic-coefficients","seed":202610097025,"semantic_template":"qf-density-eigen-characteristic-coefficients/[\"37\",\"299\"]"},"fields":{"title":"Descending Characteristic Coefficients of a Real 2×2 Matrix","stem":"Over R, M=[[12, 1], [1, 25]] is 2×2. Give its characteristic polynomial coefficients in descending order (λ²,λ,constant).","answer":{"kind":"expression","value":"(1, -37, 299)","explanation":"Use (1,−tr(M),det(M)) to obtain (1, -37, 299), not the ascending order used for the separate polynomial lists. No diagonalization or multiplicity claim is added."},"skills":["Descending Characteristic Coefficients of a Real 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-20261007-advance-20261009-r07-eigenvalue-0026","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"658311bff02c768dcace56ae96a639aea5438156dda1bc79bf431c0178010c77","family_id":"qf-density-eigen-discriminant","seed":202610097026,"semantic_template":"qf-density-eigen-discriminant/[\"0\",\"-1\"]"},"fields":{"title":"Exact Real 2×2 Characteristic Discriminant","stem":"Over R, M=[[0, -1], [1, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).","answer":{"kind":"short_text","value":"-4","explanation":"The integer discriminant is -4. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted."},"skills":["Exact Real 2×2 Characteristic Discriminant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0027","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32c070fd2d43649ff281ca7b8601e0dd13708b3cc4fa63a10fbd678269f9c6d7","family_id":"qf-density-eigen-discriminant","seed":202610097027,"semantic_template":"qf-density-eigen-discriminant/[\"0\",\"0\"]"},"fields":{"title":"Exact Real 2×2 Characteristic Discriminant","stem":"Over R, M=[[3, 1], [0, 3]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).","answer":{"kind":"short_text","value":"0","explanation":"The integer discriminant is 0. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted."},"skills":["Exact Real 2×2 Characteristic Discriminant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0028","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ee6470f66e27078201c74a866055e28e31787ba0e0d916f6165de46df3a2ebe","family_id":"qf-density-eigen-discriminant","seed":202610097028,"semantic_template":"qf-density-eigen-discriminant/[\"2\",\"2\"]"},"fields":{"title":"Exact Real 2×2 Characteristic Discriminant","stem":"Over R, M=[[2, 1], [2, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).","answer":{"kind":"short_text","value":"12","explanation":"The integer discriminant is 12. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted."},"skills":["Exact Real 2×2 Characteristic Discriminant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0029","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cfd2c3a73d44dea13b9aa699381eb0057d9c1e24c7431f4a68dcb8d9a1e9bba7","family_id":"qf-density-eigen-discriminant","seed":202610097029,"semantic_template":"qf-density-eigen-discriminant/[\"3\",\"-3\"]"},"fields":{"title":"Exact Real 2×2 Characteristic Discriminant","stem":"Over R, M=[[3, 1], [-3, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).","answer":{"kind":"short_text","value":"-3","explanation":"The integer discriminant is -3. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted."},"skills":["Exact Real 2×2 Characteristic Discriminant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0030","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb59203f03444793fd6361efd51cdbac6cd1ea2b93f695504ec8da0b4707c9bd","family_id":"qf-density-eigen-discriminant","seed":202610097030,"semantic_template":"qf-density-eigen-discriminant/[\"4\",\"4\"]"},"fields":{"title":"Exact Real 2×2 Characteristic Discriminant","stem":"Over R, M=[[4, 1], [4, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).","answer":{"kind":"short_text","value":"32","explanation":"The integer discriminant is 32. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted."},"skills":["Exact Real 2×2 Characteristic Discriminant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0031","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c20063f3d4eda15b2b7257b1bc16ac18e7580e204c5a72c930c7746c7d590c2a","family_id":"qf-density-eigen-discriminant","seed":202610097031,"semantic_template":"qf-density-eigen-discriminant/[\"5\",\"-5\"]"},"fields":{"title":"Exact Real 2×2 Characteristic Discriminant","stem":"Over R, M=[[5, 1], [-5, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).","answer":{"kind":"short_text","value":"5","explanation":"The integer discriminant is 5. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted."},"skills":["Exact Real 2×2 Characteristic Discriminant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0032","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"191be3259936bb61b1843ce503abb4c23fefc53c01a45fdbb460db54f11f0a82","family_id":"qf-density-eigen-discriminant","seed":202610097032,"semantic_template":"qf-density-eigen-discriminant/[\"6\",\"6\"]"},"fields":{"title":"Exact Real 2×2 Characteristic Discriminant","stem":"Over R, M=[[6, 1], [6, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).","answer":{"kind":"short_text","value":"60","explanation":"The integer discriminant is 60. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted."},"skills":["Exact Real 2×2 Characteristic Discriminant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0033","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e42f86b706ae4559334287ba0a7173d9629414e74bf0344e98a2da66a6a7ab9f","family_id":"qf-density-eigen-discriminant","seed":202610097033,"semantic_template":"qf-density-eigen-discriminant/[\"7\",\"-7\"]"},"fields":{"title":"Exact Real 2×2 Characteristic Discriminant","stem":"Over R, M=[[7, 1], [-7, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).","answer":{"kind":"short_text","value":"21","explanation":"The integer discriminant is 21. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted."},"skills":["Exact Real 2×2 Characteristic Discriminant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0034","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88864e1c69ff25dc0de9cbc83e7c449c336c74b080542b356777859d2fee1f83","family_id":"qf-density-eigen-discriminant","seed":202610097034,"semantic_template":"qf-density-eigen-discriminant/[\"8\",\"8\"]"},"fields":{"title":"Exact Real 2×2 Characteristic Discriminant","stem":"Over R, M=[[8, 1], [8, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).","answer":{"kind":"short_text","value":"96","explanation":"The integer discriminant is 96. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted."},"skills":["Exact Real 2×2 Characteristic Discriminant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0035","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e40fd08d8ab99f625d012cd6b1e7b8dc95b0a65746f66ee05a1e223f8ba2508d","family_id":"qf-density-eigen-discriminant","seed":202610097035,"semantic_template":"qf-density-eigen-discriminant/[\"9\",\"-9\"]"},"fields":{"title":"Exact Real 2×2 Characteristic Discriminant","stem":"Over R, M=[[9, 1], [-9, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).","answer":{"kind":"short_text","value":"45","explanation":"The integer discriminant is 45. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted."},"skills":["Exact Real 2×2 Characteristic Discriminant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0036","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5efae8665bea20301a75847742f00ae173188f14c9dab2ac4a728322aa698d4d","family_id":"qf-density-eigen-discriminant","seed":202610097036,"semantic_template":"qf-density-eigen-discriminant/[\"10\",\"10\"]"},"fields":{"title":"Exact Real 2×2 Characteristic Discriminant","stem":"Over R, M=[[10, 1], [10, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).","answer":{"kind":"short_text","value":"140","explanation":"The integer discriminant is 140. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted."},"skills":["Exact Real 2×2 Characteristic Discriminant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0037","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36b5e94233a746bae4203268b57c3f597201ea05f5350442cc777abd249230a8","family_id":"qf-density-eigen-discriminant","seed":202610097037,"semantic_template":"qf-density-eigen-discriminant/[\"11\",\"-11\"]"},"fields":{"title":"Exact Real 2×2 Characteristic Discriminant","stem":"Over R, M=[[11, 1], [-11, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).","answer":{"kind":"short_text","value":"77","explanation":"The integer discriminant is 77. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted."},"skills":["Exact Real 2×2 Characteristic Discriminant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0038","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d022309e401f5bf1911ead598a492d146d86f290e8062f24cfab5a5cb368557","family_id":"qf-density-eigen-discriminant","seed":202610097038,"semantic_template":"qf-density-eigen-discriminant/[\"12\",\"12\"]"},"fields":{"title":"Exact Real 2×2 Characteristic Discriminant","stem":"Over R, M=[[12, 1], [12, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).","answer":{"kind":"short_text","value":"192","explanation":"The integer discriminant is 192. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted."},"skills":["Exact Real 2×2 Characteristic Discriminant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0039","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e8ea4d233a29c28903f8d65d21e2e3e1ad60671d114f62057f21dbb08a92fd8","family_id":"qf-density-eigen-discriminant","seed":202610097039,"semantic_template":"qf-density-eigen-discriminant/[\"13\",\"-13\"]"},"fields":{"title":"Exact Real 2×2 Characteristic Discriminant","stem":"Over R, M=[[13, 1], [-13, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).","answer":{"kind":"short_text","value":"117","explanation":"The integer discriminant is 117. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted."},"skills":["Exact Real 2×2 Characteristic Discriminant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0040","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22fc0aa71d3f64a981d0d4e1dc161a0baf0e3eb128dbe999b116501d7fe80209","family_id":"qf-density-eigen-discriminant","seed":202610097040,"semantic_template":"qf-density-eigen-discriminant/[\"14\",\"14\"]"},"fields":{"title":"Exact Real 2×2 Characteristic Discriminant","stem":"Over R, M=[[14, 1], [14, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).","answer":{"kind":"short_text","value":"252","explanation":"The integer discriminant is 252. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted."},"skills":["Exact Real 2×2 Characteristic Discriminant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0041","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5d02b58e6b9d6401394e89c1e8491b0f3eb2fab2805cbd2e5a8b6cbb4a147f4","family_id":"qf-density-eigen-discriminant","seed":202610097041,"semantic_template":"qf-density-eigen-discriminant/[\"15\",\"-15\"]"},"fields":{"title":"Exact Real 2×2 Characteristic Discriminant","stem":"Over R, M=[[15, 1], [-15, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).","answer":{"kind":"short_text","value":"165","explanation":"The integer discriminant is 165. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted."},"skills":["Exact Real 2×2 Characteristic Discriminant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0042","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f9d8d65ebf657852fd2ba1c360e400343f52ddf3cb25884e81724a52adf8f990","family_id":"qf-density-eigen-discriminant","seed":202610097042,"semantic_template":"qf-density-eigen-discriminant/[\"16\",\"16\"]"},"fields":{"title":"Exact Real 2×2 Characteristic Discriminant","stem":"Over R, M=[[16, 1], [16, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).","answer":{"kind":"short_text","value":"320","explanation":"The integer discriminant is 320. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted."},"skills":["Exact Real 2×2 Characteristic Discriminant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0043","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f6c12fb0a78321eac97415c209fa1fc8fbdb7af64267b70746e61a4ed0b5b60","family_id":"qf-density-eigen-discriminant","seed":202610097043,"semantic_template":"qf-density-eigen-discriminant/[\"17\",\"-17\"]"},"fields":{"title":"Exact Real 2×2 Characteristic Discriminant","stem":"Over R, M=[[17, 1], [-17, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).","answer":{"kind":"short_text","value":"221","explanation":"The integer discriminant is 221. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted."},"skills":["Exact Real 2×2 Characteristic Discriminant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0044","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42ad80d0ed7b414611df4b993ac32d713a9e937aa42fdb8133c207d2a1018c68","family_id":"qf-density-eigen-discriminant","seed":202610097044,"semantic_template":"qf-density-eigen-discriminant/[\"18\",\"18\"]"},"fields":{"title":"Exact Real 2×2 Characteristic Discriminant","stem":"Over R, M=[[18, 1], [18, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).","answer":{"kind":"short_text","value":"396","explanation":"The integer discriminant is 396. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted."},"skills":["Exact Real 2×2 Characteristic Discriminant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0045","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b666fa83e42afc6e9859c296abe088fed81dc45753fbf8ecef923df760ada07a","family_id":"qf-density-eigen-discriminant","seed":202610097045,"semantic_template":"qf-density-eigen-discriminant/[\"19\",\"-19\"]"},"fields":{"title":"Exact Real 2×2 Characteristic Discriminant","stem":"Over R, M=[[19, 1], [-19, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).","answer":{"kind":"short_text","value":"285","explanation":"The integer discriminant is 285. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted."},"skills":["Exact Real 2×2 Characteristic Discriminant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0046","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1cb95449d5e64745a30e749aea320708406f7d94689d3db81347e6e915005786","family_id":"qf-density-eigen-discriminant","seed":202610097046,"semantic_template":"qf-density-eigen-discriminant/[\"20\",\"20\"]"},"fields":{"title":"Exact Real 2×2 Characteristic Discriminant","stem":"Over R, M=[[20, 1], [20, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).","answer":{"kind":"short_text","value":"480","explanation":"The integer discriminant is 480. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted."},"skills":["Exact Real 2×2 Characteristic Discriminant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0047","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"977b34c84bcaa10e26d9a3d9e76bab7ac2e945e7cf79da32df0267e725b14502","family_id":"qf-density-eigen-discriminant","seed":202610097047,"semantic_template":"qf-density-eigen-discriminant/[\"21\",\"-21\"]"},"fields":{"title":"Exact Real 2×2 Characteristic Discriminant","stem":"Over R, M=[[21, 1], [-21, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).","answer":{"kind":"short_text","value":"357","explanation":"The integer discriminant is 357. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted."},"skills":["Exact Real 2×2 Characteristic Discriminant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0048","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2434d9a62122373c4e47ca8f4e3d4949497822a1d8156dc1a4846a3f371c1693","family_id":"qf-density-eigen-discriminant","seed":202610097048,"semantic_template":"qf-density-eigen-discriminant/[\"22\",\"22\"]"},"fields":{"title":"Exact Real 2×2 Characteristic Discriminant","stem":"Over R, M=[[22, 1], [22, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).","answer":{"kind":"short_text","value":"572","explanation":"The integer discriminant is 572. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted."},"skills":["Exact Real 2×2 Characteristic Discriminant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0049","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef5636750e15b1484681506605242a9a68aabc4c2ed41c7c22f5b11f1878e063","family_id":"qf-density-eigen-discriminant","seed":202610097049,"semantic_template":"qf-density-eigen-discriminant/[\"23\",\"-23\"]"},"fields":{"title":"Exact Real 2×2 Characteristic Discriminant","stem":"Over R, M=[[23, 1], [-23, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).","answer":{"kind":"short_text","value":"437","explanation":"The integer discriminant is 437. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted."},"skills":["Exact Real 2×2 Characteristic Discriminant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0050","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"729ad1e640f44ac089fe9692d4b490f964c8d50f22fb0a8d7ab3d4c124018b9e","family_id":"qf-density-eigen-discriminant","seed":202610097050,"semantic_template":"qf-density-eigen-discriminant/[\"24\",\"24\"]"},"fields":{"title":"Exact Real 2×2 Characteristic Discriminant","stem":"Over R, M=[[24, 1], [24, 0]] is 2×2. Compute the exact characteristic discriminant tr(M)²−4det(M).","answer":{"kind":"short_text","value":"672","explanation":"The integer discriminant is 672. Its sign controls the number of distinct real eigenvalues; no floating roots or complex-spectrum expansion are asserted."},"skills":["Exact Real 2×2 Characteristic Discriminant"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0051","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0199bfc1b36b0cd664372f1a00dfa1b5a4c036516a86b04a908f288b1efe5886","family_id":"qf-density-eigen-kernel-residual","seed":202610097051,"semantic_template":"qf-density-eigen-kernel-residual/[\"zero_vector_residual\"]"},"fields":{"title":"Ordered Residual of a Specified Kernel Equation","stem":"Over R, M=[[2, 1], [0, -3]] is 2×2, λ=4, and v=[0, 0]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.","answer":{"kind":"expression","value":"(0, 0)","explanation":"Exact multiplication and subtraction yield (0, 0). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector."},"skills":["Ordered Residual of a Specified Kernel Equation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0052","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b43cffb846e9a966a4738a3895ca09edbb5d33bd69db889fa92ecea0acd7a31","family_id":"qf-density-eigen-kernel-residual","seed":202610097052,"semantic_template":"qf-density-eigen-kernel-residual/[[[2,0],[0,3]],2,[1,0]]"},"fields":{"title":"Ordered Residual of a Specified Kernel Equation","stem":"Over R, M=[[2, 0], [0, 3]] is 2×2, λ=2, and v=[1, 0]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.","answer":{"kind":"expression","value":"(0, 0)","explanation":"Exact multiplication and subtraction yield (0, 0). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector."},"skills":["Ordered Residual of a Specified Kernel Equation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0053","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24bdd07c0f31765ef66f8f66c5f8aeb2eba1dc437e6d640af51187eb957d993d","family_id":"qf-density-eigen-kernel-residual","seed":202610097053,"semantic_template":"qf-density-eigen-kernel-residual/[[[3,3],[1,0]],0,[3,0]]"},"fields":{"title":"Ordered Residual of a Specified Kernel Equation","stem":"Over R, M=[[3, 3], [1, 0]] is 2×2, λ=0, and v=[3, 0]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.","answer":{"kind":"expression","value":"(9, 3)","explanation":"Exact multiplication and subtraction yield (9, 3). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector."},"skills":["Ordered Residual of a Specified Kernel Equation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0054","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07274d7b37d385d835898e718eeec1e07c14cdd80effe70804d2932af6f5663a","family_id":"qf-density-eigen-kernel-residual","seed":202610097054,"semantic_template":"qf-density-eigen-kernel-residual/[[[4,1],[-1,-1]],1,[1,-1]]"},"fields":{"title":"Ordered Residual of a Specified Kernel Equation","stem":"Over R, M=[[4, 1], [-1, -1]] is 2×2, λ=1, and v=[1, -1]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.","answer":{"kind":"expression","value":"(2, 1)","explanation":"Exact multiplication and subtraction yield (2, 1). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector."},"skills":["Ordered Residual of a Specified Kernel Equation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0055","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63c4ad1d5601a0688cd0174d301dcfa387929383e639b881e0e3355e4f296444","family_id":"qf-density-eigen-kernel-residual","seed":202610097055,"semantic_template":"qf-density-eigen-kernel-residual/[[[5,2],[1,-2]],2,[2,-2]]"},"fields":{"title":"Ordered Residual of a Specified Kernel Equation","stem":"Over R, M=[[5, 2], [1, -2]] is 2×2, λ=2, and v=[2, -2]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.","answer":{"kind":"expression","value":"(2, 10)","explanation":"Exact multiplication and subtraction yield (2, 10). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector."},"skills":["Ordered Residual of a Specified Kernel Equation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0056","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e03a6f0e427c4626cb0a031bee2327f0bc61db508f4cd067e07fa9c416fb5e6b","family_id":"qf-density-eigen-kernel-residual","seed":202610097056,"semantic_template":"qf-density-eigen-kernel-residual/[[[6,3],[-1,-3]],-2,[3,2]]"},"fields":{"title":"Ordered Residual of a Specified Kernel Equation","stem":"Over R, M=[[6, 3], [-1, -3]] is 2×2, λ=-2, and v=[3, 2]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.","answer":{"kind":"expression","value":"(30, -5)","explanation":"Exact multiplication and subtraction yield (30, -5). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector."},"skills":["Ordered Residual of a Specified Kernel Equation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0057","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e3f65ec39873d7456babe8af7c5703264cead3198d0f7f6ec2e9c265e82a3558","family_id":"qf-density-eigen-kernel-residual","seed":202610097057,"semantic_template":"qf-density-eigen-kernel-residual/[[[7,1],[1,-4]],-1,[1,1]]"},"fields":{"title":"Ordered Residual of a Specified Kernel Equation","stem":"Over R, M=[[7, 1], [1, -4]] is 2×2, λ=-1, and v=[1, 1]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.","answer":{"kind":"expression","value":"(9, -2)","explanation":"Exact multiplication and subtraction yield (9, -2). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector."},"skills":["Ordered Residual of a Specified Kernel Equation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0058","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f48cc76e29e4ccda3eaf2dd68bfd10756fc742cee9bdeab30187ebd5de7d243e","family_id":"qf-density-eigen-kernel-residual","seed":202610097058,"semantic_template":"qf-density-eigen-kernel-residual/[[[8,2],[-1,-5]],0,[2,0]]"},"fields":{"title":"Ordered Residual of a Specified Kernel Equation","stem":"Over R, M=[[8, 2], [-1, -5]] is 2×2, λ=0, and v=[2, 0]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.","answer":{"kind":"expression","value":"(16, -2)","explanation":"Exact multiplication and subtraction yield (16, -2). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector."},"skills":["Ordered Residual of a Specified Kernel Equation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0059","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"541e4549d48f3dcf78db6f57f23bd618707912c4e97cb7fb2985cf47ce2c3821","family_id":"qf-density-eigen-kernel-residual","seed":202610097059,"semantic_template":"qf-density-eigen-kernel-residual/[[[9,3],[1,-6]],1,[3,-1]]"},"fields":{"title":"Ordered Residual of a Specified Kernel Equation","stem":"Over R, M=[[9, 3], [1, -6]] is 2×2, λ=1, and v=[3, -1]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.","answer":{"kind":"expression","value":"(21, 10)","explanation":"Exact multiplication and subtraction yield (21, 10). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector."},"skills":["Ordered Residual of a Specified Kernel Equation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0060","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e71e55c3d065a5752f8d3ece49b7a8afacac6875add896764c804f1a3f671912","family_id":"qf-density-eigen-kernel-residual","seed":202610097060,"semantic_template":"qf-density-eigen-kernel-residual/[[[10,1],[-1,-7]],2,[1,-2]]"},"fields":{"title":"Ordered Residual of a Specified Kernel Equation","stem":"Over R, M=[[10, 1], [-1, -7]] is 2×2, λ=2, and v=[1, -2]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.","answer":{"kind":"expression","value":"(6, 17)","explanation":"Exact multiplication and subtraction yield (6, 17). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector."},"skills":["Ordered Residual of a Specified Kernel Equation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0061","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1dae9b41782d8ebe8f58a42d75b56058d450a9750e164d333218350174cdd2ae","family_id":"qf-density-eigen-kernel-residual","seed":202610097061,"semantic_template":"qf-density-eigen-kernel-residual/[[[11,2],[1,-8]],-2,[2,2]]"},"fields":{"title":"Ordered Residual of a Specified Kernel Equation","stem":"Over R, M=[[11, 2], [1, -8]] is 2×2, λ=-2, and v=[2, 2]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.","answer":{"kind":"expression","value":"(30, -10)","explanation":"Exact multiplication and subtraction yield (30, -10). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector."},"skills":["Ordered Residual of a Specified Kernel Equation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0062","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6527bf2f19832ff0a676e2da6636da533c8742e75522a7bd73b6d817752ef81","family_id":"qf-density-eigen-kernel-residual","seed":202610097062,"semantic_template":"qf-density-eigen-kernel-residual/[[[12,3],[-1,-9]],-1,[3,1]]"},"fields":{"title":"Ordered Residual of a Specified Kernel Equation","stem":"Over R, M=[[12, 3], [-1, -9]] is 2×2, λ=-1, and v=[3, 1]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.","answer":{"kind":"expression","value":"(42, -11)","explanation":"Exact multiplication and subtraction yield (42, -11). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector."},"skills":["Ordered Residual of a Specified Kernel Equation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0063","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eed0e3026906eb0f8658c96dc5e026ce36cafea80985c180bb6aa8f6c9b48f4b","family_id":"qf-density-eigen-kernel-residual","seed":202610097063,"semantic_template":"qf-density-eigen-kernel-residual/[[[13,1],[1,-10]],0,[1,0]]"},"fields":{"title":"Ordered Residual of a Specified Kernel Equation","stem":"Over R, M=[[13, 1], [1, -10]] is 2×2, λ=0, and v=[1, 0]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.","answer":{"kind":"expression","value":"(13, 1)","explanation":"Exact multiplication and subtraction yield (13, 1). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector."},"skills":["Ordered Residual of a Specified Kernel Equation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0064","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d2468ba6a47f9039efb855fba91ef82f2ec52045008a3c0dcdaa2210f0dc85f","family_id":"qf-density-eigen-kernel-residual","seed":202610097064,"semantic_template":"qf-density-eigen-kernel-residual/[[[14,2],[-1,-11]],1,[2,-1]]"},"fields":{"title":"Ordered Residual of a Specified Kernel Equation","stem":"Over R, M=[[14, 2], [-1, -11]] is 2×2, λ=1, and v=[2, -1]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.","answer":{"kind":"expression","value":"(24, 10)","explanation":"Exact multiplication and subtraction yield (24, 10). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector."},"skills":["Ordered Residual of a Specified Kernel Equation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0065","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a5462093a8015c4b79b0c0feb274d305b32416d839b8298432258d196affc0f","family_id":"qf-density-eigen-kernel-residual","seed":202610097065,"semantic_template":"qf-density-eigen-kernel-residual/[[[15,3],[1,-12]],2,[3,-2]]"},"fields":{"title":"Ordered Residual of a Specified Kernel Equation","stem":"Over R, M=[[15, 3], [1, -12]] is 2×2, λ=2, and v=[3, -2]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.","answer":{"kind":"expression","value":"(33, 31)","explanation":"Exact multiplication and subtraction yield (33, 31). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector."},"skills":["Ordered Residual of a Specified Kernel Equation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0066","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2b7f3c2d06ba8034a7b3632df473618f6e0e7c796079ca323b4dd8b4f325960","family_id":"qf-density-eigen-kernel-residual","seed":202610097066,"semantic_template":"qf-density-eigen-kernel-residual/[[[16,1],[-1,-13]],-2,[1,2]]"},"fields":{"title":"Ordered Residual of a Specified Kernel Equation","stem":"Over R, M=[[16, 1], [-1, -13]] is 2×2, λ=-2, and v=[1, 2]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.","answer":{"kind":"expression","value":"(20, -23)","explanation":"Exact multiplication and subtraction yield (20, -23). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector."},"skills":["Ordered Residual of a Specified Kernel Equation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0067","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce24551f2b3ff237b4f5231fb1ddad5a83a88dafcd7a3badcb1d4f8feecd3ac6","family_id":"qf-density-eigen-kernel-residual","seed":202610097067,"semantic_template":"qf-density-eigen-kernel-residual/[[[17,2],[1,-14]],-1,[2,1]]"},"fields":{"title":"Ordered Residual of a Specified Kernel Equation","stem":"Over R, M=[[17, 2], [1, -14]] is 2×2, λ=-1, and v=[2, 1]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.","answer":{"kind":"expression","value":"(38, -11)","explanation":"Exact multiplication and subtraction yield (38, -11). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector."},"skills":["Ordered Residual of a Specified Kernel Equation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0068","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee9e02a23c6e6d7481d7257dc0910c40e71f466b76a3ea26570983c3ef5d7ac5","family_id":"qf-density-eigen-kernel-residual","seed":202610097068,"semantic_template":"qf-density-eigen-kernel-residual/[[[18,3],[-1,-15]],0,[3,0]]"},"fields":{"title":"Ordered Residual of a Specified Kernel Equation","stem":"Over R, M=[[18, 3], [-1, -15]] is 2×2, λ=0, and v=[3, 0]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.","answer":{"kind":"expression","value":"(54, -3)","explanation":"Exact multiplication and subtraction yield (54, -3). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector."},"skills":["Ordered Residual of a Specified Kernel Equation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0069","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28c0bd3313bea2be06cd4e1b959c9672c6c3f6016fb16e3a4743aadd68846b5c","family_id":"qf-density-eigen-kernel-residual","seed":202610097069,"semantic_template":"qf-density-eigen-kernel-residual/[[[19,1],[1,-16]],1,[1,-1]]"},"fields":{"title":"Ordered Residual of a Specified Kernel Equation","stem":"Over R, M=[[19, 1], [1, -16]] is 2×2, λ=1, and v=[1, -1]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.","answer":{"kind":"expression","value":"(17, 18)","explanation":"Exact multiplication and subtraction yield (17, 18). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector."},"skills":["Ordered Residual of a Specified Kernel Equation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0070","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e62dbb92a08ab75099f0810201b2fc76066af265750894ba00c57fdf2621a56","family_id":"qf-density-eigen-kernel-residual","seed":202610097070,"semantic_template":"qf-density-eigen-kernel-residual/[[[20,2],[-1,-17]],2,[2,-2]]"},"fields":{"title":"Ordered Residual of a Specified Kernel Equation","stem":"Over R, M=[[20, 2], [-1, -17]] is 2×2, λ=2, and v=[2, -2]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.","answer":{"kind":"expression","value":"(32, 36)","explanation":"Exact multiplication and subtraction yield (32, 36). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector."},"skills":["Ordered Residual of a Specified Kernel Equation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0071","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13be4a8b6f0d31e57ce382c393d9a03351712c1b550c69e7389b064d3a65c397","family_id":"qf-density-eigen-kernel-residual","seed":202610097071,"semantic_template":"qf-density-eigen-kernel-residual/[[[21,3],[1,-18]],-2,[3,2]]"},"fields":{"title":"Ordered Residual of a Specified Kernel Equation","stem":"Over R, M=[[21, 3], [1, -18]] is 2×2, λ=-2, and v=[3, 2]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.","answer":{"kind":"expression","value":"(75, -29)","explanation":"Exact multiplication and subtraction yield (75, -29). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector."},"skills":["Ordered Residual of a Specified Kernel Equation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0072","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9692adc666eb1ce939ae1f366b786ed5312cd9eb504d6856e8e9668b10daa100","family_id":"qf-density-eigen-kernel-residual","seed":202610097072,"semantic_template":"qf-density-eigen-kernel-residual/[[[22,1],[-1,-19]],-1,[1,1]]"},"fields":{"title":"Ordered Residual of a Specified Kernel Equation","stem":"Over R, M=[[22, 1], [-1, -19]] is 2×2, λ=-1, and v=[1, 1]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.","answer":{"kind":"expression","value":"(24, -19)","explanation":"Exact multiplication and subtraction yield (24, -19). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector."},"skills":["Ordered Residual of a Specified Kernel Equation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0073","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b002518ace701a068112d474222219305d7e11e24cf4311a02b0e10ab0ec860c","family_id":"qf-density-eigen-kernel-residual","seed":202610097073,"semantic_template":"qf-density-eigen-kernel-residual/[[[23,2],[1,-20]],0,[2,0]]"},"fields":{"title":"Ordered Residual of a Specified Kernel Equation","stem":"Over R, M=[[23, 2], [1, -20]] is 2×2, λ=0, and v=[2, 0]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.","answer":{"kind":"expression","value":"(46, 2)","explanation":"Exact multiplication and subtraction yield (46, 2). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector."},"skills":["Ordered Residual of a Specified Kernel Equation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0074","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39efb8dcbd748a47c88ccddefbed3c698f6b664b863b91336e15b955339e6a8b","family_id":"qf-density-eigen-kernel-residual","seed":202610097074,"semantic_template":"qf-density-eigen-kernel-residual/[[[24,3],[-1,-21]],1,[3,-1]]"},"fields":{"title":"Ordered Residual of a Specified Kernel Equation","stem":"Over R, M=[[24, 3], [-1, -21]] is 2×2, λ=1, and v=[3, -1]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.","answer":{"kind":"expression","value":"(66, 19)","explanation":"Exact multiplication and subtraction yield (66, 19). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector."},"skills":["Ordered Residual of a Specified Kernel Equation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0075","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f510edbcfb55520c506652792adb4993f7e1994da8d8be41a590ac192f57f90","family_id":"qf-density-eigen-kernel-residual","seed":202610097075,"semantic_template":"qf-density-eigen-kernel-residual/[[[25,1],[1,-22]],2,[1,-2]]"},"fields":{"title":"Ordered Residual of a Specified Kernel Equation","stem":"Over R, M=[[25, 1], [1, -22]] is 2×2, λ=2, and v=[1, -2]. Give the ordered residual Mv−λv, without assuming λ is an eigenvalue.","answer":{"kind":"expression","value":"(21, 49)","explanation":"Exact multiplication and subtraction yield (21, 49). A zero vector always has zero residual, which alone does not establish an eigenvalue or an eligible eigenvector."},"skills":["Ordered Residual of a Specified Kernel Equation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0076","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"015c0af77b6df2608109116cc0485d02d9c3828b279436a4c9a80ca9f4132ad0","family_id":"qf-density-eigen-first-mismatch","seed":202610097076,"semantic_template":"qf-density-eigen-first-mismatch/[[[1,1],[0,2]],2,[1,1]]"},"fields":{"title":"First Mismatch of a Nonzero Eigenvector Candidate","stem":"Over R, M=[[1, 1], [0, 2]] is 2×2, λ=2, and the nonzero candidate v=[1, 1]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.","answer":{"kind":"short_text","value":"none","explanation":"The first mismatch is none. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory."},"skills":["First Mismatch of a Nonzero Eigenvector Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0077","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95e7d9c95e7323087bd82d9a2b778468cc7db231ebf0b7be654157f2e7802382","family_id":"qf-density-eigen-first-mismatch","seed":202610097077,"semantic_template":"qf-density-eigen-first-mismatch/[[[0,2],[1,2]],0,[0,1]]"},"fields":{"title":"First Mismatch of a Nonzero Eigenvector Candidate","stem":"Over R, M=[[0, 2], [1, 2]] is 2×2, λ=0, and the nonzero candidate v=[0, 1]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.","answer":{"kind":"short_text","value":"1","explanation":"The first mismatch is 1. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory."},"skills":["First Mismatch of a Nonzero Eigenvector Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0078","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b79bcc3ac657d6cf5fe34ca7a2cdeae6e4a1dcdaea578288345e4e1eabd54a5","family_id":"qf-density-eigen-first-mismatch","seed":202610097078,"semantic_template":"qf-density-eigen-first-mismatch/[[[3,0],[0,8]],3,[1,1]]"},"fields":{"title":"First Mismatch of a Nonzero Eigenvector Candidate","stem":"Over R, M=[[3, 0], [0, 8]] is 2×2, λ=3, and the nonzero candidate v=[1, 1]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.","answer":{"kind":"short_text","value":"2","explanation":"The first mismatch is 2. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory."},"skills":["First Mismatch of a Nonzero Eigenvector Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0079","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8213c3bf914d6829791b48e544bf5b848c1e39de5c8c3fd5ecd3c444e8cb2a6","family_id":"qf-density-eigen-first-mismatch","seed":202610097079,"semantic_template":"qf-density-eigen-first-mismatch/[[[1,1],[0,5]],5,[1,4]]"},"fields":{"title":"First Mismatch of a Nonzero Eigenvector Candidate","stem":"Over R, M=[[1, 1], [0, 5]] is 2×2, λ=5, and the nonzero candidate v=[1, 4]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.","answer":{"kind":"short_text","value":"none","explanation":"The first mismatch is none. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory."},"skills":["First Mismatch of a Nonzero Eigenvector Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0080","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a132b1bb11cf8a8a48a1b237e895817b9be97414d8bdf0c93c43c3e2ce4ff62c","family_id":"qf-density-eigen-first-mismatch","seed":202610097080,"semantic_template":"qf-density-eigen-first-mismatch/[[[0,5],[1,2]],0,[0,1]]"},"fields":{"title":"First Mismatch of a Nonzero Eigenvector Candidate","stem":"Over R, M=[[0, 5], [1, 2]] is 2×2, λ=0, and the nonzero candidate v=[0, 1]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.","answer":{"kind":"short_text","value":"1","explanation":"The first mismatch is 1. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory."},"skills":["First Mismatch of a Nonzero Eigenvector Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0081","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"539ff7f7ebf9a9a8333f7a871ba0559933285a8da6612b7a094644fd8b7f5b18","family_id":"qf-density-eigen-first-mismatch","seed":202610097081,"semantic_template":"qf-density-eigen-first-mismatch/[[[6,0],[0,14]],6,[1,1]]"},"fields":{"title":"First Mismatch of a Nonzero Eigenvector Candidate","stem":"Over R, M=[[6, 0], [0, 14]] is 2×2, λ=6, and the nonzero candidate v=[1, 1]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.","answer":{"kind":"short_text","value":"2","explanation":"The first mismatch is 2. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory."},"skills":["First Mismatch of a Nonzero Eigenvector Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0082","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c45cb17899b45aa37f5474db1fc4236404fdfed02f36078727fe7dc2f67bcfb5","family_id":"qf-density-eigen-first-mismatch","seed":202610097082,"semantic_template":"qf-density-eigen-first-mismatch/[[[1,1],[0,8]],8,[1,7]]"},"fields":{"title":"First Mismatch of a Nonzero Eigenvector Candidate","stem":"Over R, M=[[1, 1], [0, 8]] is 2×2, λ=8, and the nonzero candidate v=[1, 7]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.","answer":{"kind":"short_text","value":"none","explanation":"The first mismatch is none. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory."},"skills":["First Mismatch of a Nonzero Eigenvector Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0083","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65ca5e08e3497dd7f517a771688d82519081342016c04bd2ad94aaef6e045535","family_id":"qf-density-eigen-first-mismatch","seed":202610097083,"semantic_template":"qf-density-eigen-first-mismatch/[[[0,8],[1,2]],0,[0,1]]"},"fields":{"title":"First Mismatch of a Nonzero Eigenvector Candidate","stem":"Over R, M=[[0, 8], [1, 2]] is 2×2, λ=0, and the nonzero candidate v=[0, 1]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.","answer":{"kind":"short_text","value":"1","explanation":"The first mismatch is 1. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory."},"skills":["First Mismatch of a Nonzero Eigenvector Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0084","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a7713945380e0deac9e1fa3fc8f3b86463d646b15ecff5b1514072df391fb7c","family_id":"qf-density-eigen-first-mismatch","seed":202610097084,"semantic_template":"qf-density-eigen-first-mismatch/[[[9,0],[0,20]],9,[1,1]]"},"fields":{"title":"First Mismatch of a Nonzero Eigenvector Candidate","stem":"Over R, M=[[9, 0], [0, 20]] is 2×2, λ=9, and the nonzero candidate v=[1, 1]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.","answer":{"kind":"short_text","value":"2","explanation":"The first mismatch is 2. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory."},"skills":["First Mismatch of a Nonzero Eigenvector Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0085","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f9db79d58baed5da321a9131e2d35e4bc7079e0fdf3ad0ec83688e11b2a2a28","family_id":"qf-density-eigen-first-mismatch","seed":202610097085,"semantic_template":"qf-density-eigen-first-mismatch/[[[1,1],[0,11]],11,[1,10]]"},"fields":{"title":"First Mismatch of a Nonzero Eigenvector Candidate","stem":"Over R, M=[[1, 1], [0, 11]] is 2×2, λ=11, and the nonzero candidate v=[1, 10]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.","answer":{"kind":"short_text","value":"none","explanation":"The first mismatch is none. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory."},"skills":["First Mismatch of a Nonzero Eigenvector Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0086","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9603b3e6c559aae8b5e28a2cccafd98faa01935ed3a0c0f205ac88ad15d8983e","family_id":"qf-density-eigen-first-mismatch","seed":202610097086,"semantic_template":"qf-density-eigen-first-mismatch/[[[0,11],[1,2]],0,[0,1]]"},"fields":{"title":"First Mismatch of a Nonzero Eigenvector Candidate","stem":"Over R, M=[[0, 11], [1, 2]] is 2×2, λ=0, and the nonzero candidate v=[0, 1]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.","answer":{"kind":"short_text","value":"1","explanation":"The first mismatch is 1. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory."},"skills":["First Mismatch of a Nonzero Eigenvector Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0087","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca226afd4364a217aa9a7114d834197f9a5db9e4e933e9feeb77a881ef95a443","family_id":"qf-density-eigen-first-mismatch","seed":202610097087,"semantic_template":"qf-density-eigen-first-mismatch/[[[12,0],[0,26]],12,[1,1]]"},"fields":{"title":"First Mismatch of a Nonzero Eigenvector Candidate","stem":"Over R, M=[[12, 0], [0, 26]] is 2×2, λ=12, and the nonzero candidate v=[1, 1]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.","answer":{"kind":"short_text","value":"2","explanation":"The first mismatch is 2. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory."},"skills":["First Mismatch of a Nonzero Eigenvector Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0088","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49a8d6ed43fb90e23c3591daebd66c9db026e0488644a0a126f9f856afb2c67e","family_id":"qf-density-eigen-first-mismatch","seed":202610097088,"semantic_template":"qf-density-eigen-first-mismatch/[[[1,1],[0,14]],14,[1,13]]"},"fields":{"title":"First Mismatch of a Nonzero Eigenvector Candidate","stem":"Over R, M=[[1, 1], [0, 14]] is 2×2, λ=14, and the nonzero candidate v=[1, 13]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.","answer":{"kind":"short_text","value":"none","explanation":"The first mismatch is none. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory."},"skills":["First Mismatch of a Nonzero Eigenvector Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0089","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c2c666633a75f769c2bb8f63d55b0eb1e823107247af48ed483dc85aa89d198","family_id":"qf-density-eigen-first-mismatch","seed":202610097089,"semantic_template":"qf-density-eigen-first-mismatch/[[[0,14],[1,2]],0,[0,1]]"},"fields":{"title":"First Mismatch of a Nonzero Eigenvector Candidate","stem":"Over R, M=[[0, 14], [1, 2]] is 2×2, λ=0, and the nonzero candidate v=[0, 1]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.","answer":{"kind":"short_text","value":"1","explanation":"The first mismatch is 1. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory."},"skills":["First Mismatch of a Nonzero Eigenvector Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0090","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14b6795966a549367030040c98b2c14684296e93548a0e129e5e37c72b781646","family_id":"qf-density-eigen-first-mismatch","seed":202610097090,"semantic_template":"qf-density-eigen-first-mismatch/[[[15,0],[0,32]],15,[1,1]]"},"fields":{"title":"First Mismatch of a Nonzero Eigenvector Candidate","stem":"Over R, M=[[15, 0], [0, 32]] is 2×2, λ=15, and the nonzero candidate v=[1, 1]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.","answer":{"kind":"short_text","value":"2","explanation":"The first mismatch is 2. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory."},"skills":["First Mismatch of a Nonzero Eigenvector Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0091","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a62e7739143bb996dd2721466a0cef1e307ad56a76fed17ac254d63fc0e4eb5","family_id":"qf-density-eigen-first-mismatch","seed":202610097091,"semantic_template":"qf-density-eigen-first-mismatch/[[[1,1],[0,17]],17,[1,16]]"},"fields":{"title":"First Mismatch of a Nonzero Eigenvector Candidate","stem":"Over R, M=[[1, 1], [0, 17]] is 2×2, λ=17, and the nonzero candidate v=[1, 16]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.","answer":{"kind":"short_text","value":"none","explanation":"The first mismatch is none. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory."},"skills":["First Mismatch of a Nonzero Eigenvector Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0092","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76a1da7525fb3826781c99edf66e99917770521c7de9e1adfad99be83fc9c182","family_id":"qf-density-eigen-first-mismatch","seed":202610097092,"semantic_template":"qf-density-eigen-first-mismatch/[[[0,17],[1,2]],0,[0,1]]"},"fields":{"title":"First Mismatch of a Nonzero Eigenvector Candidate","stem":"Over R, M=[[0, 17], [1, 2]] is 2×2, λ=0, and the nonzero candidate v=[0, 1]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.","answer":{"kind":"short_text","value":"1","explanation":"The first mismatch is 1. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory."},"skills":["First Mismatch of a Nonzero Eigenvector Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0093","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"884b1227478030e56129c9816e1a5482ea692ad221dfeda3858bc135d9bfeffc","family_id":"qf-density-eigen-first-mismatch","seed":202610097093,"semantic_template":"qf-density-eigen-first-mismatch/[[[18,0],[0,38]],18,[1,1]]"},"fields":{"title":"First Mismatch of a Nonzero Eigenvector Candidate","stem":"Over R, M=[[18, 0], [0, 38]] is 2×2, λ=18, and the nonzero candidate v=[1, 1]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.","answer":{"kind":"short_text","value":"2","explanation":"The first mismatch is 2. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory."},"skills":["First Mismatch of a Nonzero Eigenvector Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0094","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"075f43765d0ec4dd93e58bdd35c410df56d283e72ae6771a0aa091425792eaab","family_id":"qf-density-eigen-first-mismatch","seed":202610097094,"semantic_template":"qf-density-eigen-first-mismatch/[[[1,1],[0,20]],20,[1,19]]"},"fields":{"title":"First Mismatch of a Nonzero Eigenvector Candidate","stem":"Over R, M=[[1, 1], [0, 20]] is 2×2, λ=20, and the nonzero candidate v=[1, 19]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.","answer":{"kind":"short_text","value":"none","explanation":"The first mismatch is none. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory."},"skills":["First Mismatch of a Nonzero Eigenvector Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0095","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7cfd40b7d1ec99f885f0218a0059df95e4929cdc83892854547e66f179a397ae","family_id":"qf-density-eigen-first-mismatch","seed":202610097095,"semantic_template":"qf-density-eigen-first-mismatch/[[[0,20],[1,2]],0,[0,1]]"},"fields":{"title":"First Mismatch of a Nonzero Eigenvector Candidate","stem":"Over R, M=[[0, 20], [1, 2]] is 2×2, λ=0, and the nonzero candidate v=[0, 1]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.","answer":{"kind":"short_text","value":"1","explanation":"The first mismatch is 1. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory."},"skills":["First Mismatch of a Nonzero Eigenvector Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0096","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe5386669797b1cbbbc4332f3d49d162e03bf8df283053738121e04e3ca32ecd","family_id":"qf-density-eigen-first-mismatch","seed":202610097096,"semantic_template":"qf-density-eigen-first-mismatch/[[[21,0],[0,44]],21,[1,1]]"},"fields":{"title":"First Mismatch of a Nonzero Eigenvector Candidate","stem":"Over R, M=[[21, 0], [0, 44]] is 2×2, λ=21, and the nonzero candidate v=[1, 1]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.","answer":{"kind":"short_text","value":"2","explanation":"The first mismatch is 2. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory."},"skills":["First Mismatch of a Nonzero Eigenvector Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0097","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"394e5349358b9fdaf6c537b860e041cbb373f8d4ddd973677eef49fb1b058d0f","family_id":"qf-density-eigen-first-mismatch","seed":202610097097,"semantic_template":"qf-density-eigen-first-mismatch/[[[1,1],[0,23]],23,[1,22]]"},"fields":{"title":"First Mismatch of a Nonzero Eigenvector Candidate","stem":"Over R, M=[[1, 1], [0, 23]] is 2×2, λ=23, and the nonzero candidate v=[1, 22]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.","answer":{"kind":"short_text","value":"none","explanation":"The first mismatch is none. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory."},"skills":["First Mismatch of a Nonzero Eigenvector Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0098","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"219b065cb04bea44d1f60d41907627952ef3ff0dcfcb3dcdf032e33a7f354fa9","family_id":"qf-density-eigen-first-mismatch","seed":202610097098,"semantic_template":"qf-density-eigen-first-mismatch/[[[0,23],[1,2]],0,[0,1]]"},"fields":{"title":"First Mismatch of a Nonzero Eigenvector Candidate","stem":"Over R, M=[[0, 23], [1, 2]] is 2×2, λ=0, and the nonzero candidate v=[0, 1]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.","answer":{"kind":"short_text","value":"1","explanation":"The first mismatch is 1. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory."},"skills":["First Mismatch of a Nonzero Eigenvector Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0099","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"835ee2a84638cf94c53344a533db53433b9e21d85681d9d09354c66a081d2386","family_id":"qf-density-eigen-first-mismatch","seed":202610097099,"semantic_template":"qf-density-eigen-first-mismatch/[[[24,0],[0,50]],24,[1,1]]"},"fields":{"title":"First Mismatch of a Nonzero Eigenvector Candidate","stem":"Over R, M=[[24, 0], [0, 50]] is 2×2, λ=24, and the nonzero candidate v=[1, 1]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.","answer":{"kind":"short_text","value":"2","explanation":"The first mismatch is 2. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory."},"skills":["First Mismatch of a Nonzero Eigenvector Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261007-advance-20261009-r07-eigenvalue-0100","batch_id":"batch-20261007-advance-20261009-r07-eigenvalue-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f9c37a2eba5c7b1337f2f1245a5e4557cd954dd6e22b48290e61798f0425f3c","family_id":"qf-density-eigen-first-mismatch","seed":202610097100,"semantic_template":"qf-density-eigen-first-mismatch/[[[1,1],[0,26]],26,[1,25]]"},"fields":{"title":"First Mismatch of a Nonzero Eigenvector Candidate","stem":"Over R, M=[[1, 1], [0, 26]] is 2×2, λ=26, and the nonzero candidate v=[1, 25]. Check residual components in order 1,2 and report the first nonzero component index, or none if both vanish.","answer":{"kind":"short_text","value":"none","explanation":"The first mismatch is none. Only with the stated nonzero-vector premise does no mismatch support this eigenpair; this does not establish diagonalization or general spectral theory."},"skills":["First Mismatch of a Nonzero Eigenvector Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}