{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0001","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"208842ce3b740dda1cb2a73b9dd294d02bbfc7b580f31a7aaca68971af527f1a","family_id":"qf-mod23-multivar-evaluate","seed":202610237001,"semantic_template":"qf-mod23-multivar-evaluate/{\"coefficients\":[-12,-2,-3,1,-1,-1],\"x\":-2,\"y\":-3}"},"fields":{"title":"Evaluate with Fixed Bivariate Coefficient Roles","stem":"Use the declared real polynomial R²→R with coefficients=[-12, -2, -3, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=-2, y=-3. Give (coefficients,x,y,exact polynomial value).","answer":{"kind":"expression","value":"([-12, -2, -3, 1, -1, -1], -2, -3, -10)","explanation":"Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-12, -2, -3, 1, -1, -1], -2, -3, -10)."},"skills":["Evaluate with Fixed Bivariate Coefficient Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0002","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dff88df65f58c3a825dd2ac3ba06422fd9578ec6982abbc62a2ddfebf461d5c7","family_id":"qf-mod23-multivar-evaluate","seed":202610237002,"semantic_template":"qf-mod23-multivar-evaluate/{\"coefficients\":[-11,-1,-2,1,0,-1],\"x\":-1,\"y\":-2}"},"fields":{"title":"Evaluate with Fixed Bivariate Coefficient Roles","stem":"Use the declared real polynomial R²→R with coefficients=[-11, -1, -2, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=-1, y=-2. Give (coefficients,x,y,exact polynomial value).","answer":{"kind":"expression","value":"([-11, -1, -2, 1, 0, -1], -1, -2, -9)","explanation":"Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-11, -1, -2, 1, 0, -1], -1, -2, -9)."},"skills":["Evaluate with Fixed Bivariate Coefficient Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0003","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32957462f218a6eb3b07458f0f69c24720cc778583e6ac30e91159c7a56e26f8","family_id":"qf-mod23-multivar-evaluate","seed":202610237003,"semantic_template":"qf-mod23-multivar-evaluate/{\"coefficients\":[-10,0,-1,1,1,-1],\"x\":0,\"y\":-1}"},"fields":{"title":"Evaluate with Fixed Bivariate Coefficient Roles","stem":"Use the declared real polynomial R²→R with coefficients=[-10, 0, -1, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=0, y=-1. Give (coefficients,x,y,exact polynomial value).","answer":{"kind":"expression","value":"([-10, 0, -1, 1, 1, -1], 0, -1, -10)","explanation":"Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 0, -1, 1, 1, -1], 0, -1, -10)."},"skills":["Evaluate with Fixed Bivariate Coefficient Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0004","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"072ab2b70fe9452cc766c5bffbc8889301803387f2d91f63f5d8479995d5c45b","family_id":"qf-mod23-multivar-evaluate","seed":202610237004,"semantic_template":"qf-mod23-multivar-evaluate/{\"coefficients\":[-9,1,0,1,-1,-1],\"x\":1,\"y\":0}"},"fields":{"title":"Evaluate with Fixed Bivariate Coefficient Roles","stem":"Use the declared real polynomial R²→R with coefficients=[-9, 1, 0, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=1, y=0. Give (coefficients,x,y,exact polynomial value).","answer":{"kind":"expression","value":"([-9, 1, 0, 1, -1, -1], 1, 0, -7)","explanation":"Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 1, 0, 1, -1, -1], 1, 0, -7)."},"skills":["Evaluate with Fixed Bivariate Coefficient Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0005","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7e3a20b1f55f007033668cf531fd5c852f14eef6f87cfc293f8932da29a8509","family_id":"qf-mod23-multivar-evaluate","seed":202610237005,"semantic_template":"qf-mod23-multivar-evaluate/{\"coefficients\":[-8,2,1,1,0,-1],\"x\":2,\"y\":1}"},"fields":{"title":"Evaluate with Fixed Bivariate Coefficient Roles","stem":"Use the declared real polynomial R²→R with coefficients=[-8, 2, 1, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=2, y=1. Give (coefficients,x,y,exact polynomial value).","answer":{"kind":"expression","value":"([-8, 2, 1, 1, 0, -1], 2, 1, 0)","explanation":"Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 2, 1, 1, 0, -1], 2, 1, 0)."},"skills":["Evaluate with Fixed Bivariate Coefficient Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0006","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7ffde34d684f2d999c7f0e52b83628d2954b0c08e8ac960a2d49b1971aec2dd","family_id":"qf-mod23-multivar-evaluate","seed":202610237006,"semantic_template":"qf-mod23-multivar-evaluate/{\"coefficients\":[-7,-2,2,1,1,-1],\"x\":-2,\"y\":2}"},"fields":{"title":"Evaluate with Fixed Bivariate Coefficient Roles","stem":"Use the declared real polynomial R²→R with coefficients=[-7, -2, 2, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=-2, y=2. Give (coefficients,x,y,exact polynomial value).","answer":{"kind":"expression","value":"([-7, -2, 2, 1, 1, -1], -2, 2, -3)","explanation":"Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-7, -2, 2, 1, 1, -1], -2, 2, -3)."},"skills":["Evaluate with Fixed Bivariate Coefficient Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0007","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7cc6b11450795b8658612f263a378a5d4ec7315d3f7a6849188d13e71a5f7c1d","family_id":"qf-mod23-multivar-evaluate","seed":202610237007,"semantic_template":"qf-mod23-multivar-evaluate/{\"coefficients\":[-6,-1,3,1,-1,-1],\"x\":-1,\"y\":3}"},"fields":{"title":"Evaluate with Fixed Bivariate Coefficient Roles","stem":"Use the declared real polynomial R²→R with coefficients=[-6, -1, 3, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=-1, y=3. Give (coefficients,x,y,exact polynomial value).","answer":{"kind":"expression","value":"([-6, -1, 3, 1, -1, -1], -1, 3, -1)","explanation":"Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-6, -1, 3, 1, -1, -1], -1, 3, -1)."},"skills":["Evaluate with Fixed Bivariate Coefficient Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0008","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b462ff16521fe8a126816f6e1abba7443d3c11d32ca3ac731ada1ac99a85383f","family_id":"qf-mod23-multivar-evaluate","seed":202610237008,"semantic_template":"qf-mod23-multivar-evaluate/{\"coefficients\":[-5,0,-3,1,0,-1],\"x\":0,\"y\":-3}"},"fields":{"title":"Evaluate with Fixed Bivariate Coefficient Roles","stem":"Use the declared real polynomial R²→R with coefficients=[-5, 0, -3, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=0, y=-3. Give (coefficients,x,y,exact polynomial value).","answer":{"kind":"expression","value":"([-5, 0, -3, 1, 0, -1], 0, -3, -5)","explanation":"Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 0, -3, 1, 0, -1], 0, -3, -5)."},"skills":["Evaluate with Fixed Bivariate Coefficient Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0009","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"25bd633a06088e1095e34ffbe09896efc6f599686e94925929c220497779f1d1","family_id":"qf-mod23-multivar-evaluate","seed":202610237009,"semantic_template":"qf-mod23-multivar-evaluate/{\"coefficients\":[-4,1,-2,1,1,-1],\"x\":1,\"y\":-2}"},"fields":{"title":"Evaluate with Fixed Bivariate Coefficient Roles","stem":"Use the declared real polynomial R²→R with coefficients=[-4, 1, -2, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=1, y=-2. Give (coefficients,x,y,exact polynomial value).","answer":{"kind":"expression","value":"([-4, 1, -2, 1, 1, -1], 1, -2, -4)","explanation":"Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 1, -2, 1, 1, -1], 1, -2, -4)."},"skills":["Evaluate with Fixed Bivariate Coefficient Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0010","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f15e46e8791ae7525c9d9808f9db54ef2e24891858c389af0f9bdc3d42172abe","family_id":"qf-mod23-multivar-evaluate","seed":202610237010,"semantic_template":"qf-mod23-multivar-evaluate/{\"coefficients\":[-3,2,-1,1,-1,-1],\"x\":2,\"y\":-1}"},"fields":{"title":"Evaluate with Fixed Bivariate Coefficient Roles","stem":"Use the declared real polynomial R²→R with coefficients=[-3, 2, -1, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=2, y=-1. Give (coefficients,x,y,exact polynomial value).","answer":{"kind":"expression","value":"([-3, 2, -1, 1, -1, -1], 2, -1, 7)","explanation":"Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 2, -1, 1, -1, -1], 2, -1, 7)."},"skills":["Evaluate with Fixed Bivariate Coefficient Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0011","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90f74572f4577e78fb8d69310142e3e1326ccdc84ba99f9d982393840c984500","family_id":"qf-mod23-multivar-evaluate","seed":202610237011,"semantic_template":"qf-mod23-multivar-evaluate/{\"coefficients\":[-2,-2,0,1,0,-1],\"x\":-2,\"y\":0}"},"fields":{"title":"Evaluate with Fixed Bivariate Coefficient Roles","stem":"Use the declared real polynomial R²→R with coefficients=[-2, -2, 0, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=-2, y=0. Give (coefficients,x,y,exact polynomial value).","answer":{"kind":"expression","value":"([-2, -2, 0, 1, 0, -1], -2, 0, 6)","explanation":"Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-2, -2, 0, 1, 0, -1], -2, 0, 6)."},"skills":["Evaluate with Fixed Bivariate Coefficient Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0012","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd79279c029de3d149b63ac56cb92b3136c9a303b8c5046b29bfa959d4864d34","family_id":"qf-mod23-multivar-evaluate","seed":202610237012,"semantic_template":"qf-mod23-multivar-evaluate/{\"coefficients\":[-1,-1,1,1,1,-1],\"x\":-1,\"y\":1}"},"fields":{"title":"Evaluate with Fixed Bivariate Coefficient Roles","stem":"Use the declared real polynomial R²→R with coefficients=[-1, -1, 1, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=-1, y=1. Give (coefficients,x,y,exact polynomial value).","answer":{"kind":"expression","value":"([-1, -1, 1, 1, 1, -1], -1, 1, 0)","explanation":"Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -1, 1, 1, 1, -1], -1, 1, 0)."},"skills":["Evaluate with Fixed Bivariate Coefficient Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0013","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6fdb4b73095537d18828ba7064f4443fb84ad5eb309015c14ec4cd3f7e6adefb","family_id":"qf-mod23-multivar-evaluate","seed":202610237013,"semantic_template":"qf-mod23-multivar-evaluate/{\"coefficients\":[0,0,2,1,-1,-1],\"x\":0,\"y\":2}"},"fields":{"title":"Evaluate with Fixed Bivariate Coefficient Roles","stem":"Use the declared real polynomial R²→R with coefficients=[0, 0, 2, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=0, y=2. Give (coefficients,x,y,exact polynomial value).","answer":{"kind":"expression","value":"([0, 0, 2, 1, -1, -1], 0, 2, 0)","explanation":"Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 2, 1, -1, -1], 0, 2, 0)."},"skills":["Evaluate with Fixed Bivariate Coefficient Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0014","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"658b58c3dede29930c67870565969795bc34c4c266a9ab491a78573bca6fc7ec","family_id":"qf-mod23-multivar-evaluate","seed":202610237014,"semantic_template":"qf-mod23-multivar-evaluate/{\"coefficients\":[1,1,3,1,0,-1],\"x\":1,\"y\":3}"},"fields":{"title":"Evaluate with Fixed Bivariate Coefficient Roles","stem":"Use the declared real polynomial R²→R with coefficients=[1, 1, 3, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=1, y=3. Give (coefficients,x,y,exact polynomial value).","answer":{"kind":"expression","value":"([1, 1, 3, 1, 0, -1], 1, 3, 3)","explanation":"Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 3, 1, 0, -1], 1, 3, 3)."},"skills":["Evaluate with Fixed Bivariate Coefficient Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0015","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b25c51032c92dbb8e2ee59eaa7e15db30800aa9249f56ab8555fd185f5db794a","family_id":"qf-mod23-multivar-evaluate","seed":202610237015,"semantic_template":"qf-mod23-multivar-evaluate/{\"coefficients\":[2,2,-3,1,1,-1],\"x\":2,\"y\":-3}"},"fields":{"title":"Evaluate with Fixed Bivariate Coefficient Roles","stem":"Use the declared real polynomial R²→R with coefficients=[2, 2, -3, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=2, y=-3. Give (coefficients,x,y,exact polynomial value).","answer":{"kind":"expression","value":"([2, 2, -3, 1, 1, -1], 2, -3, 4)","explanation":"Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, -3, 1, 1, -1], 2, -3, 4)."},"skills":["Evaluate with Fixed Bivariate Coefficient Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0016","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"435800824749188d7ffea3afa1af8a4c4e1a7ef9d15410084ad39d332d759825","family_id":"qf-mod23-multivar-evaluate","seed":202610237016,"semantic_template":"qf-mod23-multivar-evaluate/{\"coefficients\":[3,-2,-2,1,-1,-1],\"x\":-2,\"y\":-2}"},"fields":{"title":"Evaluate with Fixed Bivariate Coefficient Roles","stem":"Use the declared real polynomial R²→R with coefficients=[3, -2, -2, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=-2, y=-2. Give (coefficients,x,y,exact polynomial value).","answer":{"kind":"expression","value":"([3, -2, -2, 1, -1, -1], -2, -2, 7)","explanation":"Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, -2, -2, 1, -1, -1], -2, -2, 7)."},"skills":["Evaluate with Fixed Bivariate Coefficient Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0017","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f9a677ee9088ce0e0cb7de84b489654695453781a744ee93abf12ae1409cd7ad","family_id":"qf-mod23-multivar-evaluate","seed":202610237017,"semantic_template":"qf-mod23-multivar-evaluate/{\"coefficients\":[4,-1,-1,1,0,-1],\"x\":-1,\"y\":-1}"},"fields":{"title":"Evaluate with Fixed Bivariate Coefficient Roles","stem":"Use the declared real polynomial R²→R with coefficients=[4, -1, -1, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=-1, y=-1. Give (coefficients,x,y,exact polynomial value).","answer":{"kind":"expression","value":"([4, -1, -1, 1, 0, -1], -1, -1, 6)","explanation":"Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1, -1, 1, 0, -1], -1, -1, 6)."},"skills":["Evaluate with Fixed Bivariate Coefficient Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0018","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d12d2070b75bdfeafb5e079cace981cf7974e380761c7a9a220dfbc513d7b101","family_id":"qf-mod23-multivar-evaluate","seed":202610237018,"semantic_template":"qf-mod23-multivar-evaluate/{\"coefficients\":[5,0,0,1,1,-1],\"x\":0,\"y\":0}"},"fields":{"title":"Evaluate with Fixed Bivariate Coefficient Roles","stem":"Use the declared real polynomial R²→R with coefficients=[5, 0, 0, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=0, y=0. Give (coefficients,x,y,exact polynomial value).","answer":{"kind":"expression","value":"([5, 0, 0, 1, 1, -1], 0, 0, 5)","explanation":"Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0, 0, 1, 1, -1], 0, 0, 5)."},"skills":["Evaluate with Fixed Bivariate Coefficient Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0019","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c479373b40160149279c38bd10dfaf461226bb2eaa2a2b51e7499a6113e52f2","family_id":"qf-mod23-multivar-evaluate","seed":202610237019,"semantic_template":"qf-mod23-multivar-evaluate/{\"coefficients\":[6,1,1,1,-1,-1],\"x\":1,\"y\":1}"},"fields":{"title":"Evaluate with Fixed Bivariate Coefficient Roles","stem":"Use the declared real polynomial R²→R with coefficients=[6, 1, 1, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=1, y=1. Give (coefficients,x,y,exact polynomial value).","answer":{"kind":"expression","value":"([6, 1, 1, 1, -1, -1], 1, 1, 7)","explanation":"Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([6, 1, 1, 1, -1, -1], 1, 1, 7)."},"skills":["Evaluate with Fixed Bivariate Coefficient Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0020","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a0a24d6d75cc069a4d302e9b9c28753fdea79331c22c7a4abb97beb5b32c1a1","family_id":"qf-mod23-multivar-evaluate","seed":202610237020,"semantic_template":"qf-mod23-multivar-evaluate/{\"coefficients\":[7,2,2,1,0,-1],\"x\":2,\"y\":2}"},"fields":{"title":"Evaluate with Fixed Bivariate Coefficient Roles","stem":"Use the declared real polynomial R²→R with coefficients=[7, 2, 2, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=2, y=2. Give (coefficients,x,y,exact polynomial value).","answer":{"kind":"expression","value":"([7, 2, 2, 1, 0, -1], 2, 2, 15)","explanation":"Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([7, 2, 2, 1, 0, -1], 2, 2, 15)."},"skills":["Evaluate with Fixed Bivariate Coefficient Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0021","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52c91e6eaf8d95ab92b1ac39bdeb2a1587e6b18b605ef3ade41a8c60bc743da3","family_id":"qf-mod23-multivar-evaluate","seed":202610237021,"semantic_template":"qf-mod23-multivar-evaluate/{\"coefficients\":[8,-2,3,1,1,-1],\"x\":-2,\"y\":3}"},"fields":{"title":"Evaluate with Fixed Bivariate Coefficient Roles","stem":"Use the declared real polynomial R²→R with coefficients=[8, -2, 3, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=-2, y=3. Give (coefficients,x,y,exact polynomial value).","answer":{"kind":"expression","value":"([8, -2, 3, 1, 1, -1], -2, 3, 10)","explanation":"Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([8, -2, 3, 1, 1, -1], -2, 3, 10)."},"skills":["Evaluate with Fixed Bivariate Coefficient Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0022","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b5747044c6b00b1dd9da2237a1d9af8e8c5194a4c55791fcfe76019bc9f6e3d","family_id":"qf-mod23-multivar-evaluate","seed":202610237022,"semantic_template":"qf-mod23-multivar-evaluate/{\"coefficients\":[9,-1,-3,1,-1,-1],\"x\":-1,\"y\":-3}"},"fields":{"title":"Evaluate with Fixed Bivariate Coefficient Roles","stem":"Use the declared real polynomial R²→R with coefficients=[9, -1, -3, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=-1, y=-3. Give (coefficients,x,y,exact polynomial value).","answer":{"kind":"expression","value":"([9, -1, -3, 1, -1, -1], -1, -3, 8)","explanation":"Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([9, -1, -3, 1, -1, -1], -1, -3, 8)."},"skills":["Evaluate with Fixed Bivariate Coefficient Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0023","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e9bab6597b7eb137a8e3046bb717734e471f15ae129e05669ea637b0632515a","family_id":"qf-mod23-multivar-evaluate","seed":202610237023,"semantic_template":"qf-mod23-multivar-evaluate/{\"coefficients\":[10,0,-2,1,0,-1],\"x\":0,\"y\":-2}"},"fields":{"title":"Evaluate with Fixed Bivariate Coefficient Roles","stem":"Use the declared real polynomial R²→R with coefficients=[10, 0, -2, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=0, y=-2. Give (coefficients,x,y,exact polynomial value).","answer":{"kind":"expression","value":"([10, 0, -2, 1, 0, -1], 0, -2, 10)","explanation":"Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([10, 0, -2, 1, 0, -1], 0, -2, 10)."},"skills":["Evaluate with Fixed Bivariate Coefficient Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0024","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ae2296ba5bfc1dee5fefb2daeb3f7830b31da35b7f2be3574d3961652ad948c","family_id":"qf-mod23-multivar-evaluate","seed":202610237024,"semantic_template":"qf-mod23-multivar-evaluate/{\"coefficients\":[11,1,-1,1,1,-1],\"x\":1,\"y\":-1}"},"fields":{"title":"Evaluate with Fixed Bivariate Coefficient Roles","stem":"Use the declared real polynomial R²→R with coefficients=[11, 1, -1, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=1, y=-1. Give (coefficients,x,y,exact polynomial value).","answer":{"kind":"expression","value":"([11, 1, -1, 1, 1, -1], 1, -1, 12)","explanation":"Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([11, 1, -1, 1, 1, -1], 1, -1, 12)."},"skills":["Evaluate with Fixed Bivariate Coefficient Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0025","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a34af9c882cc50d91e87249ae00ce6936ddd24f73e7c809e29360d7d337d5646","family_id":"qf-mod23-multivar-evaluate","seed":202610237025,"semantic_template":"qf-mod23-multivar-evaluate/{\"coefficients\":[12,2,0,1,-1,-1],\"x\":2,\"y\":0}"},"fields":{"title":"Evaluate with Fixed Bivariate Coefficient Roles","stem":"Use the declared real polynomial R²→R with coefficients=[12, 2, 0, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=2, y=0. Give (coefficients,x,y,exact polynomial value).","answer":{"kind":"expression","value":"([12, 2, 0, 1, -1, -1], 2, 0, 20)","explanation":"Substitute each coordinate into its declared monomial; swapping x and y changes the semantic task. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([12, 2, 0, 1, -1, -1], 2, 0, 20)."},"skills":["Evaluate with Fixed Bivariate Coefficient Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0026","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da4ceef186f314cdda69db6a10e49e154da7f16cfe61d80af5538cccbca79fcf","family_id":"qf-mod23-multivar-rectangle-domain-audit","seed":202610237026,"semantic_template":"qf-mod23-multivar-rectangle-domain-audit/{\"bounds\":[-2,2,-1,3],\"point\":[-3,-2]}"},"fields":{"title":"Inclusive Membership in a Declared Closed Rectangle","stem":"Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-2, 2, -1, 3] and point [x,y]=[-3, -2]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.","answer":{"kind":"expression","value":"([-2, 2, -1, 3], [-3, -2], No)","explanation":"Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-2, 2, -1, 3], [-3, -2], No)."},"skills":["Inclusive Membership in a Declared Closed Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0027","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6cae6864a707afa7780c50e7d7fc170d5d03f936373e53eb8f02a59fd2f7fa26","family_id":"qf-mod23-multivar-rectangle-domain-audit","seed":202610237027,"semantic_template":"qf-mod23-multivar-rectangle-domain-audit/{\"bounds\":[-3,3,-2,4],\"point\":[-2,-1]}"},"fields":{"title":"Inclusive Membership in a Declared Closed Rectangle","stem":"Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-3, 3, -2, 4] and point [x,y]=[-2, -1]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.","answer":{"kind":"expression","value":"([-3, 3, -2, 4], [-2, -1], Yes)","explanation":"Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 3, -2, 4], [-2, -1], Yes)."},"skills":["Inclusive Membership in a Declared Closed Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0028","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a295143c4c09efd61927ddeda7598b1f16aaedbcb13c19ee26f4d559ff5c78bf","family_id":"qf-mod23-multivar-rectangle-domain-audit","seed":202610237028,"semantic_template":"qf-mod23-multivar-rectangle-domain-audit/{\"bounds\":[-4,4,-3,5],\"point\":[-1,0]}"},"fields":{"title":"Inclusive Membership in a Declared Closed Rectangle","stem":"Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-4, 4, -3, 5] and point [x,y]=[-1, 0]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.","answer":{"kind":"expression","value":"([-4, 4, -3, 5], [-1, 0], Yes)","explanation":"Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 4, -3, 5], [-1, 0], Yes)."},"skills":["Inclusive Membership in a Declared Closed Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0029","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45ea73a9f7900a2e24f17361d5a12d1932c7e8fef37978743744e5287d78b610","family_id":"qf-mod23-multivar-rectangle-domain-audit","seed":202610237029,"semantic_template":"qf-mod23-multivar-rectangle-domain-audit/{\"bounds\":[-5,5,-4,6],\"point\":[0,1]}"},"fields":{"title":"Inclusive Membership in a Declared Closed Rectangle","stem":"Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-5, 5, -4, 6] and point [x,y]=[0, 1]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.","answer":{"kind":"expression","value":"([-5, 5, -4, 6], [0, 1], Yes)","explanation":"Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 5, -4, 6], [0, 1], Yes)."},"skills":["Inclusive Membership in a Declared Closed Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0030","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d619876f1bb57d7ce7c72b63e4db17f15b9f51172ba1f402b94b286c7dcca9d","family_id":"qf-mod23-multivar-rectangle-domain-audit","seed":202610237030,"semantic_template":"qf-mod23-multivar-rectangle-domain-audit/{\"bounds\":[-6,6,-5,7],\"point\":[1,2]}"},"fields":{"title":"Inclusive Membership in a Declared Closed Rectangle","stem":"Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-6, 6, -5, 7] and point [x,y]=[1, 2]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.","answer":{"kind":"expression","value":"([-6, 6, -5, 7], [1, 2], Yes)","explanation":"Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-6, 6, -5, 7], [1, 2], Yes)."},"skills":["Inclusive Membership in a Declared Closed Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0031","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"09835a4bce8b8cfcb1e0b8b4b30657e33b65aef8effab929f55cacde2c25b566","family_id":"qf-mod23-multivar-rectangle-domain-audit","seed":202610237031,"semantic_template":"qf-mod23-multivar-rectangle-domain-audit/{\"bounds\":[-7,7,-6,8],\"point\":[2,-2]}"},"fields":{"title":"Inclusive Membership in a Declared Closed Rectangle","stem":"Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-7, 7, -6, 8] and point [x,y]=[2, -2]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.","answer":{"kind":"expression","value":"([-7, 7, -6, 8], [2, -2], Yes)","explanation":"Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-7, 7, -6, 8], [2, -2], Yes)."},"skills":["Inclusive Membership in a Declared Closed Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0032","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47168b6a554a99331430999cd00d6f96668bb7a1efce43c5045dff4a4303a15b","family_id":"qf-mod23-multivar-rectangle-domain-audit","seed":202610237032,"semantic_template":"qf-mod23-multivar-rectangle-domain-audit/{\"bounds\":[-8,8,-7,9],\"point\":[3,-1]}"},"fields":{"title":"Inclusive Membership in a Declared Closed Rectangle","stem":"Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-8, 8, -7, 9] and point [x,y]=[3, -1]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.","answer":{"kind":"expression","value":"([-8, 8, -7, 9], [3, -1], Yes)","explanation":"Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 8, -7, 9], [3, -1], Yes)."},"skills":["Inclusive Membership in a Declared Closed Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0033","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e35dcd501cc9cecc6e13e7addff6fbfab32d925f691ef1ea21a34d35015a9bc","family_id":"qf-mod23-multivar-rectangle-domain-audit","seed":202610237033,"semantic_template":"qf-mod23-multivar-rectangle-domain-audit/{\"bounds\":[-9,9,-8,10],\"point\":[-3,0]}"},"fields":{"title":"Inclusive Membership in a Declared Closed Rectangle","stem":"Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-9, 9, -8, 10] and point [x,y]=[-3, 0]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.","answer":{"kind":"expression","value":"([-9, 9, -8, 10], [-3, 0], Yes)","explanation":"Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 9, -8, 10], [-3, 0], Yes)."},"skills":["Inclusive Membership in a Declared Closed Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0034","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58938dbd212642fc329ca69111b1655ba566e291214bed545f7de1bc63f358cd","family_id":"qf-mod23-multivar-rectangle-domain-audit","seed":202610237034,"semantic_template":"qf-mod23-multivar-rectangle-domain-audit/{\"bounds\":[-10,10,-9,11],\"point\":[-2,1]}"},"fields":{"title":"Inclusive Membership in a Declared Closed Rectangle","stem":"Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-10, 10, -9, 11] and point [x,y]=[-2, 1]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.","answer":{"kind":"expression","value":"([-10, 10, -9, 11], [-2, 1], Yes)","explanation":"Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 10, -9, 11], [-2, 1], Yes)."},"skills":["Inclusive Membership in a Declared Closed Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0035","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64f41cdc9c9df46ae7555cc27c698a01ded515573deb7708efa7af6c3bd318f4","family_id":"qf-mod23-multivar-rectangle-domain-audit","seed":202610237035,"semantic_template":"qf-mod23-multivar-rectangle-domain-audit/{\"bounds\":[-11,11,-10,12],\"point\":[-1,2]}"},"fields":{"title":"Inclusive Membership in a Declared Closed Rectangle","stem":"Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-11, 11, -10, 12] and point [x,y]=[-1, 2]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.","answer":{"kind":"expression","value":"([-11, 11, -10, 12], [-1, 2], Yes)","explanation":"Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-11, 11, -10, 12], [-1, 2], Yes)."},"skills":["Inclusive Membership in a Declared Closed Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0036","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5336fc5cd59d83411a297e3fd8ac87f988114cd0e625d506b2be7178706c496d","family_id":"qf-mod23-multivar-rectangle-domain-audit","seed":202610237036,"semantic_template":"qf-mod23-multivar-rectangle-domain-audit/{\"bounds\":[-12,12,-11,13],\"point\":[0,-2]}"},"fields":{"title":"Inclusive Membership in a Declared Closed Rectangle","stem":"Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-12, 12, -11, 13] and point [x,y]=[0, -2]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.","answer":{"kind":"expression","value":"([-12, 12, -11, 13], [0, -2], Yes)","explanation":"Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-12, 12, -11, 13], [0, -2], Yes)."},"skills":["Inclusive Membership in a Declared Closed Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0037","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb768522ee1536cdd8479fc06dcc8e6924b1ab9b61081f6b5fe068de9cfe3db3","family_id":"qf-mod23-multivar-rectangle-domain-audit","seed":202610237037,"semantic_template":"qf-mod23-multivar-rectangle-domain-audit/{\"bounds\":[-13,13,-12,14],\"point\":[1,-1]}"},"fields":{"title":"Inclusive Membership in a Declared Closed Rectangle","stem":"Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-13, 13, -12, 14] and point [x,y]=[1, -1]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.","answer":{"kind":"expression","value":"([-13, 13, -12, 14], [1, -1], Yes)","explanation":"Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-13, 13, -12, 14], [1, -1], Yes)."},"skills":["Inclusive Membership in a Declared Closed Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0038","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"56d26195216ec626aed9cd9cbc8fdda5fcae0392cc0856b3264a1f405cc40ac6","family_id":"qf-mod23-multivar-rectangle-domain-audit","seed":202610237038,"semantic_template":"qf-mod23-multivar-rectangle-domain-audit/{\"bounds\":[-14,14,-13,15],\"point\":[2,0]}"},"fields":{"title":"Inclusive Membership in a Declared Closed Rectangle","stem":"Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-14, 14, -13, 15] and point [x,y]=[2, 0]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.","answer":{"kind":"expression","value":"([-14, 14, -13, 15], [2, 0], Yes)","explanation":"Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-14, 14, -13, 15], [2, 0], Yes)."},"skills":["Inclusive Membership in a Declared Closed Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0039","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5043ba650b7797255b33135901f1e2a9e9e4f9deb05841055998cbffd91c2a8","family_id":"qf-mod23-multivar-rectangle-domain-audit","seed":202610237039,"semantic_template":"qf-mod23-multivar-rectangle-domain-audit/{\"bounds\":[-15,15,-14,16],\"point\":[3,1]}"},"fields":{"title":"Inclusive Membership in a Declared Closed Rectangle","stem":"Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-15, 15, -14, 16] and point [x,y]=[3, 1]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.","answer":{"kind":"expression","value":"([-15, 15, -14, 16], [3, 1], Yes)","explanation":"Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-15, 15, -14, 16], [3, 1], Yes)."},"skills":["Inclusive Membership in a Declared Closed Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0040","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d02394611cb268fef8b41b97d3703533d046f0abfedccca03244565b57916898","family_id":"qf-mod23-multivar-rectangle-domain-audit","seed":202610237040,"semantic_template":"qf-mod23-multivar-rectangle-domain-audit/{\"bounds\":[-16,16,-15,17],\"point\":[-3,2]}"},"fields":{"title":"Inclusive Membership in a Declared Closed Rectangle","stem":"Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-16, 16, -15, 17] and point [x,y]=[-3, 2]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.","answer":{"kind":"expression","value":"([-16, 16, -15, 17], [-3, 2], Yes)","explanation":"Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-16, 16, -15, 17], [-3, 2], Yes)."},"skills":["Inclusive Membership in a Declared Closed Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0041","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b462660157526e803f9ab1004a9d97708ee37e0e71aa4edbb94028899a845a5","family_id":"qf-mod23-multivar-rectangle-domain-audit","seed":202610237041,"semantic_template":"qf-mod23-multivar-rectangle-domain-audit/{\"bounds\":[-17,17,-16,18],\"point\":[-2,-2]}"},"fields":{"title":"Inclusive Membership in a Declared Closed Rectangle","stem":"Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-17, 17, -16, 18] and point [x,y]=[-2, -2]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.","answer":{"kind":"expression","value":"([-17, 17, -16, 18], [-2, -2], Yes)","explanation":"Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-17, 17, -16, 18], [-2, -2], Yes)."},"skills":["Inclusive Membership in a Declared Closed Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0042","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16d458e5e63c8c9899c9de2f64c83a70dfc0bc4766e8401f71382649e7f3d20b","family_id":"qf-mod23-multivar-rectangle-domain-audit","seed":202610237042,"semantic_template":"qf-mod23-multivar-rectangle-domain-audit/{\"bounds\":[-18,18,-17,19],\"point\":[-1,-1]}"},"fields":{"title":"Inclusive Membership in a Declared Closed Rectangle","stem":"Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-18, 18, -17, 19] and point [x,y]=[-1, -1]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.","answer":{"kind":"expression","value":"([-18, 18, -17, 19], [-1, -1], Yes)","explanation":"Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-18, 18, -17, 19], [-1, -1], Yes)."},"skills":["Inclusive Membership in a Declared Closed Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0043","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16ca590af0e89d65c46f5e3ee0e0c065a3ba2d8dca49b0c2dd3c42b07cb8ddb4","family_id":"qf-mod23-multivar-rectangle-domain-audit","seed":202610237043,"semantic_template":"qf-mod23-multivar-rectangle-domain-audit/{\"bounds\":[-19,19,-18,20],\"point\":[0,0]}"},"fields":{"title":"Inclusive Membership in a Declared Closed Rectangle","stem":"Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-19, 19, -18, 20] and point [x,y]=[0, 0]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.","answer":{"kind":"expression","value":"([-19, 19, -18, 20], [0, 0], Yes)","explanation":"Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-19, 19, -18, 20], [0, 0], Yes)."},"skills":["Inclusive Membership in a Declared Closed Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0044","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75492cf1f76fbe8b188bc42af11e146048a89f943bb67ab2d3b587780d2a98c2","family_id":"qf-mod23-multivar-rectangle-domain-audit","seed":202610237044,"semantic_template":"qf-mod23-multivar-rectangle-domain-audit/{\"bounds\":[-20,20,-19,21],\"point\":[1,1]}"},"fields":{"title":"Inclusive Membership in a Declared Closed Rectangle","stem":"Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-20, 20, -19, 21] and point [x,y]=[1, 1]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.","answer":{"kind":"expression","value":"([-20, 20, -19, 21], [1, 1], Yes)","explanation":"Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-20, 20, -19, 21], [1, 1], Yes)."},"skills":["Inclusive Membership in a Declared Closed Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0045","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a3a2e776f14b4f9b2b19e27d1a1f7ea4afe91dfc653c41291e163036e2f89f2","family_id":"qf-mod23-multivar-rectangle-domain-audit","seed":202610237045,"semantic_template":"qf-mod23-multivar-rectangle-domain-audit/{\"bounds\":[-21,21,-20,22],\"point\":[2,2]}"},"fields":{"title":"Inclusive Membership in a Declared Closed Rectangle","stem":"Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-21, 21, -20, 22] and point [x,y]=[2, 2]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.","answer":{"kind":"expression","value":"([-21, 21, -20, 22], [2, 2], Yes)","explanation":"Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-21, 21, -20, 22], [2, 2], Yes)."},"skills":["Inclusive Membership in a Declared Closed Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0046","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a60a3b1a410b4a2530d5f27809275a20fbbfb3fef7fee29a5fade253d72c8ef4","family_id":"qf-mod23-multivar-rectangle-domain-audit","seed":202610237046,"semantic_template":"qf-mod23-multivar-rectangle-domain-audit/{\"bounds\":[-22,22,-21,23],\"point\":[3,-2]}"},"fields":{"title":"Inclusive Membership in a Declared Closed Rectangle","stem":"Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-22, 22, -21, 23] and point [x,y]=[3, -2]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.","answer":{"kind":"expression","value":"([-22, 22, -21, 23], [3, -2], Yes)","explanation":"Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-22, 22, -21, 23], [3, -2], Yes)."},"skills":["Inclusive Membership in a Declared Closed Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0047","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3ed4bf78b4ecffefc466b377c9faf36ab52c0d7eceaf62745a830c24b8c5b56","family_id":"qf-mod23-multivar-rectangle-domain-audit","seed":202610237047,"semantic_template":"qf-mod23-multivar-rectangle-domain-audit/{\"bounds\":[-23,23,-22,24],\"point\":[-3,-1]}"},"fields":{"title":"Inclusive Membership in a Declared Closed Rectangle","stem":"Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-23, 23, -22, 24] and point [x,y]=[-3, -1]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.","answer":{"kind":"expression","value":"([-23, 23, -22, 24], [-3, -1], Yes)","explanation":"Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-23, 23, -22, 24], [-3, -1], Yes)."},"skills":["Inclusive Membership in a Declared Closed Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0048","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a272ec35b2edab9e459d6cb76504cd27c5db59e9dd30d5e21511cb28a59eb47b","family_id":"qf-mod23-multivar-rectangle-domain-audit","seed":202610237048,"semantic_template":"qf-mod23-multivar-rectangle-domain-audit/{\"bounds\":[-24,24,-23,25],\"point\":[-2,0]}"},"fields":{"title":"Inclusive Membership in a Declared Closed Rectangle","stem":"Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-24, 24, -23, 25] and point [x,y]=[-2, 0]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.","answer":{"kind":"expression","value":"([-24, 24, -23, 25], [-2, 0], Yes)","explanation":"Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-24, 24, -23, 25], [-2, 0], Yes)."},"skills":["Inclusive Membership in a Declared Closed Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0049","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0bd9595031217e45f605ec9ca0972998802f0adfe4f3e86bd7913b647ed41f5","family_id":"qf-mod23-multivar-rectangle-domain-audit","seed":202610237049,"semantic_template":"qf-mod23-multivar-rectangle-domain-audit/{\"bounds\":[-25,25,-24,26],\"point\":[-1,1]}"},"fields":{"title":"Inclusive Membership in a Declared Closed Rectangle","stem":"Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-25, 25, -24, 26] and point [x,y]=[-1, 1]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.","answer":{"kind":"expression","value":"([-25, 25, -24, 26], [-1, 1], Yes)","explanation":"Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-25, 25, -24, 26], [-1, 1], Yes)."},"skills":["Inclusive Membership in a Declared Closed Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0050","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f9291e4958969a33ad65f5ab3c3cf32859732236e74ba5df53bafcdbef82569","family_id":"qf-mod23-multivar-rectangle-domain-audit","seed":202610237050,"semantic_template":"qf-mod23-multivar-rectangle-domain-audit/{\"bounds\":[-26,26,-25,27],\"point\":[0,2]}"},"fields":{"title":"Inclusive Membership in a Declared Closed Rectangle","stem":"Use the explicitly declared closed rectangle with bounds [x_min,x_max,y_min,y_max]=[-26, 26, -25, 27] and point [x,y]=[0, 2]. Require ordered bounds, allowing equal endpoints. Give (bounds,point,in this declared domain?). A polynomial’s possible global definition does not override a task’s restricted domain.","answer":{"kind":"expression","value":"([-26, 26, -25, 27], [0, 2], Yes)","explanation":"Check both coordinate intervals with inclusive inequalities; this is only a domain-membership audit, not continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-26, 26, -25, 27], [0, 2], Yes)."},"skills":["Inclusive Membership in a Declared Closed Rectangle"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0051","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e1698afb7a4dd24a4fe9211778d5b5a58cd2562ab1a7e59068c4cdb629e828c","family_id":"qf-mod23-multivar-x-slice-table","seed":202610237051,"semantic_template":"qf-mod23-multivar-x-slice-table/{\"coefficients\":[-12,-2,-3,1,-1,-1],\"xs\":[-3,-1,0],\"y\":-2}"},"fields":{"title":"A Fixed-Second-Coordinate Slice on Listed Inputs Only","stem":"Use the declared real polynomial R²→R with coefficients=[-12, -2, -3, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=-2, x input list=[-1, 0, -3] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.","answer":{"kind":"expression","value":"([-12, -2, -3, 1, -1, -1], -2, [-3, -1, 0], [[-3, -1], [-1, -9], [0, -10]])","explanation":"Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-12, -2, -3, 1, -1, -1], -2, [-3, -1, 0], [[-3, -1], [-1, -9], [0, -10]])."},"skills":["A Fixed-Second-Coordinate Slice on Listed Inputs Only"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0052","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"776efafc6d1b7789f683afaf126d4aad76d10c5b8a4ca1dc5f4edeace0a9f6de","family_id":"qf-mod23-multivar-x-slice-table","seed":202610237052,"semantic_template":"qf-mod23-multivar-x-slice-table/{\"coefficients\":[-11,-1,-2,1,0,-1],\"xs\":[-2,-1,0],\"y\":-1}"},"fields":{"title":"A Fixed-Second-Coordinate Slice on Listed Inputs Only","stem":"Use the declared real polynomial R²→R with coefficients=[-11, -1, -2, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=-1, x input list=[-1, 0, -2] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.","answer":{"kind":"expression","value":"([-11, -1, -2, 1, 0, -1], -1, [-2, -1, 0], [[-2, -4], [-1, -8], [0, -10]])","explanation":"Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-11, -1, -2, 1, 0, -1], -1, [-2, -1, 0], [[-2, -4], [-1, -8], [0, -10]])."},"skills":["A Fixed-Second-Coordinate Slice on Listed Inputs Only"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0053","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ff6e2ae47ffbb20495d6e4097b80f84fd925fc355bbe07310bc88412aad8378","family_id":"qf-mod23-multivar-x-slice-table","seed":202610237053,"semantic_template":"qf-mod23-multivar-x-slice-table/{\"coefficients\":[-10,0,-1,1,1,-1],\"xs\":[-1,0],\"y\":0}"},"fields":{"title":"A Fixed-Second-Coordinate Slice on Listed Inputs Only","stem":"Use the declared real polynomial R²→R with coefficients=[-10, 0, -1, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=0, x input list=[-1, 0, -1] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.","answer":{"kind":"expression","value":"([-10, 0, -1, 1, 1, -1], 0, [-1, 0], [[-1, -9], [0, -10]])","explanation":"Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 0, -1, 1, 1, -1], 0, [-1, 0], [[-1, -9], [0, -10]])."},"skills":["A Fixed-Second-Coordinate Slice on Listed Inputs Only"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0054","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a406b44daf9ad46d8caa75318dc421cf8fb5eebf2b7e929a6f7e370493b26dc2","family_id":"qf-mod23-multivar-x-slice-table","seed":202610237054,"semantic_template":"qf-mod23-multivar-x-slice-table/{\"coefficients\":[-9,1,0,1,-1,-1],\"xs\":[-1,0],\"y\":1}"},"fields":{"title":"A Fixed-Second-Coordinate Slice on Listed Inputs Only","stem":"Use the declared real polynomial R²→R with coefficients=[-9, 1, 0, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=1, x input list=[-1, 0, 0] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.","answer":{"kind":"expression","value":"([-9, 1, 0, 1, -1, -1], 1, [-1, 0], [[-1, -9], [0, -10]])","explanation":"Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 1, 0, 1, -1, -1], 1, [-1, 0], [[-1, -9], [0, -10]])."},"skills":["A Fixed-Second-Coordinate Slice on Listed Inputs Only"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0055","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97e51d697ab3b7a6776fe009e41fefdab61149472e2ba59c63c2e8471da7ddf4","family_id":"qf-mod23-multivar-x-slice-table","seed":202610237055,"semantic_template":"qf-mod23-multivar-x-slice-table/{\"coefficients\":[-8,2,1,1,0,-1],\"xs\":[-1,0,1],\"y\":2}"},"fields":{"title":"A Fixed-Second-Coordinate Slice on Listed Inputs Only","stem":"Use the declared real polynomial R²→R with coefficients=[-8, 2, 1, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=2, x input list=[-1, 0, 1] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.","answer":{"kind":"expression","value":"([-8, 2, 1, 1, 0, -1], 2, [-1, 0, 1], [[-1, -11], [0, -10], [1, -7]])","explanation":"Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 2, 1, 1, 0, -1], 2, [-1, 0, 1], [[-1, -11], [0, -10], [1, -7]])."},"skills":["A Fixed-Second-Coordinate Slice on Listed Inputs Only"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0056","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a83056a8caee9774dcf0dd93b34514e0c4ab6ee3ef676b4c5e324cb8c3e7d9b","family_id":"qf-mod23-multivar-x-slice-table","seed":202610237056,"semantic_template":"qf-mod23-multivar-x-slice-table/{\"coefficients\":[-7,-2,2,1,1,-1],\"xs\":[-1,0,2],\"y\":-2}"},"fields":{"title":"A Fixed-Second-Coordinate Slice on Listed Inputs Only","stem":"Use the declared real polynomial R²→R with coefficients=[-7, -2, 2, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=-2, x input list=[-1, 0, 2] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.","answer":{"kind":"expression","value":"([-7, -2, 2, 1, 1, -1], -2, [-1, 0, 2], [[-1, -10], [0, -15], [2, -19]])","explanation":"Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-7, -2, 2, 1, 1, -1], -2, [-1, 0, 2], [[-1, -10], [0, -15], [2, -19]])."},"skills":["A Fixed-Second-Coordinate Slice on Listed Inputs Only"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0057","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f07c234272ea93073c3cd2a555771472644ddab72115b1ffff4e72bfa540b8a","family_id":"qf-mod23-multivar-x-slice-table","seed":202610237057,"semantic_template":"qf-mod23-multivar-x-slice-table/{\"coefficients\":[-6,-1,3,1,-1,-1],\"xs\":[-1,0,3],\"y\":-1}"},"fields":{"title":"A Fixed-Second-Coordinate Slice on Listed Inputs Only","stem":"Use the declared real polynomial R²→R with coefficients=[-6, -1, 3, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=-1, x input list=[-1, 0, 3] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.","answer":{"kind":"expression","value":"([-6, -1, 3, 1, -1, -1], -1, [-1, 0, 3], [[-1, -9], [0, -10], [3, -1]])","explanation":"Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-6, -1, 3, 1, -1, -1], -1, [-1, 0, 3], [[-1, -9], [0, -10], [3, -1]])."},"skills":["A Fixed-Second-Coordinate Slice on Listed Inputs Only"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0058","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae3f9c6016601b665a4259674057b21fa5919ba5f115d36c558aaef198180b2c","family_id":"qf-mod23-multivar-x-slice-table","seed":202610237058,"semantic_template":"qf-mod23-multivar-x-slice-table/{\"coefficients\":[-5,0,-3,1,0,-1],\"xs\":[-3,-1,0],\"y\":0}"},"fields":{"title":"A Fixed-Second-Coordinate Slice on Listed Inputs Only","stem":"Use the declared real polynomial R²→R with coefficients=[-5, 0, -3, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=0, x input list=[-1, 0, -3] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.","answer":{"kind":"expression","value":"([-5, 0, -3, 1, 0, -1], 0, [-3, -1, 0], [[-3, 4], [-1, -4], [0, -5]])","explanation":"Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 0, -3, 1, 0, -1], 0, [-3, -1, 0], [[-3, 4], [-1, -4], [0, -5]])."},"skills":["A Fixed-Second-Coordinate Slice on Listed Inputs Only"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0059","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a1f160bdfe97bbea558e61a290144a862c27f4b1b7ed4441479fcb6b22b9274","family_id":"qf-mod23-multivar-x-slice-table","seed":202610237059,"semantic_template":"qf-mod23-multivar-x-slice-table/{\"coefficients\":[-4,1,-2,1,1,-1],\"xs\":[-2,-1,0],\"y\":1}"},"fields":{"title":"A Fixed-Second-Coordinate Slice on Listed Inputs Only","stem":"Use the declared real polynomial R²→R with coefficients=[-4, 1, -2, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=1, x input list=[-1, 0, -2] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.","answer":{"kind":"expression","value":"([-4, 1, -2, 1, 1, -1], 1, [-2, -1, 0], [[-2, -7], [-1, -8], [0, -7]])","explanation":"Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 1, -2, 1, 1, -1], 1, [-2, -1, 0], [[-2, -7], [-1, -8], [0, -7]])."},"skills":["A Fixed-Second-Coordinate Slice on Listed Inputs Only"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0060","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6f14060c701cae1e0f1e18f001ca1b803d631ae99c4dd06af9afe1294eddf78","family_id":"qf-mod23-multivar-x-slice-table","seed":202610237060,"semantic_template":"qf-mod23-multivar-x-slice-table/{\"coefficients\":[-3,2,-1,1,-1,-1],\"xs\":[-1,0],\"y\":2}"},"fields":{"title":"A Fixed-Second-Coordinate Slice on Listed Inputs Only","stem":"Use the declared real polynomial R²→R with coefficients=[-3, 2, -1, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=2, x input list=[-1, 0, -1] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.","answer":{"kind":"expression","value":"([-3, 2, -1, 1, -1, -1], 2, [-1, 0], [[-1, -8], [0, -9]])","explanation":"Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 2, -1, 1, -1, -1], 2, [-1, 0], [[-1, -8], [0, -9]])."},"skills":["A Fixed-Second-Coordinate Slice on Listed Inputs Only"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0061","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17b2f5ee5e7839af3088c0e53f76aa86880d3be8f0f5e77a915be011f04bfec8","family_id":"qf-mod23-multivar-x-slice-table","seed":202610237061,"semantic_template":"qf-mod23-multivar-x-slice-table/{\"coefficients\":[-2,-2,0,1,0,-1],\"xs\":[-1,0],\"y\":-2}"},"fields":{"title":"A Fixed-Second-Coordinate Slice on Listed Inputs Only","stem":"Use the declared real polynomial R²→R with coefficients=[-2, -2, 0, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=-2, x input list=[-1, 0, 0] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.","answer":{"kind":"expression","value":"([-2, -2, 0, 1, 0, -1], -2, [-1, 0], [[-1, -3], [0, -6]])","explanation":"Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-2, -2, 0, 1, 0, -1], -2, [-1, 0], [[-1, -3], [0, -6]])."},"skills":["A Fixed-Second-Coordinate Slice on Listed Inputs Only"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0062","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66ed17e074d5a776de540275ce34640167e50821d4e28151378b9e070698c6cc","family_id":"qf-mod23-multivar-x-slice-table","seed":202610237062,"semantic_template":"qf-mod23-multivar-x-slice-table/{\"coefficients\":[-1,-1,1,1,1,-1],\"xs\":[-1,0,1],\"y\":-1}"},"fields":{"title":"A Fixed-Second-Coordinate Slice on Listed Inputs Only","stem":"Use the declared real polynomial R²→R with coefficients=[-1, -1, 1, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=-1, x input list=[-1, 0, 1] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.","answer":{"kind":"expression","value":"([-1, -1, 1, 1, 1, -1], -1, [-1, 0, 1], [[-1, 0], [0, -3], [1, -4]])","explanation":"Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -1, 1, 1, 1, -1], -1, [-1, 0, 1], [[-1, 0], [0, -3], [1, -4]])."},"skills":["A Fixed-Second-Coordinate Slice on Listed Inputs Only"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0063","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2bdd4633776a164e268879589d1a07422ccb95aa0713d2d9404460b13a664a43","family_id":"qf-mod23-multivar-x-slice-table","seed":202610237063,"semantic_template":"qf-mod23-multivar-x-slice-table/{\"coefficients\":[0,0,2,1,-1,-1],\"xs\":[-1,0,2],\"y\":0}"},"fields":{"title":"A Fixed-Second-Coordinate Slice on Listed Inputs Only","stem":"Use the declared real polynomial R²→R with coefficients=[0, 0, 2, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=0, x input list=[-1, 0, 2] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.","answer":{"kind":"expression","value":"([0, 0, 2, 1, -1, -1], 0, [-1, 0, 2], [[-1, 1], [0, 0], [2, 4]])","explanation":"Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 2, 1, -1, -1], 0, [-1, 0, 2], [[-1, 1], [0, 0], [2, 4]])."},"skills":["A Fixed-Second-Coordinate Slice on Listed Inputs Only"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0064","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bedccce037f7d0bcddb967198b17d90410834c57678328f5b29f8a6f9ab8aa3f","family_id":"qf-mod23-multivar-x-slice-table","seed":202610237064,"semantic_template":"qf-mod23-multivar-x-slice-table/{\"coefficients\":[1,1,3,1,0,-1],\"xs\":[-1,0,3],\"y\":1}"},"fields":{"title":"A Fixed-Second-Coordinate Slice on Listed Inputs Only","stem":"Use the declared real polynomial R²→R with coefficients=[1, 1, 3, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=1, x input list=[-1, 0, 3] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.","answer":{"kind":"expression","value":"([1, 1, 3, 1, 0, -1], 1, [-1, 0, 3], [[-1, 3], [0, 3], [3, 15]])","explanation":"Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 3, 1, 0, -1], 1, [-1, 0, 3], [[-1, 3], [0, 3], [3, 15]])."},"skills":["A Fixed-Second-Coordinate Slice on Listed Inputs Only"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0065","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"903284b91c0d8557bc28e8ea4a294f2ee0f58875cbec4bfb06da4dc228282256","family_id":"qf-mod23-multivar-x-slice-table","seed":202610237065,"semantic_template":"qf-mod23-multivar-x-slice-table/{\"coefficients\":[2,2,-3,1,1,-1],\"xs\":[-3,-1,0],\"y\":2}"},"fields":{"title":"A Fixed-Second-Coordinate Slice on Listed Inputs Only","stem":"Use the declared real polynomial R²→R with coefficients=[2, 2, -3, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=2, x input list=[-1, 0, -3] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.","answer":{"kind":"expression","value":"([2, 2, -3, 1, 1, -1], 2, [-3, -1, 0], [[-3, -11], [-1, -11], [0, -8]])","explanation":"Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, -3, 1, 1, -1], 2, [-3, -1, 0], [[-3, -11], [-1, -11], [0, -8]])."},"skills":["A Fixed-Second-Coordinate Slice on Listed Inputs Only"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0066","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3946435ee67df9aaa342183713793aded9a29b1f2a7c6194494d04f958d045d","family_id":"qf-mod23-multivar-x-slice-table","seed":202610237066,"semantic_template":"qf-mod23-multivar-x-slice-table/{\"coefficients\":[3,-2,-2,1,-1,-1],\"xs\":[-2,-1,0],\"y\":-2}"},"fields":{"title":"A Fixed-Second-Coordinate Slice on Listed Inputs Only","stem":"Use the declared real polynomial R²→R with coefficients=[3, -2, -2, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=-2, x input list=[-1, 0, -2] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.","answer":{"kind":"expression","value":"([3, -2, -2, 1, -1, -1], -2, [-2, -1, 0], [[-2, 7], [-1, 4], [0, 3]])","explanation":"Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, -2, -2, 1, -1, -1], -2, [-2, -1, 0], [[-2, 7], [-1, 4], [0, 3]])."},"skills":["A Fixed-Second-Coordinate Slice on Listed Inputs Only"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0067","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d18105ca34f84dc84df4777f6df304db097547ec462b715c350b6b36ae96ac76","family_id":"qf-mod23-multivar-x-slice-table","seed":202610237067,"semantic_template":"qf-mod23-multivar-x-slice-table/{\"coefficients\":[4,-1,-1,1,0,-1],\"xs\":[-1,0],\"y\":-1}"},"fields":{"title":"A Fixed-Second-Coordinate Slice on Listed Inputs Only","stem":"Use the declared real polynomial R²→R with coefficients=[4, -1, -1, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=-1, x input list=[-1, 0, -1] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.","answer":{"kind":"expression","value":"([4, -1, -1, 1, 0, -1], -1, [-1, 0], [[-1, 6], [0, 4]])","explanation":"Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1, -1, 1, 0, -1], -1, [-1, 0], [[-1, 6], [0, 4]])."},"skills":["A Fixed-Second-Coordinate Slice on Listed Inputs Only"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0068","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ccfc1f8749d5968b3b23aff35a0668adc6d801247da2550273d0026b5223680b","family_id":"qf-mod23-multivar-x-slice-table","seed":202610237068,"semantic_template":"qf-mod23-multivar-x-slice-table/{\"coefficients\":[5,0,0,1,1,-1],\"xs\":[-1,0],\"y\":0}"},"fields":{"title":"A Fixed-Second-Coordinate Slice on Listed Inputs Only","stem":"Use the declared real polynomial R²→R with coefficients=[5, 0, 0, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=0, x input list=[-1, 0, 0] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.","answer":{"kind":"expression","value":"([5, 0, 0, 1, 1, -1], 0, [-1, 0], [[-1, 6], [0, 5]])","explanation":"Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0, 0, 1, 1, -1], 0, [-1, 0], [[-1, 6], [0, 5]])."},"skills":["A Fixed-Second-Coordinate Slice on Listed Inputs Only"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0069","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ce77a551f551c68f5abcebbd0eff29f504bf1f709f6099d2bff503c1187dd3a","family_id":"qf-mod23-multivar-x-slice-table","seed":202610237069,"semantic_template":"qf-mod23-multivar-x-slice-table/{\"coefficients\":[6,1,1,1,-1,-1],\"xs\":[-1,0,1],\"y\":1}"},"fields":{"title":"A Fixed-Second-Coordinate Slice on Listed Inputs Only","stem":"Use the declared real polynomial R²→R with coefficients=[6, 1, 1, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=1, x input list=[-1, 0, 1] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.","answer":{"kind":"expression","value":"([6, 1, 1, 1, -1, -1], 1, [-1, 0, 1], [[-1, 7], [0, 6], [1, 7]])","explanation":"Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([6, 1, 1, 1, -1, -1], 1, [-1, 0, 1], [[-1, 7], [0, 6], [1, 7]])."},"skills":["A Fixed-Second-Coordinate Slice on Listed Inputs Only"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0070","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1248b751922ae7dc956ce3b7d3199080bdacf7149479b96ebb2e74a16e100115","family_id":"qf-mod23-multivar-x-slice-table","seed":202610237070,"semantic_template":"qf-mod23-multivar-x-slice-table/{\"coefficients\":[7,2,2,1,0,-1],\"xs\":[-1,0,2],\"y\":2}"},"fields":{"title":"A Fixed-Second-Coordinate Slice on Listed Inputs Only","stem":"Use the declared real polynomial R²→R with coefficients=[7, 2, 2, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=2, x input list=[-1, 0, 2] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.","answer":{"kind":"expression","value":"([7, 2, 2, 1, 0, -1], 2, [-1, 0, 2], [[-1, 6], [0, 7], [2, 15]])","explanation":"Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([7, 2, 2, 1, 0, -1], 2, [-1, 0, 2], [[-1, 6], [0, 7], [2, 15]])."},"skills":["A Fixed-Second-Coordinate Slice on Listed Inputs Only"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0071","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"986df5c6a6f5e4b0c014ebe5e58837db907fc489f1b8b6f887ae7bfcaf0c4ad5","family_id":"qf-mod23-multivar-x-slice-table","seed":202610237071,"semantic_template":"qf-mod23-multivar-x-slice-table/{\"coefficients\":[8,-2,3,1,1,-1],\"xs\":[-1,0,3],\"y\":-2}"},"fields":{"title":"A Fixed-Second-Coordinate Slice on Listed Inputs Only","stem":"Use the declared real polynomial R²→R with coefficients=[8, -2, 3, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=-2, x input list=[-1, 0, 3] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.","answer":{"kind":"expression","value":"([8, -2, 3, 1, 1, -1], -2, [-1, 0, 3], [[-1, 3], [0, -2], [3, -5]])","explanation":"Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([8, -2, 3, 1, 1, -1], -2, [-1, 0, 3], [[-1, 3], [0, -2], [3, -5]])."},"skills":["A Fixed-Second-Coordinate Slice on Listed Inputs Only"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0072","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2258309a3c900ddb21f03bf55ebd4431dffeffbfc404dc18c7e659950fc7ad42","family_id":"qf-mod23-multivar-x-slice-table","seed":202610237072,"semantic_template":"qf-mod23-multivar-x-slice-table/{\"coefficients\":[9,-1,-3,1,-1,-1],\"xs\":[-3,-1,0],\"y\":-1}"},"fields":{"title":"A Fixed-Second-Coordinate Slice on Listed Inputs Only","stem":"Use the declared real polynomial R²→R with coefficients=[9, -1, -3, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=-1, x input list=[-1, 0, -3] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.","answer":{"kind":"expression","value":"([9, -1, -3, 1, -1, -1], -1, [-3, -1, 0], [[-3, 20], [-1, 12], [0, 11]])","explanation":"Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([9, -1, -3, 1, -1, -1], -1, [-3, -1, 0], [[-3, 20], [-1, 12], [0, 11]])."},"skills":["A Fixed-Second-Coordinate Slice on Listed Inputs Only"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0073","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e510807f4ff5bb52a28633465a530c0baa4d2df1d3ee17cc434be323f9acf9ff","family_id":"qf-mod23-multivar-x-slice-table","seed":202610237073,"semantic_template":"qf-mod23-multivar-x-slice-table/{\"coefficients\":[10,0,-2,1,0,-1],\"xs\":[-2,-1,0],\"y\":0}"},"fields":{"title":"A Fixed-Second-Coordinate Slice on Listed Inputs Only","stem":"Use the declared real polynomial R²→R with coefficients=[10, 0, -2, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=0, x input list=[-1, 0, -2] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.","answer":{"kind":"expression","value":"([10, 0, -2, 1, 0, -1], 0, [-2, -1, 0], [[-2, 14], [-1, 11], [0, 10]])","explanation":"Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([10, 0, -2, 1, 0, -1], 0, [-2, -1, 0], [[-2, 14], [-1, 11], [0, 10]])."},"skills":["A Fixed-Second-Coordinate Slice on Listed Inputs Only"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0074","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"971d7a0ee729ee81a5e896cffe5fa3bef249500aec0bdded3efd19aa893ce696","family_id":"qf-mod23-multivar-x-slice-table","seed":202610237074,"semantic_template":"qf-mod23-multivar-x-slice-table/{\"coefficients\":[11,1,-1,1,1,-1],\"xs\":[-1,0],\"y\":1}"},"fields":{"title":"A Fixed-Second-Coordinate Slice on Listed Inputs Only","stem":"Use the declared real polynomial R²→R with coefficients=[11, 1, -1, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=1, x input list=[-1, 0, -1] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.","answer":{"kind":"expression","value":"([11, 1, -1, 1, 1, -1], 1, [-1, 0], [[-1, 8], [0, 9]])","explanation":"Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([11, 1, -1, 1, 1, -1], 1, [-1, 0], [[-1, 8], [0, 9]])."},"skills":["A Fixed-Second-Coordinate Slice on Listed Inputs Only"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0075","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed5fc24d0b113127be2d4aed531341aff8e3e7793f405a119a84ead95c38f95a","family_id":"qf-mod23-multivar-x-slice-table","seed":202610237075,"semantic_template":"qf-mod23-multivar-x-slice-table/{\"coefficients\":[12,2,0,1,-1,-1],\"xs\":[-1,0],\"y\":2}"},"fields":{"title":"A Fixed-Second-Coordinate Slice on Listed Inputs Only","stem":"Use the declared real polynomial R²→R with coefficients=[12, 2, 0, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. fixed y=2, x input list=[-1, 0, 0] normalized to sorted distinct integers. Give (coefficients,fixed y,normalized inputs,every [x,f(x,y)] row). Empty input lists are allowed.","answer":{"kind":"expression","value":"([12, 2, 0, 1, -1, -1], 2, [-1, 0], [[-1, 9], [0, 8]])","explanation":"Fix the second coordinate and evaluate the first-coordinate slice only at listed inputs. A finite or empty slice does not determine the whole function. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([12, 2, 0, 1, -1, -1], 2, [-1, 0], [[-1, 9], [0, 8]])."},"skills":["A Fixed-Second-Coordinate Slice on Listed Inputs Only"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0076","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dbde1c5ee560df8bdf2be70c5292c03ed0969167ff05b14dc5a722332f8e056a","family_id":"qf-mod23-multivar-point-table-audit","seed":202610237076,"semantic_template":"qf-mod23-multivar-point-table-audit/{\"coefficients\":[-12,-2,-3,1,-1,-1],\"points\":[[0,0],[0,1],[1,0]],\"values\":[-12,-13,-15]}"},"fields":{"title":"Preserve Original Point-Value Pairing in a Finite Audit","stem":"Use the declared real polynomial R²→R with coefficients=[-12, -2, -3, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[-12, -13, -15]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.","answer":{"kind":"expression","value":"([-12, -2, -3, 1, -1, -1], [[0, 0, -12, -12, Yes], [1, 0, -13, -13, Yes], [0, 1, -15, -16, No]], No)","explanation":"Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-12, -2, -3, 1, -1, -1], [[0, 0, -12, -12, Yes], [1, 0, -13, -13, Yes], [0, 1, -15, -16, No]], No)."},"skills":["Preserve Original Point-Value Pairing in a Finite Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0077","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d32f2098ba125647075e91faba55cfe6e0faa7b08efc53f178eae81bcb7d44d9","family_id":"qf-mod23-multivar-point-table-audit","seed":202610237077,"semantic_template":"qf-mod23-multivar-point-table-audit/{\"coefficients\":[-11,-1,-2,1,0,-1],\"points\":[[0,0],[0,1],[1,0]],\"values\":[-11,-11,-14]}"},"fields":{"title":"Preserve Original Point-Value Pairing in a Finite Audit","stem":"Use the declared real polynomial R²→R with coefficients=[-11, -1, -2, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[-11, -11, -14]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.","answer":{"kind":"expression","value":"([-11, -1, -2, 1, 0, -1], [[0, 0, -11, -11, Yes], [1, 0, -11, -11, Yes], [0, 1, -14, -14, Yes]], Yes)","explanation":"Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-11, -1, -2, 1, 0, -1], [[0, 0, -11, -11, Yes], [1, 0, -11, -11, Yes], [0, 1, -14, -14, Yes]], Yes)."},"skills":["Preserve Original Point-Value Pairing in a Finite Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0078","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f250bd151fea49d65f5c61989c001256eb5a7f116daf6bd9dc93530d84a78fe3","family_id":"qf-mod23-multivar-point-table-audit","seed":202610237078,"semantic_template":"qf-mod23-multivar-point-table-audit/{\"coefficients\":[-10,0,-1,1,1,-1],\"points\":[[0,0],[0,1],[1,0]],\"values\":[-10,-9,-12]}"},"fields":{"title":"Preserve Original Point-Value Pairing in a Finite Audit","stem":"Use the declared real polynomial R²→R with coefficients=[-10, 0, -1, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[-10, -9, -12]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.","answer":{"kind":"expression","value":"([-10, 0, -1, 1, 1, -1], [[0, 0, -10, -10, Yes], [1, 0, -9, -9, Yes], [0, 1, -12, -12, Yes]], Yes)","explanation":"Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 0, -1, 1, 1, -1], [[0, 0, -10, -10, Yes], [1, 0, -9, -9, Yes], [0, 1, -12, -12, Yes]], Yes)."},"skills":["Preserve Original Point-Value Pairing in a Finite Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0079","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"391940b98770eaabf631a4f8294753c535e23e54d4d6431de5853c7e9ba5e41f","family_id":"qf-mod23-multivar-point-table-audit","seed":202610237079,"semantic_template":"qf-mod23-multivar-point-table-audit/{\"coefficients\":[-9,1,0,1,-1,-1],\"points\":[[0,0],[0,1],[1,0]],\"values\":[-9,-7,-10]}"},"fields":{"title":"Preserve Original Point-Value Pairing in a Finite Audit","stem":"Use the declared real polynomial R²→R with coefficients=[-9, 1, 0, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[-9, -7, -10]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.","answer":{"kind":"expression","value":"([-9, 1, 0, 1, -1, -1], [[0, 0, -9, -9, Yes], [1, 0, -7, -7, Yes], [0, 1, -10, -10, Yes]], Yes)","explanation":"Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 1, 0, 1, -1, -1], [[0, 0, -9, -9, Yes], [1, 0, -7, -7, Yes], [0, 1, -10, -10, Yes]], Yes)."},"skills":["Preserve Original Point-Value Pairing in a Finite Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0080","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b9e083c85d61323f276fff92d6f3afcca6f7b87096bd78e73f2dbb6e9f091c0","family_id":"qf-mod23-multivar-point-table-audit","seed":202610237080,"semantic_template":"qf-mod23-multivar-point-table-audit/{\"coefficients\":[-8,2,1,1,0,-1],\"points\":[[0,0],[0,1],[1,0]],\"values\":[-8,-5,-7]}"},"fields":{"title":"Preserve Original Point-Value Pairing in a Finite Audit","stem":"Use the declared real polynomial R²→R with coefficients=[-8, 2, 1, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[-8, -5, -7]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.","answer":{"kind":"expression","value":"([-8, 2, 1, 1, 0, -1], [[0, 0, -8, -8, Yes], [1, 0, -5, -5, Yes], [0, 1, -7, -8, No]], No)","explanation":"Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 2, 1, 1, 0, -1], [[0, 0, -8, -8, Yes], [1, 0, -5, -5, Yes], [0, 1, -7, -8, No]], No)."},"skills":["Preserve Original Point-Value Pairing in a Finite Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0081","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2015906dfdcf1607cbd201855645ec15a8b985b16c085fc7738e6ac667d250ed","family_id":"qf-mod23-multivar-point-table-audit","seed":202610237081,"semantic_template":"qf-mod23-multivar-point-table-audit/{\"coefficients\":[-7,-2,2,1,1,-1],\"points\":[[0,0],[0,1],[1,0]],\"values\":[-7,-8,-6]}"},"fields":{"title":"Preserve Original Point-Value Pairing in a Finite Audit","stem":"Use the declared real polynomial R²→R with coefficients=[-7, -2, 2, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[-7, -8, -6]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.","answer":{"kind":"expression","value":"([-7, -2, 2, 1, 1, -1], [[0, 0, -7, -7, Yes], [1, 0, -8, -8, Yes], [0, 1, -6, -6, Yes]], Yes)","explanation":"Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-7, -2, 2, 1, 1, -1], [[0, 0, -7, -7, Yes], [1, 0, -8, -8, Yes], [0, 1, -6, -6, Yes]], Yes)."},"skills":["Preserve Original Point-Value Pairing in a Finite Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0082","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11de3a6bb4be4e2a2a1ce85792adeda17131dba47c6ae5f31911740210b9bbd6","family_id":"qf-mod23-multivar-point-table-audit","seed":202610237082,"semantic_template":"qf-mod23-multivar-point-table-audit/{\"coefficients\":[-6,-1,3,1,-1,-1],\"points\":[[0,0],[0,1],[1,0]],\"values\":[-6,-6,-4]}"},"fields":{"title":"Preserve Original Point-Value Pairing in a Finite Audit","stem":"Use the declared real polynomial R²→R with coefficients=[-6, -1, 3, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[-6, -6, -4]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.","answer":{"kind":"expression","value":"([-6, -1, 3, 1, -1, -1], [[0, 0, -6, -6, Yes], [1, 0, -6, -6, Yes], [0, 1, -4, -4, Yes]], Yes)","explanation":"Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-6, -1, 3, 1, -1, -1], [[0, 0, -6, -6, Yes], [1, 0, -6, -6, Yes], [0, 1, -4, -4, Yes]], Yes)."},"skills":["Preserve Original Point-Value Pairing in a Finite Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0083","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e9f0e14807a40922daaa2a0e62646203e329b59a531c428c983e2695c2d1032","family_id":"qf-mod23-multivar-point-table-audit","seed":202610237083,"semantic_template":"qf-mod23-multivar-point-table-audit/{\"coefficients\":[-5,0,-3,1,0,-1],\"points\":[[0,0],[0,1],[1,0]],\"values\":[-5,-4,-9]}"},"fields":{"title":"Preserve Original Point-Value Pairing in a Finite Audit","stem":"Use the declared real polynomial R²→R with coefficients=[-5, 0, -3, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[-5, -4, -9]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.","answer":{"kind":"expression","value":"([-5, 0, -3, 1, 0, -1], [[0, 0, -5, -5, Yes], [1, 0, -4, -4, Yes], [0, 1, -9, -9, Yes]], Yes)","explanation":"Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 0, -3, 1, 0, -1], [[0, 0, -5, -5, Yes], [1, 0, -4, -4, Yes], [0, 1, -9, -9, Yes]], Yes)."},"skills":["Preserve Original Point-Value Pairing in a Finite Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0084","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b94800c4a89f11d4544888034d49edd09a320330e30138bf966f85f577ebf8e","family_id":"qf-mod23-multivar-point-table-audit","seed":202610237084,"semantic_template":"qf-mod23-multivar-point-table-audit/{\"coefficients\":[-4,1,-2,1,1,-1],\"points\":[[0,0],[0,1],[1,0]],\"values\":[-4,-2,-6]}"},"fields":{"title":"Preserve Original Point-Value Pairing in a Finite Audit","stem":"Use the declared real polynomial R²→R with coefficients=[-4, 1, -2, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[-4, -2, -6]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.","answer":{"kind":"expression","value":"([-4, 1, -2, 1, 1, -1], [[0, 0, -4, -4, Yes], [1, 0, -2, -2, Yes], [0, 1, -6, -7, No]], No)","explanation":"Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 1, -2, 1, 1, -1], [[0, 0, -4, -4, Yes], [1, 0, -2, -2, Yes], [0, 1, -6, -7, No]], No)."},"skills":["Preserve Original Point-Value Pairing in a Finite Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0085","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"043dcb20617cdb27d5b4c6a0f60f252a26492e7dbcf5cec95470f3347337fbe9","family_id":"qf-mod23-multivar-point-table-audit","seed":202610237085,"semantic_template":"qf-mod23-multivar-point-table-audit/{\"coefficients\":[-3,2,-1,1,-1,-1],\"points\":[[0,0],[0,1],[1,0]],\"values\":[-3,0,-5]}"},"fields":{"title":"Preserve Original Point-Value Pairing in a Finite Audit","stem":"Use the declared real polynomial R²→R with coefficients=[-3, 2, -1, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[-3, 0, -5]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.","answer":{"kind":"expression","value":"([-3, 2, -1, 1, -1, -1], [[0, 0, -3, -3, Yes], [1, 0, 0, 0, Yes], [0, 1, -5, -5, Yes]], Yes)","explanation":"Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 2, -1, 1, -1, -1], [[0, 0, -3, -3, Yes], [1, 0, 0, 0, Yes], [0, 1, -5, -5, Yes]], Yes)."},"skills":["Preserve Original Point-Value Pairing in a Finite Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0086","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e1b202402f2c4951fb0e0c6230604b891edd3570083589749e21be88223c08b","family_id":"qf-mod23-multivar-point-table-audit","seed":202610237086,"semantic_template":"qf-mod23-multivar-point-table-audit/{\"coefficients\":[-2,-2,0,1,0,-1],\"points\":[[0,0],[0,1],[1,0]],\"values\":[-2,-3,-3]}"},"fields":{"title":"Preserve Original Point-Value Pairing in a Finite Audit","stem":"Use the declared real polynomial R²→R with coefficients=[-2, -2, 0, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[-2, -3, -3]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.","answer":{"kind":"expression","value":"([-2, -2, 0, 1, 0, -1], [[0, 0, -2, -2, Yes], [1, 0, -3, -3, Yes], [0, 1, -3, -3, Yes]], Yes)","explanation":"Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-2, -2, 0, 1, 0, -1], [[0, 0, -2, -2, Yes], [1, 0, -3, -3, Yes], [0, 1, -3, -3, Yes]], Yes)."},"skills":["Preserve Original Point-Value Pairing in a Finite Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0087","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6439e37bdbbe1af6ad03d3f5b2b518275f09a3c4a6c1056149b4c6f1c6bc87d7","family_id":"qf-mod23-multivar-point-table-audit","seed":202610237087,"semantic_template":"qf-mod23-multivar-point-table-audit/{\"coefficients\":[-1,-1,1,1,1,-1],\"points\":[[0,0],[0,1],[1,0]],\"values\":[-1,-1,-1]}"},"fields":{"title":"Preserve Original Point-Value Pairing in a Finite Audit","stem":"Use the declared real polynomial R²→R with coefficients=[-1, -1, 1, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[-1, -1, -1]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.","answer":{"kind":"expression","value":"([-1, -1, 1, 1, 1, -1], [[0, 0, -1, -1, Yes], [1, 0, -1, -1, Yes], [0, 1, -1, -1, Yes]], Yes)","explanation":"Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -1, 1, 1, 1, -1], [[0, 0, -1, -1, Yes], [1, 0, -1, -1, Yes], [0, 1, -1, -1, Yes]], Yes)."},"skills":["Preserve Original Point-Value Pairing in a Finite Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0088","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8fdb6d81402b1d8213e383c35d8afcd14739b56f8f992dff3568fc8c77250f4","family_id":"qf-mod23-multivar-point-table-audit","seed":202610237088,"semantic_template":"qf-mod23-multivar-point-table-audit/{\"coefficients\":[0,0,2,1,-1,-1],\"points\":[[0,0],[0,1],[1,0]],\"values\":[0,1,2]}"},"fields":{"title":"Preserve Original Point-Value Pairing in a Finite Audit","stem":"Use the declared real polynomial R²→R with coefficients=[0, 0, 2, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[0, 1, 2]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.","answer":{"kind":"expression","value":"([0, 0, 2, 1, -1, -1], [[0, 0, 0, 0, Yes], [1, 0, 1, 1, Yes], [0, 1, 2, 1, No]], No)","explanation":"Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 2, 1, -1, -1], [[0, 0, 0, 0, Yes], [1, 0, 1, 1, Yes], [0, 1, 2, 1, No]], No)."},"skills":["Preserve Original Point-Value Pairing in a Finite Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0089","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8f38bbe17b2d3218026731658a49e4cc589d1915c87c17473153b3bcc68a73f","family_id":"qf-mod23-multivar-point-table-audit","seed":202610237089,"semantic_template":"qf-mod23-multivar-point-table-audit/{\"coefficients\":[1,1,3,1,0,-1],\"points\":[[0,0],[0,1],[1,0]],\"values\":[1,3,3]}"},"fields":{"title":"Preserve Original Point-Value Pairing in a Finite Audit","stem":"Use the declared real polynomial R²→R with coefficients=[1, 1, 3, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[1, 3, 3]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.","answer":{"kind":"expression","value":"([1, 1, 3, 1, 0, -1], [[0, 0, 1, 1, Yes], [1, 0, 3, 3, Yes], [0, 1, 3, 3, Yes]], Yes)","explanation":"Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 3, 1, 0, -1], [[0, 0, 1, 1, Yes], [1, 0, 3, 3, Yes], [0, 1, 3, 3, Yes]], Yes)."},"skills":["Preserve Original Point-Value Pairing in a Finite Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0090","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a18de7c21e4ef11dd0e91991488879e0e73806c22a048b86547b72db9c0a069","family_id":"qf-mod23-multivar-point-table-audit","seed":202610237090,"semantic_template":"qf-mod23-multivar-point-table-audit/{\"coefficients\":[2,2,-3,1,1,-1],\"points\":[[0,0],[0,1],[1,0]],\"values\":[2,5,-2]}"},"fields":{"title":"Preserve Original Point-Value Pairing in a Finite Audit","stem":"Use the declared real polynomial R²→R with coefficients=[2, 2, -3, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[2, 5, -2]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.","answer":{"kind":"expression","value":"([2, 2, -3, 1, 1, -1], [[0, 0, 2, 2, Yes], [1, 0, 5, 5, Yes], [0, 1, -2, -2, Yes]], Yes)","explanation":"Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, -3, 1, 1, -1], [[0, 0, 2, 2, Yes], [1, 0, 5, 5, Yes], [0, 1, -2, -2, Yes]], Yes)."},"skills":["Preserve Original Point-Value Pairing in a Finite Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0091","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c96b0f5d6fad4f679cb2d9d5f971f2c6ba995113b47f48dc86247d9c34625446","family_id":"qf-mod23-multivar-point-table-audit","seed":202610237091,"semantic_template":"qf-mod23-multivar-point-table-audit/{\"coefficients\":[3,-2,-2,1,-1,-1],\"points\":[[0,0],[0,1],[1,0]],\"values\":[3,2,0]}"},"fields":{"title":"Preserve Original Point-Value Pairing in a Finite Audit","stem":"Use the declared real polynomial R²→R with coefficients=[3, -2, -2, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[3, 2, 0]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.","answer":{"kind":"expression","value":"([3, -2, -2, 1, -1, -1], [[0, 0, 3, 3, Yes], [1, 0, 2, 2, Yes], [0, 1, 0, 0, Yes]], Yes)","explanation":"Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, -2, -2, 1, -1, -1], [[0, 0, 3, 3, Yes], [1, 0, 2, 2, Yes], [0, 1, 0, 0, Yes]], Yes)."},"skills":["Preserve Original Point-Value Pairing in a Finite Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0092","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba7a6f18d17779f2852485b14668a5131774c1017ebb10de744a8910304d6b75","family_id":"qf-mod23-multivar-point-table-audit","seed":202610237092,"semantic_template":"qf-mod23-multivar-point-table-audit/{\"coefficients\":[4,-1,-1,1,0,-1],\"points\":[[0,0],[0,1],[1,0]],\"values\":[4,4,3]}"},"fields":{"title":"Preserve Original Point-Value Pairing in a Finite Audit","stem":"Use the declared real polynomial R²→R with coefficients=[4, -1, -1, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[4, 4, 3]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.","answer":{"kind":"expression","value":"([4, -1, -1, 1, 0, -1], [[0, 0, 4, 4, Yes], [1, 0, 4, 4, Yes], [0, 1, 3, 2, No]], No)","explanation":"Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1, -1, 1, 0, -1], [[0, 0, 4, 4, Yes], [1, 0, 4, 4, Yes], [0, 1, 3, 2, No]], No)."},"skills":["Preserve Original Point-Value Pairing in a Finite Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0093","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df033bb9f6901aadeeeb093a67aab7ef9801cfcc43e2ee5b57d375a6378510c6","family_id":"qf-mod23-multivar-point-table-audit","seed":202610237093,"semantic_template":"qf-mod23-multivar-point-table-audit/{\"coefficients\":[5,0,0,1,1,-1],\"points\":[[0,0],[0,1],[1,0]],\"values\":[5,6,4]}"},"fields":{"title":"Preserve Original Point-Value Pairing in a Finite Audit","stem":"Use the declared real polynomial R²→R with coefficients=[5, 0, 0, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[5, 6, 4]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.","answer":{"kind":"expression","value":"([5, 0, 0, 1, 1, -1], [[0, 0, 5, 5, Yes], [1, 0, 6, 6, Yes], [0, 1, 4, 4, Yes]], Yes)","explanation":"Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0, 0, 1, 1, -1], [[0, 0, 5, 5, Yes], [1, 0, 6, 6, Yes], [0, 1, 4, 4, Yes]], Yes)."},"skills":["Preserve Original Point-Value Pairing in a Finite Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0094","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0492a464afd5544d241c0396d5d4ef74ce6f25f63d2947fe954f30bf2e86b78","family_id":"qf-mod23-multivar-point-table-audit","seed":202610237094,"semantic_template":"qf-mod23-multivar-point-table-audit/{\"coefficients\":[6,1,1,1,-1,-1],\"points\":[[0,0],[0,1],[1,0]],\"values\":[6,8,6]}"},"fields":{"title":"Preserve Original Point-Value Pairing in a Finite Audit","stem":"Use the declared real polynomial R²→R with coefficients=[6, 1, 1, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[6, 8, 6]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.","answer":{"kind":"expression","value":"([6, 1, 1, 1, -1, -1], [[0, 0, 6, 6, Yes], [1, 0, 8, 8, Yes], [0, 1, 6, 6, Yes]], Yes)","explanation":"Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([6, 1, 1, 1, -1, -1], [[0, 0, 6, 6, Yes], [1, 0, 8, 8, Yes], [0, 1, 6, 6, Yes]], Yes)."},"skills":["Preserve Original Point-Value Pairing in a Finite Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0095","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c5b72727e04fb4c50caf3422d50d29d5388ae08b7f4fa0bb4b2b816a425c62d","family_id":"qf-mod23-multivar-point-table-audit","seed":202610237095,"semantic_template":"qf-mod23-multivar-point-table-audit/{\"coefficients\":[7,2,2,1,0,-1],\"points\":[[0,0],[0,1],[1,0]],\"values\":[7,10,8]}"},"fields":{"title":"Preserve Original Point-Value Pairing in a Finite Audit","stem":"Use the declared real polynomial R²→R with coefficients=[7, 2, 2, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[7, 10, 8]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.","answer":{"kind":"expression","value":"([7, 2, 2, 1, 0, -1], [[0, 0, 7, 7, Yes], [1, 0, 10, 10, Yes], [0, 1, 8, 8, Yes]], Yes)","explanation":"Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([7, 2, 2, 1, 0, -1], [[0, 0, 7, 7, Yes], [1, 0, 10, 10, Yes], [0, 1, 8, 8, Yes]], Yes)."},"skills":["Preserve Original Point-Value Pairing in a Finite Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0096","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb63961ef7c1d2f6af46ea96d4a34cf4f0ac4792eaffd71cebc84582eabf3ee0","family_id":"qf-mod23-multivar-point-table-audit","seed":202610237096,"semantic_template":"qf-mod23-multivar-point-table-audit/{\"coefficients\":[8,-2,3,1,1,-1],\"points\":[[0,0],[0,1],[1,0]],\"values\":[8,7,11]}"},"fields":{"title":"Preserve Original Point-Value Pairing in a Finite Audit","stem":"Use the declared real polynomial R²→R with coefficients=[8, -2, 3, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[8, 7, 11]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.","answer":{"kind":"expression","value":"([8, -2, 3, 1, 1, -1], [[0, 0, 8, 8, Yes], [1, 0, 7, 7, Yes], [0, 1, 11, 10, No]], No)","explanation":"Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([8, -2, 3, 1, 1, -1], [[0, 0, 8, 8, Yes], [1, 0, 7, 7, Yes], [0, 1, 11, 10, No]], No)."},"skills":["Preserve Original Point-Value Pairing in a Finite Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0097","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7668c5220d38aa295c644a75d7e833a0669762fd35dd235b1f9e7a8e6e9174da","family_id":"qf-mod23-multivar-point-table-audit","seed":202610237097,"semantic_template":"qf-mod23-multivar-point-table-audit/{\"coefficients\":[9,-1,-3,1,-1,-1],\"points\":[[0,0],[0,1],[1,0]],\"values\":[9,9,5]}"},"fields":{"title":"Preserve Original Point-Value Pairing in a Finite Audit","stem":"Use the declared real polynomial R²→R with coefficients=[9, -1, -3, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[9, 9, 5]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.","answer":{"kind":"expression","value":"([9, -1, -3, 1, -1, -1], [[0, 0, 9, 9, Yes], [1, 0, 9, 9, Yes], [0, 1, 5, 5, Yes]], Yes)","explanation":"Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([9, -1, -3, 1, -1, -1], [[0, 0, 9, 9, Yes], [1, 0, 9, 9, Yes], [0, 1, 5, 5, Yes]], Yes)."},"skills":["Preserve Original Point-Value Pairing in a Finite Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0098","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a17bd46b7ffa5e2feff1ade540425f1cd21d511cd85e0fb54bc811a0759b3234","family_id":"qf-mod23-multivar-point-table-audit","seed":202610237098,"semantic_template":"qf-mod23-multivar-point-table-audit/{\"coefficients\":[10,0,-2,1,0,-1],\"points\":[[0,0],[0,1],[1,0]],\"values\":[10,11,7]}"},"fields":{"title":"Preserve Original Point-Value Pairing in a Finite Audit","stem":"Use the declared real polynomial R²→R with coefficients=[10, 0, -2, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[10, 11, 7]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.","answer":{"kind":"expression","value":"([10, 0, -2, 1, 0, -1], [[0, 0, 10, 10, Yes], [1, 0, 11, 11, Yes], [0, 1, 7, 7, Yes]], Yes)","explanation":"Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([10, 0, -2, 1, 0, -1], [[0, 0, 10, 10, Yes], [1, 0, 11, 11, Yes], [0, 1, 7, 7, Yes]], Yes)."},"skills":["Preserve Original Point-Value Pairing in a Finite Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0099","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ede93c875e816a3c1c9c06881a1c06f9fe666c831e98815cbb44b6d05574621","family_id":"qf-mod23-multivar-point-table-audit","seed":202610237099,"semantic_template":"qf-mod23-multivar-point-table-audit/{\"coefficients\":[11,1,-1,1,1,-1],\"points\":[[0,0],[0,1],[1,0]],\"values\":[11,13,9]}"},"fields":{"title":"Preserve Original Point-Value Pairing in a Finite Audit","stem":"Use the declared real polynomial R²→R with coefficients=[11, 1, -1, 1, 1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[11, 13, 9]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.","answer":{"kind":"expression","value":"([11, 1, -1, 1, 1, -1], [[0, 0, 11, 11, Yes], [1, 0, 13, 13, Yes], [0, 1, 9, 9, Yes]], Yes)","explanation":"Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([11, 1, -1, 1, 1, -1], [[0, 0, 11, 11, Yes], [1, 0, 13, 13, Yes], [0, 1, 9, 9, Yes]], Yes)."},"skills":["Preserve Original Point-Value Pairing in a Finite Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r07-multivariable-real-function-0100","batch_id":"batch-20261013-advance-20261023-r07-multivariable-real-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc79180e21f8310315c88eca6407017db9c67288d0611aeafd7fea569b9eaba8","family_id":"qf-mod23-multivar-point-table-audit","seed":202610237100,"semantic_template":"qf-mod23-multivar-point-table-audit/{\"coefficients\":[12,2,0,1,-1,-1],\"points\":[[0,0],[0,1],[1,0]],\"values\":[12,15,12]}"},"fields":{"title":"Preserve Original Point-Value Pairing in a Finite Audit","stem":"Use the declared real polynomial R²→R with coefficients=[12, 2, 0, 1, -1, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. original points=[[0, 0], [1, 0], [0, 1]], aligned claimed values=[12, 15, 12]. Give (coefficients,every [x,y,claimed value,actual value,match?] row,all listed rows match?). Preserve point order, repeated points and value alignment; an empty table has a vacuously true all-match flag.","answer":{"kind":"expression","value":"([12, 2, 0, 1, -1, -1], [[0, 0, 12, 12, Yes], [1, 0, 15, 15, Yes], [0, 1, 12, 11, No]], No)","explanation":"Recompute every declared point without sorting away its claimed value. Even all-match does not prove global function equivalence, continuity or differentiability. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([12, 2, 0, 1, -1, -1], [[0, 0, 12, 12, Yes], [1, 0, 15, 15, Yes], [0, 1, 12, 11, No]], No)."},"skills":["Preserve Original Point-Value Pairing in a Finite Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}