{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0001","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b20cb22cfc9984f84f030d94610815962d43cdc93bc9ee721e77428f10bbb3e5","family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136001,"semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":2,\"terms\":[[[1,1],[0,0]],[[1,1],[2,1]],[[2,1],[0,1]]]}"},"fields":{"title":"The Symbolic Gradient Keeps Every Euclidean Coordinate in Order","stem":"Use dimension=2, raw polynomial terms=[[1, [2, 1]], [2, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [2, [0, 1]], [1, [2, 1]]], [[[2, [1, 1]]], [[2, [0, 0]], [1, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation":"Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [2, [0, 1]], [1, [2, 1]]], [[[2, [1, 1]]], [[2, [0, 0]], [1, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)."},"skills":["The Symbolic Gradient Keeps Every Euclidean Coordinate in Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0002","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ee050d7323a7ba42acba54c7b60bdc31bba0e220d311914bf61bb44cd523b35","family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136002,"semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":3,\"terms\":[[[1,1],[0,0,0]],[[2,1],[2,1,0]],[[3,1],[0,1,1]]]}"},"fields":{"title":"The Symbolic Gradient Keeps Every Euclidean Coordinate in Order","stem":"Use dimension=3, raw polynomial terms=[[2, [2, 1, 0]], [3, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [3, [0, 1, 1]], [2, [2, 1, 0]]], [[[4, [1, 1, 0]]], [[3, [0, 0, 1]], [2, [2, 0, 0]]], [[3, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation":"Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [3, [0, 1, 1]], [2, [2, 1, 0]]], [[[4, [1, 1, 0]]], [[3, [0, 0, 1]], [2, [2, 0, 0]]], [[3, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)."},"skills":["The Symbolic Gradient Keeps Every Euclidean Coordinate in Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0003","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a70bcace4119a0af64f36c6802ce11ab0422b9308a749521ac1e5b03416f1310","family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136003,"semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":2,\"terms\":[[[1,1],[0,0]],[[3,1],[2,1]],[[4,1],[0,1]]]}"},"fields":{"title":"The Symbolic Gradient Keeps Every Euclidean Coordinate in Order","stem":"Use dimension=2, raw polynomial terms=[[3, [2, 1]], [4, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [4, [0, 1]], [3, [2, 1]]], [[[6, [1, 1]]], [[4, [0, 0]], [3, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation":"Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [4, [0, 1]], [3, [2, 1]]], [[[6, [1, 1]]], [[4, [0, 0]], [3, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)."},"skills":["The Symbolic Gradient Keeps Every Euclidean Coordinate in Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0004","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ecd7aaa7ae7964d5a3205898b203e32458b6506c5044c35bacce5aa8ad628fc3","family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136004,"semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":3,\"terms\":[[[1,1],[0,0,0]],[[4,1],[2,1,0]],[[5,1],[0,1,1]]]}"},"fields":{"title":"The Symbolic Gradient Keeps Every Euclidean Coordinate in Order","stem":"Use dimension=3, raw polynomial terms=[[4, [2, 1, 0]], [5, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [5, [0, 1, 1]], [4, [2, 1, 0]]], [[[8, [1, 1, 0]]], [[5, [0, 0, 1]], [4, [2, 0, 0]]], [[5, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation":"Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [5, [0, 1, 1]], [4, [2, 1, 0]]], [[[8, [1, 1, 0]]], [[5, [0, 0, 1]], [4, [2, 0, 0]]], [[5, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)."},"skills":["The Symbolic Gradient Keeps Every Euclidean Coordinate in Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0005","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03a6ac25dbff563f559e34d4796f35e699d7319f1aae376619d694b0770d0987","family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136005,"semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":2,\"terms\":[[[1,1],[0,0]],[[5,1],[2,1]],[[6,1],[0,1]]]}"},"fields":{"title":"The Symbolic Gradient Keeps Every Euclidean Coordinate in Order","stem":"Use dimension=2, raw polynomial terms=[[5, [2, 1]], [6, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [6, [0, 1]], [5, [2, 1]]], [[[10, [1, 1]]], [[6, [0, 0]], [5, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation":"Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [6, [0, 1]], [5, [2, 1]]], [[[10, [1, 1]]], [[6, [0, 0]], [5, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)."},"skills":["The Symbolic Gradient Keeps Every Euclidean Coordinate in Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0006","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"721a8a56729bbc97045be7062a34353c37a6b61306b2f0e836cf3d348c563770","family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136006,"semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":3,\"terms\":[[[1,1],[0,0,0]],[[6,1],[2,1,0]],[[7,1],[0,1,1]]]}"},"fields":{"title":"The Symbolic Gradient Keeps Every Euclidean Coordinate in Order","stem":"Use dimension=3, raw polynomial terms=[[6, [2, 1, 0]], [7, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [7, [0, 1, 1]], [6, [2, 1, 0]]], [[[12, [1, 1, 0]]], [[7, [0, 0, 1]], [6, [2, 0, 0]]], [[7, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation":"Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [7, [0, 1, 1]], [6, [2, 1, 0]]], [[[12, [1, 1, 0]]], [[7, [0, 0, 1]], [6, [2, 0, 0]]], [[7, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)."},"skills":["The Symbolic Gradient Keeps Every Euclidean Coordinate in Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0007","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81613e92b84f84f52219559ad5ef17c9c0a0b62e4644b6f64ae2ec204047426d","family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136007,"semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":2,\"terms\":[[[1,1],[0,0]],[[7,1],[2,1]],[[8,1],[0,1]]]}"},"fields":{"title":"The Symbolic Gradient Keeps Every Euclidean Coordinate in Order","stem":"Use dimension=2, raw polynomial terms=[[7, [2, 1]], [8, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [8, [0, 1]], [7, [2, 1]]], [[[14, [1, 1]]], [[8, [0, 0]], [7, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation":"Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [8, [0, 1]], [7, [2, 1]]], [[[14, [1, 1]]], [[8, [0, 0]], [7, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)."},"skills":["The Symbolic Gradient Keeps Every Euclidean Coordinate in Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0008","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"353577c1ff5082c3080f268ea0efd5ee5086b08548c43a4d9613abf6f8571ec0","family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136008,"semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":3,\"terms\":[[[1,1],[0,0,0]],[[8,1],[2,1,0]],[[9,1],[0,1,1]]]}"},"fields":{"title":"The Symbolic Gradient Keeps Every Euclidean Coordinate in Order","stem":"Use dimension=3, raw polynomial terms=[[8, [2, 1, 0]], [9, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [9, [0, 1, 1]], [8, [2, 1, 0]]], [[[16, [1, 1, 0]]], [[9, [0, 0, 1]], [8, [2, 0, 0]]], [[9, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation":"Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [9, [0, 1, 1]], [8, [2, 1, 0]]], [[[16, [1, 1, 0]]], [[9, [0, 0, 1]], [8, [2, 0, 0]]], [[9, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)."},"skills":["The Symbolic Gradient Keeps Every Euclidean Coordinate in Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0009","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b65ef7ed7d0cba1932694fbc7e1965ad985ca33d4b61a5c703f88e2413587695","family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136009,"semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":2,\"terms\":[[[1,1],[0,0]],[[10,1],[0,1]],[[9,1],[2,1]]]}"},"fields":{"title":"The Symbolic Gradient Keeps Every Euclidean Coordinate in Order","stem":"Use dimension=2, raw polynomial terms=[[9, [2, 1]], [10, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [10, [0, 1]], [9, [2, 1]]], [[[18, [1, 1]]], [[10, [0, 0]], [9, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation":"Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [10, [0, 1]], [9, [2, 1]]], [[[18, [1, 1]]], [[10, [0, 0]], [9, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)."},"skills":["The Symbolic Gradient Keeps Every Euclidean Coordinate in Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0010","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"80ecb758f50318ce2a55abd095484db16bf08eb982844c94c45b679276bb4844","family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136010,"semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":3,\"terms\":[[[1,1],[0,0,0]],[[10,1],[2,1,0]],[[11,1],[0,1,1]]]}"},"fields":{"title":"The Symbolic Gradient Keeps Every Euclidean Coordinate in Order","stem":"Use dimension=3, raw polynomial terms=[[10, [2, 1, 0]], [11, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [11, [0, 1, 1]], [10, [2, 1, 0]]], [[[20, [1, 1, 0]]], [[11, [0, 0, 1]], [10, [2, 0, 0]]], [[11, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation":"Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [11, [0, 1, 1]], [10, [2, 1, 0]]], [[[20, [1, 1, 0]]], [[11, [0, 0, 1]], [10, [2, 0, 0]]], [[11, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)."},"skills":["The Symbolic Gradient Keeps Every Euclidean Coordinate in Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0011","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fffe51ee54a609a6f72c197f50d584cc0be23fb33fd29f339ffde0d8e48c0002","family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136011,"semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":2,\"terms\":[[[1,1],[0,0]],[[11,1],[2,1]],[[12,1],[0,1]]]}"},"fields":{"title":"The Symbolic Gradient Keeps Every Euclidean Coordinate in Order","stem":"Use dimension=2, raw polynomial terms=[[11, [2, 1]], [12, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [12, [0, 1]], [11, [2, 1]]], [[[22, [1, 1]]], [[12, [0, 0]], [11, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation":"Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [12, [0, 1]], [11, [2, 1]]], [[[22, [1, 1]]], [[12, [0, 0]], [11, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)."},"skills":["The Symbolic Gradient Keeps Every Euclidean Coordinate in Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0012","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04174a0b82ff0ae6b8512f0b6391e25736cadb9a4e48dc8437997df473dad3bb","family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136012,"semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":3,\"terms\":[[[1,1],[0,0,0]],[[12,1],[2,1,0]],[[13,1],[0,1,1]]]}"},"fields":{"title":"The Symbolic Gradient Keeps Every Euclidean Coordinate in Order","stem":"Use dimension=3, raw polynomial terms=[[12, [2, 1, 0]], [13, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [13, [0, 1, 1]], [12, [2, 1, 0]]], [[[24, [1, 1, 0]]], [[13, [0, 0, 1]], [12, [2, 0, 0]]], [[13, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation":"Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [13, [0, 1, 1]], [12, [2, 1, 0]]], [[[24, [1, 1, 0]]], [[13, [0, 0, 1]], [12, [2, 0, 0]]], [[13, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)."},"skills":["The Symbolic Gradient Keeps Every Euclidean Coordinate in Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0013","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe8b37bfb56782b8be44354bd37d78a9d06150e10f7878358c9f71384ea2e5d9","family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136013,"semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":2,\"terms\":[[[1,1],[0,0]],[[13,1],[2,1]],[[14,1],[0,1]]]}"},"fields":{"title":"The Symbolic Gradient Keeps Every Euclidean Coordinate in Order","stem":"Use dimension=2, raw polynomial terms=[[13, [2, 1]], [14, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [14, [0, 1]], [13, [2, 1]]], [[[26, [1, 1]]], [[14, [0, 0]], [13, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation":"Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [14, [0, 1]], [13, [2, 1]]], [[[26, [1, 1]]], [[14, [0, 0]], [13, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)."},"skills":["The Symbolic Gradient Keeps Every Euclidean Coordinate in Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0014","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88a881d85690a57337f97eca75a863c28c1b0f9e69748ef85e130ab608387e94","family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136014,"semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":3,\"terms\":[[[1,1],[0,0,0]],[[14,1],[2,1,0]],[[15,1],[0,1,1]]]}"},"fields":{"title":"The Symbolic Gradient Keeps Every Euclidean Coordinate in Order","stem":"Use dimension=3, raw polynomial terms=[[14, [2, 1, 0]], [15, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [15, [0, 1, 1]], [14, [2, 1, 0]]], [[[28, [1, 1, 0]]], [[15, [0, 0, 1]], [14, [2, 0, 0]]], [[15, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation":"Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [15, [0, 1, 1]], [14, [2, 1, 0]]], [[[28, [1, 1, 0]]], [[15, [0, 0, 1]], [14, [2, 0, 0]]], [[15, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)."},"skills":["The Symbolic Gradient Keeps Every Euclidean Coordinate in Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0015","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f72d89625483e0c6a056d4efda55da0c4cdd8e9f41d6400a41425f93b9ddda8e","family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136015,"semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":2,\"terms\":[[[1,1],[0,0]],[[15,1],[2,1]],[[16,1],[0,1]]]}"},"fields":{"title":"The Symbolic Gradient Keeps Every Euclidean Coordinate in Order","stem":"Use dimension=2, raw polynomial terms=[[15, [2, 1]], [16, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [16, [0, 1]], [15, [2, 1]]], [[[30, [1, 1]]], [[16, [0, 0]], [15, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation":"Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [16, [0, 1]], [15, [2, 1]]], [[[30, [1, 1]]], [[16, [0, 0]], [15, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)."},"skills":["The Symbolic Gradient Keeps Every Euclidean Coordinate in Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0016","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"730d40fcc20251740a3d406dc86fa6eb6db60d9fd4558649ba853b539ba8182b","family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136016,"semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":3,\"terms\":[[[1,1],[0,0,0]],[[16,1],[2,1,0]],[[17,1],[0,1,1]]]}"},"fields":{"title":"The Symbolic Gradient Keeps Every Euclidean Coordinate in Order","stem":"Use dimension=3, raw polynomial terms=[[16, [2, 1, 0]], [17, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [17, [0, 1, 1]], [16, [2, 1, 0]]], [[[32, [1, 1, 0]]], [[17, [0, 0, 1]], [16, [2, 0, 0]]], [[17, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation":"Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [17, [0, 1, 1]], [16, [2, 1, 0]]], [[[32, [1, 1, 0]]], [[17, [0, 0, 1]], [16, [2, 0, 0]]], [[17, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)."},"skills":["The Symbolic Gradient Keeps Every Euclidean Coordinate in Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0017","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"016317d2f342cd04bd0fc4ec3c86e10ad774dd5bca722722eb2c8c710ba5dc7b","family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136017,"semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":2,\"terms\":[[[1,1],[0,0]],[[17,1],[2,1]],[[18,1],[0,1]]]}"},"fields":{"title":"The Symbolic Gradient Keeps Every Euclidean Coordinate in Order","stem":"Use dimension=2, raw polynomial terms=[[17, [2, 1]], [18, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [18, [0, 1]], [17, [2, 1]]], [[[34, [1, 1]]], [[18, [0, 0]], [17, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation":"Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [18, [0, 1]], [17, [2, 1]]], [[[34, [1, 1]]], [[18, [0, 0]], [17, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)."},"skills":["The Symbolic Gradient Keeps Every Euclidean Coordinate in Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0018","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02987fd4d14bea3018572913eb203b24b647d973c3b4ee902b3eed49feb8d6bd","family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136018,"semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":3,\"terms\":[[[1,1],[0,0,0]],[[18,1],[2,1,0]],[[19,1],[0,1,1]]]}"},"fields":{"title":"The Symbolic Gradient Keeps Every Euclidean Coordinate in Order","stem":"Use dimension=3, raw polynomial terms=[[18, [2, 1, 0]], [19, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [19, [0, 1, 1]], [18, [2, 1, 0]]], [[[36, [1, 1, 0]]], [[19, [0, 0, 1]], [18, [2, 0, 0]]], [[19, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation":"Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [19, [0, 1, 1]], [18, [2, 1, 0]]], [[[36, [1, 1, 0]]], [[19, [0, 0, 1]], [18, [2, 0, 0]]], [[19, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)."},"skills":["The Symbolic Gradient Keeps Every Euclidean Coordinate in Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0019","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab2374219d662474ebefc0c66121900eecc361a47097b582fc5f484b93e2e8ec","family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136019,"semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":2,\"terms\":[[[1,1],[0,0]],[[19,1],[2,1]],[[20,1],[0,1]]]}"},"fields":{"title":"The Symbolic Gradient Keeps Every Euclidean Coordinate in Order","stem":"Use dimension=2, raw polynomial terms=[[19, [2, 1]], [20, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [20, [0, 1]], [19, [2, 1]]], [[[38, [1, 1]]], [[20, [0, 0]], [19, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation":"Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [20, [0, 1]], [19, [2, 1]]], [[[38, [1, 1]]], [[20, [0, 0]], [19, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)."},"skills":["The Symbolic Gradient Keeps Every Euclidean Coordinate in Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0020","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fed686b90534b89de3daa406aba0621a0a204af39ee33ac4499720bdec2c2ddb","family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136020,"semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":3,\"terms\":[[[1,1],[0,0,0]],[[20,1],[2,1,0]],[[21,1],[0,1,1]]]}"},"fields":{"title":"The Symbolic Gradient Keeps Every Euclidean Coordinate in Order","stem":"Use dimension=3, raw polynomial terms=[[20, [2, 1, 0]], [21, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [21, [0, 1, 1]], [20, [2, 1, 0]]], [[[40, [1, 1, 0]]], [[21, [0, 0, 1]], [20, [2, 0, 0]]], [[21, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation":"Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [21, [0, 1, 1]], [20, [2, 1, 0]]], [[[40, [1, 1, 0]]], [[21, [0, 0, 1]], [20, [2, 0, 0]]], [[21, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)."},"skills":["The Symbolic Gradient Keeps Every Euclidean Coordinate in Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0021","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c36f736884c299e27256b2894092f77b03cad4a429cb260b3b922880a3545f9","family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136021,"semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":2,\"terms\":[[[1,1],[0,0]],[[21,1],[2,1]],[[22,1],[0,1]]]}"},"fields":{"title":"The Symbolic Gradient Keeps Every Euclidean Coordinate in Order","stem":"Use dimension=2, raw polynomial terms=[[21, [2, 1]], [22, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [22, [0, 1]], [21, [2, 1]]], [[[42, [1, 1]]], [[22, [0, 0]], [21, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation":"Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [22, [0, 1]], [21, [2, 1]]], [[[42, [1, 1]]], [[22, [0, 0]], [21, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)."},"skills":["The Symbolic Gradient Keeps Every Euclidean Coordinate in Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0022","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ecda20ff68750abb2e8be4b1cffef974ad4d3be83017849c2644194c244dae33","family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136022,"semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":3,\"terms\":[[[1,1],[0,0,0]],[[22,1],[2,1,0]],[[23,1],[0,1,1]]]}"},"fields":{"title":"The Symbolic Gradient Keeps Every Euclidean Coordinate in Order","stem":"Use dimension=3, raw polynomial terms=[[22, [2, 1, 0]], [23, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [23, [0, 1, 1]], [22, [2, 1, 0]]], [[[44, [1, 1, 0]]], [[23, [0, 0, 1]], [22, [2, 0, 0]]], [[23, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation":"Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [23, [0, 1, 1]], [22, [2, 1, 0]]], [[[44, [1, 1, 0]]], [[23, [0, 0, 1]], [22, [2, 0, 0]]], [[23, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)."},"skills":["The Symbolic Gradient Keeps Every Euclidean Coordinate in Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0023","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52f84814a49c8856adcc330c982df9f6c0bf1fd3e34c44f0f7878d00ff5c5a4f","family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136023,"semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":2,\"terms\":[[[1,1],[0,0]],[[23,1],[2,1]],[[24,1],[0,1]]]}"},"fields":{"title":"The Symbolic Gradient Keeps Every Euclidean Coordinate in Order","stem":"Use dimension=2, raw polynomial terms=[[23, [2, 1]], [24, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [24, [0, 1]], [23, [2, 1]]], [[[46, [1, 1]]], [[24, [0, 0]], [23, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation":"Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [24, [0, 1]], [23, [2, 1]]], [[[46, [1, 1]]], [[24, [0, 0]], [23, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)."},"skills":["The Symbolic Gradient Keeps Every Euclidean Coordinate in Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0024","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11a67d7070bea1409fcaa410d2b12a16d30eaf1fc95254453e4a58fa07afc484","family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136024,"semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":3,\"terms\":[[[1,1],[0,0,0]],[[24,1],[2,1,0]],[[25,1],[0,1,1]]]}"},"fields":{"title":"The Symbolic Gradient Keeps Every Euclidean Coordinate in Order","stem":"Use dimension=3, raw polynomial terms=[[24, [2, 1, 0]], [25, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [25, [0, 1, 1]], [24, [2, 1, 0]]], [[[48, [1, 1, 0]]], [[25, [0, 0, 1]], [24, [2, 0, 0]]], [[25, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation":"Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [25, [0, 1, 1]], [24, [2, 1, 0]]], [[[48, [1, 1, 0]]], [[25, [0, 0, 1]], [24, [2, 0, 0]]], [[25, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)."},"skills":["The Symbolic Gradient Keeps Every Euclidean Coordinate in Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0025","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ade55808af4fb5a7136336e7df800e97adc5834c8ed5f144840b531d737cd120","family_id":"qf-mod44-gradient-symbolic-polynomial","seed":202611136025,"semantic_template":"qf-mod44-gradient-symbolic-polynomial/{\"dimension\":2,\"terms\":[[[1,1],[0,0]],[[25,1],[2,1]],[[26,1],[0,1]]]}"},"fields":{"title":"The Symbolic Gradient Keeps Every Euclidean Coordinate in Order","stem":"Use dimension=2, raw polynomial terms=[[25, [2, 1]], [26, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Give (dimension, canonical_terms, coordinate_derivative_polynomials, certificate).","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [26, [0, 1]], [25, [2, 1]]], [[[50, [1, 1]]], [[26, [0, 0]], [25, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)","explanation":"Differentiate the canonical polynomial once in each coordinate and report ordered derivative-polynomial term lists [coefficient,exponents]. These are symbolic component polynomials, not their values at an unspecified point. No metric/coordinate change is implicit. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [26, [0, 1]], [25, [2, 1]]], [[[50, [1, 1]]], [[26, [0, 0]], [25, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)."},"skills":["The Symbolic Gradient Keeps Every Euclidean Coordinate in Order"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0026","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"316bc36c0c6344bfd0bdf5ed92cf4c947115f94f513fdfdecd1a7f2abce36b9e","family_id":"qf-mod44-gradient-rational-point","seed":202611136026,"semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":2,\"point\":[[1,1],[1,1]],\"terms\":[[[1,1],[0,0]],[[1,1],[2,1]],[[2,1],[0,1]]]}"},"fields":{"title":"Evaluate Every Coordinate Derivative at the Same Exact Rational Point","stem":"Use dimension=2, raw polynomial terms=[[1, [2, 1]], [2, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[1/1, 2/2]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [2, [0, 1]], [1, [2, 1]]], [1, 1], [[[2, [1, 1]]], [[2, [0, 0]], [1, [2, 0]]]], [2, 3], exact_rational_coordinate_evaluation)","explanation":"Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [2, [0, 1]], [1, [2, 1]]], [1, 1], [[[2, [1, 1]]], [[2, [0, 0]], [1, [2, 0]]]], [2, 3], exact_rational_coordinate_evaluation)."},"skills":["Evaluate Every Coordinate Derivative at the Same Exact Rational Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0027","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02f18b9b9fd700930edf13c26d10cc762ecce4df39d1a6e72fe2c518f7cf458a","family_id":"qf-mod44-gradient-rational-point","seed":202611136027,"semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":3,\"point\":[[2,1],[3,2],[4,3]],\"terms\":[[[1,1],[0,0,0]],[[2,1],[2,1,0]],[[3,1],[0,1,1]]]}"},"fields":{"title":"Evaluate Every Coordinate Derivative at the Same Exact Rational Point","stem":"Use dimension=3, raw polynomial terms=[[2, [2, 1, 0]], [3, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[2/1, 3/2, 4/3]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [3, [0, 1, 1]], [2, [2, 1, 0]]], [2, 3/2, 4/3], [[[4, [1, 1, 0]]], [[3, [0, 0, 1]], [2, [2, 0, 0]]], [[3, [0, 1, 0]]]], [12, 12, 9/2], exact_rational_coordinate_evaluation)","explanation":"Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [3, [0, 1, 1]], [2, [2, 1, 0]]], [2, 3/2, 4/3], [[[4, [1, 1, 0]]], [[3, [0, 0, 1]], [2, [2, 0, 0]]], [[3, [0, 1, 0]]]], [12, 12, 9/2], exact_rational_coordinate_evaluation)."},"skills":["Evaluate Every Coordinate Derivative at the Same Exact Rational Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0028","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"084751bbeac1ca508df060186243f300692c97001c960dc03a3adb33cd38f457","family_id":"qf-mod44-gradient-rational-point","seed":202611136028,"semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":2,\"point\":[[3,1],[2,1]],\"terms\":[[[1,1],[0,0]],[[3,1],[2,1]],[[4,1],[0,1]]]}"},"fields":{"title":"Evaluate Every Coordinate Derivative at the Same Exact Rational Point","stem":"Use dimension=2, raw polynomial terms=[[3, [2, 1]], [4, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[3/1, 4/2]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [4, [0, 1]], [3, [2, 1]]], [3, 2], [[[6, [1, 1]]], [[4, [0, 0]], [3, [2, 0]]]], [36, 31], exact_rational_coordinate_evaluation)","explanation":"Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [4, [0, 1]], [3, [2, 1]]], [3, 2], [[[6, [1, 1]]], [[4, [0, 0]], [3, [2, 0]]]], [36, 31], exact_rational_coordinate_evaluation)."},"skills":["Evaluate Every Coordinate Derivative at the Same Exact Rational Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0029","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"970557ec18c27ec9fec4e9470df46a08c4f2bae69843fc2dcc74a5ffe6fe888c","family_id":"qf-mod44-gradient-rational-point","seed":202611136029,"semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":3,\"point\":[[4,1],[5,2],[2,1]],\"terms\":[[[1,1],[0,0,0]],[[4,1],[2,1,0]],[[5,1],[0,1,1]]]}"},"fields":{"title":"Evaluate Every Coordinate Derivative at the Same Exact Rational Point","stem":"Use dimension=3, raw polynomial terms=[[4, [2, 1, 0]], [5, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[4/1, 5/2, 6/3]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [5, [0, 1, 1]], [4, [2, 1, 0]]], [4, 5/2, 2], [[[8, [1, 1, 0]]], [[5, [0, 0, 1]], [4, [2, 0, 0]]], [[5, [0, 1, 0]]]], [80, 74, 25/2], exact_rational_coordinate_evaluation)","explanation":"Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [5, [0, 1, 1]], [4, [2, 1, 0]]], [4, 5/2, 2], [[[8, [1, 1, 0]]], [[5, [0, 0, 1]], [4, [2, 0, 0]]], [[5, [0, 1, 0]]]], [80, 74, 25/2], exact_rational_coordinate_evaluation)."},"skills":["Evaluate Every Coordinate Derivative at the Same Exact Rational Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0030","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe4f15c408d8508cba561513ad5c7f647c3fc1149180fd69634673ad4173d095","family_id":"qf-mod44-gradient-rational-point","seed":202611136030,"semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":2,\"point\":[[5,1],[3,1]],\"terms\":[[[1,1],[0,0]],[[5,1],[2,1]],[[6,1],[0,1]]]}"},"fields":{"title":"Evaluate Every Coordinate Derivative at the Same Exact Rational Point","stem":"Use dimension=2, raw polynomial terms=[[5, [2, 1]], [6, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[5/1, 6/2]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [6, [0, 1]], [5, [2, 1]]], [5, 3], [[[10, [1, 1]]], [[6, [0, 0]], [5, [2, 0]]]], [150, 131], exact_rational_coordinate_evaluation)","explanation":"Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [6, [0, 1]], [5, [2, 1]]], [5, 3], [[[10, [1, 1]]], [[6, [0, 0]], [5, [2, 0]]]], [150, 131], exact_rational_coordinate_evaluation)."},"skills":["Evaluate Every Coordinate Derivative at the Same Exact Rational Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0031","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"490a0dcfd685d975cd9ed2d69f9e5e4c949f0ca84fe8f90d46a12e0f15c9f63b","family_id":"qf-mod44-gradient-rational-point","seed":202611136031,"semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":3,\"point\":[[6,1],[7,2],[8,3]],\"terms\":[[[1,1],[0,0,0]],[[6,1],[2,1,0]],[[7,1],[0,1,1]]]}"},"fields":{"title":"Evaluate Every Coordinate Derivative at the Same Exact Rational Point","stem":"Use dimension=3, raw polynomial terms=[[6, [2, 1, 0]], [7, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[6/1, 7/2, 8/3]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [7, [0, 1, 1]], [6, [2, 1, 0]]], [6, 7/2, 8/3], [[[12, [1, 1, 0]]], [[7, [0, 0, 1]], [6, [2, 0, 0]]], [[7, [0, 1, 0]]]], [252, 704/3, 49/2], exact_rational_coordinate_evaluation)","explanation":"Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [7, [0, 1, 1]], [6, [2, 1, 0]]], [6, 7/2, 8/3], [[[12, [1, 1, 0]]], [[7, [0, 0, 1]], [6, [2, 0, 0]]], [[7, [0, 1, 0]]]], [252, 704/3, 49/2], exact_rational_coordinate_evaluation)."},"skills":["Evaluate Every Coordinate Derivative at the Same Exact Rational Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0032","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ffe49fe2d930ea7536e8d52466441922f1e33318b4b97f67587c04bad211699","family_id":"qf-mod44-gradient-rational-point","seed":202611136032,"semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":2,\"point\":[[7,1],[4,1]],\"terms\":[[[1,1],[0,0]],[[7,1],[2,1]],[[8,1],[0,1]]]}"},"fields":{"title":"Evaluate Every Coordinate Derivative at the Same Exact Rational Point","stem":"Use dimension=2, raw polynomial terms=[[7, [2, 1]], [8, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[7/1, 8/2]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [8, [0, 1]], [7, [2, 1]]], [7, 4], [[[14, [1, 1]]], [[8, [0, 0]], [7, [2, 0]]]], [392, 351], exact_rational_coordinate_evaluation)","explanation":"Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [8, [0, 1]], [7, [2, 1]]], [7, 4], [[[14, [1, 1]]], [[8, [0, 0]], [7, [2, 0]]]], [392, 351], exact_rational_coordinate_evaluation)."},"skills":["Evaluate Every Coordinate Derivative at the Same Exact Rational Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0033","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3246ae517fdab9ffd70df0f3eb77aa7866596d1a7e8dce52b49716eac3b355b5","family_id":"qf-mod44-gradient-rational-point","seed":202611136033,"semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":3,\"point\":[[8,1],[9,2],[10,3]],\"terms\":[[[1,1],[0,0,0]],[[8,1],[2,1,0]],[[9,1],[0,1,1]]]}"},"fields":{"title":"Evaluate Every Coordinate Derivative at the Same Exact Rational Point","stem":"Use dimension=3, raw polynomial terms=[[8, [2, 1, 0]], [9, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[8/1, 9/2, 10/3]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [9, [0, 1, 1]], [8, [2, 1, 0]]], [8, 9/2, 10/3], [[[16, [1, 1, 0]]], [[9, [0, 0, 1]], [8, [2, 0, 0]]], [[9, [0, 1, 0]]]], [576, 542, 81/2], exact_rational_coordinate_evaluation)","explanation":"Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [9, [0, 1, 1]], [8, [2, 1, 0]]], [8, 9/2, 10/3], [[[16, [1, 1, 0]]], [[9, [0, 0, 1]], [8, [2, 0, 0]]], [[9, [0, 1, 0]]]], [576, 542, 81/2], exact_rational_coordinate_evaluation)."},"skills":["Evaluate Every Coordinate Derivative at the Same Exact Rational Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0034","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32ec743aa7961beb66344ca97f638e343925313d9f15fe88ec9fdc7b9e31b0c1","family_id":"qf-mod44-gradient-rational-point","seed":202611136034,"semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":2,\"point\":[[9,1],[5,1]],\"terms\":[[[1,1],[0,0]],[[10,1],[0,1]],[[9,1],[2,1]]]}"},"fields":{"title":"Evaluate Every Coordinate Derivative at the Same Exact Rational Point","stem":"Use dimension=2, raw polynomial terms=[[9, [2, 1]], [10, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[9/1, 10/2]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [10, [0, 1]], [9, [2, 1]]], [9, 5], [[[18, [1, 1]]], [[10, [0, 0]], [9, [2, 0]]]], [810, 739], exact_rational_coordinate_evaluation)","explanation":"Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [10, [0, 1]], [9, [2, 1]]], [9, 5], [[[18, [1, 1]]], [[10, [0, 0]], [9, [2, 0]]]], [810, 739], exact_rational_coordinate_evaluation)."},"skills":["Evaluate Every Coordinate Derivative at the Same Exact Rational Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0035","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"769aed6d3f3189e15b108b4accdcb8b5ad2de95bc3c03f320a32dd2653b475fd","family_id":"qf-mod44-gradient-rational-point","seed":202611136035,"semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":3,\"point\":[[10,1],[11,2],[4,1]],\"terms\":[[[1,1],[0,0,0]],[[10,1],[2,1,0]],[[11,1],[0,1,1]]]}"},"fields":{"title":"Evaluate Every Coordinate Derivative at the Same Exact Rational Point","stem":"Use dimension=3, raw polynomial terms=[[10, [2, 1, 0]], [11, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[10/1, 11/2, 12/3]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [11, [0, 1, 1]], [10, [2, 1, 0]]], [10, 11/2, 4], [[[20, [1, 1, 0]]], [[11, [0, 0, 1]], [10, [2, 0, 0]]], [[11, [0, 1, 0]]]], [1100, 1044, 121/2], exact_rational_coordinate_evaluation)","explanation":"Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [11, [0, 1, 1]], [10, [2, 1, 0]]], [10, 11/2, 4], [[[20, [1, 1, 0]]], [[11, [0, 0, 1]], [10, [2, 0, 0]]], [[11, [0, 1, 0]]]], [1100, 1044, 121/2], exact_rational_coordinate_evaluation)."},"skills":["Evaluate Every Coordinate Derivative at the Same Exact Rational Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0036","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ede52c854bc2cf856c22751f075ab298e694b296371893a10426b0ce6568c388","family_id":"qf-mod44-gradient-rational-point","seed":202611136036,"semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":2,\"point\":[[11,1],[6,1]],\"terms\":[[[1,1],[0,0]],[[11,1],[2,1]],[[12,1],[0,1]]]}"},"fields":{"title":"Evaluate Every Coordinate Derivative at the Same Exact Rational Point","stem":"Use dimension=2, raw polynomial terms=[[11, [2, 1]], [12, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[11/1, 12/2]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [12, [0, 1]], [11, [2, 1]]], [11, 6], [[[22, [1, 1]]], [[12, [0, 0]], [11, [2, 0]]]], [1452, 1343], exact_rational_coordinate_evaluation)","explanation":"Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [12, [0, 1]], [11, [2, 1]]], [11, 6], [[[22, [1, 1]]], [[12, [0, 0]], [11, [2, 0]]]], [1452, 1343], exact_rational_coordinate_evaluation)."},"skills":["Evaluate Every Coordinate Derivative at the Same Exact Rational Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0037","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f62da81913ac1ed65f2cc771979f273b5ab509ec139492d60305011a1e7829c","family_id":"qf-mod44-gradient-rational-point","seed":202611136037,"semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":3,\"point\":[[12,1],[13,2],[14,3]],\"terms\":[[[1,1],[0,0,0]],[[12,1],[2,1,0]],[[13,1],[0,1,1]]]}"},"fields":{"title":"Evaluate Every Coordinate Derivative at the Same Exact Rational Point","stem":"Use dimension=3, raw polynomial terms=[[12, [2, 1, 0]], [13, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[12/1, 13/2, 14/3]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [13, [0, 1, 1]], [12, [2, 1, 0]]], [12, 13/2, 14/3], [[[24, [1, 1, 0]]], [[13, [0, 0, 1]], [12, [2, 0, 0]]], [[13, [0, 1, 0]]]], [1872, 5366/3, 169/2], exact_rational_coordinate_evaluation)","explanation":"Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [13, [0, 1, 1]], [12, [2, 1, 0]]], [12, 13/2, 14/3], [[[24, [1, 1, 0]]], [[13, [0, 0, 1]], [12, [2, 0, 0]]], [[13, [0, 1, 0]]]], [1872, 5366/3, 169/2], exact_rational_coordinate_evaluation)."},"skills":["Evaluate Every Coordinate Derivative at the Same Exact Rational Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0038","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3085ae111f3925fff01de71552c0c4b1253f055d3984294030f1a1aed2c6716c","family_id":"qf-mod44-gradient-rational-point","seed":202611136038,"semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":2,\"point\":[[13,1],[7,1]],\"terms\":[[[1,1],[0,0]],[[13,1],[2,1]],[[14,1],[0,1]]]}"},"fields":{"title":"Evaluate Every Coordinate Derivative at the Same Exact Rational Point","stem":"Use dimension=2, raw polynomial terms=[[13, [2, 1]], [14, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[13/1, 14/2]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [14, [0, 1]], [13, [2, 1]]], [13, 7], [[[26, [1, 1]]], [[14, [0, 0]], [13, [2, 0]]]], [2366, 2211], exact_rational_coordinate_evaluation)","explanation":"Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [14, [0, 1]], [13, [2, 1]]], [13, 7], [[[26, [1, 1]]], [[14, [0, 0]], [13, [2, 0]]]], [2366, 2211], exact_rational_coordinate_evaluation)."},"skills":["Evaluate Every Coordinate Derivative at the Same Exact Rational Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0039","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f280cd22d77538096303de0299cf73f0d1d7f8408715bd0a5d765636eac51c56","family_id":"qf-mod44-gradient-rational-point","seed":202611136039,"semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":3,\"point\":[[14,1],[15,2],[16,3]],\"terms\":[[[1,1],[0,0,0]],[[14,1],[2,1,0]],[[15,1],[0,1,1]]]}"},"fields":{"title":"Evaluate Every Coordinate Derivative at the Same Exact Rational Point","stem":"Use dimension=3, raw polynomial terms=[[14, [2, 1, 0]], [15, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[14/1, 15/2, 16/3]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [15, [0, 1, 1]], [14, [2, 1, 0]]], [14, 15/2, 16/3], [[[28, [1, 1, 0]]], [[15, [0, 0, 1]], [14, [2, 0, 0]]], [[15, [0, 1, 0]]]], [2940, 2824, 225/2], exact_rational_coordinate_evaluation)","explanation":"Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [15, [0, 1, 1]], [14, [2, 1, 0]]], [14, 15/2, 16/3], [[[28, [1, 1, 0]]], [[15, [0, 0, 1]], [14, [2, 0, 0]]], [[15, [0, 1, 0]]]], [2940, 2824, 225/2], exact_rational_coordinate_evaluation)."},"skills":["Evaluate Every Coordinate Derivative at the Same Exact Rational Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0040","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e7284c1a14ec6642c95488f0fa340855e2b368a25f3f5b79cc58b7791e589da","family_id":"qf-mod44-gradient-rational-point","seed":202611136040,"semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":2,\"point\":[[15,1],[8,1]],\"terms\":[[[1,1],[0,0]],[[15,1],[2,1]],[[16,1],[0,1]]]}"},"fields":{"title":"Evaluate Every Coordinate Derivative at the Same Exact Rational Point","stem":"Use dimension=2, raw polynomial terms=[[15, [2, 1]], [16, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[15/1, 16/2]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [16, [0, 1]], [15, [2, 1]]], [15, 8], [[[30, [1, 1]]], [[16, [0, 0]], [15, [2, 0]]]], [3600, 3391], exact_rational_coordinate_evaluation)","explanation":"Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [16, [0, 1]], [15, [2, 1]]], [15, 8], [[[30, [1, 1]]], [[16, [0, 0]], [15, [2, 0]]]], [3600, 3391], exact_rational_coordinate_evaluation)."},"skills":["Evaluate Every Coordinate Derivative at the Same Exact Rational Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0041","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe5dac65a45cf475342bc18aacfeba201c27e141f01fb36a9cf2e36ed670d5ce","family_id":"qf-mod44-gradient-rational-point","seed":202611136041,"semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":3,\"point\":[[16,1],[17,2],[6,1]],\"terms\":[[[1,1],[0,0,0]],[[16,1],[2,1,0]],[[17,1],[0,1,1]]]}"},"fields":{"title":"Evaluate Every Coordinate Derivative at the Same Exact Rational Point","stem":"Use dimension=3, raw polynomial terms=[[16, [2, 1, 0]], [17, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[16/1, 17/2, 18/3]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [17, [0, 1, 1]], [16, [2, 1, 0]]], [16, 17/2, 6], [[[32, [1, 1, 0]]], [[17, [0, 0, 1]], [16, [2, 0, 0]]], [[17, [0, 1, 0]]]], [4352, 4198, 289/2], exact_rational_coordinate_evaluation)","explanation":"Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [17, [0, 1, 1]], [16, [2, 1, 0]]], [16, 17/2, 6], [[[32, [1, 1, 0]]], [[17, [0, 0, 1]], [16, [2, 0, 0]]], [[17, [0, 1, 0]]]], [4352, 4198, 289/2], exact_rational_coordinate_evaluation)."},"skills":["Evaluate Every Coordinate Derivative at the Same Exact Rational Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0042","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b763da005e7322632c9c87d2b4d8d15646a56c8169c7de160ae469d41bf2b92c","family_id":"qf-mod44-gradient-rational-point","seed":202611136042,"semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":2,\"point\":[[17,1],[9,1]],\"terms\":[[[1,1],[0,0]],[[17,1],[2,1]],[[18,1],[0,1]]]}"},"fields":{"title":"Evaluate Every Coordinate Derivative at the Same Exact Rational Point","stem":"Use dimension=2, raw polynomial terms=[[17, [2, 1]], [18, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[17/1, 18/2]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [18, [0, 1]], [17, [2, 1]]], [17, 9], [[[34, [1, 1]]], [[18, [0, 0]], [17, [2, 0]]]], [5202, 4931], exact_rational_coordinate_evaluation)","explanation":"Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [18, [0, 1]], [17, [2, 1]]], [17, 9], [[[34, [1, 1]]], [[18, [0, 0]], [17, [2, 0]]]], [5202, 4931], exact_rational_coordinate_evaluation)."},"skills":["Evaluate Every Coordinate Derivative at the Same Exact Rational Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0043","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f12bfc74d0ec9961bfe0b05844a436c15efc7e7b827f2cb7f78bd78b4f06f1a","family_id":"qf-mod44-gradient-rational-point","seed":202611136043,"semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":3,\"point\":[[18,1],[19,2],[20,3]],\"terms\":[[[1,1],[0,0,0]],[[18,1],[2,1,0]],[[19,1],[0,1,1]]]}"},"fields":{"title":"Evaluate Every Coordinate Derivative at the Same Exact Rational Point","stem":"Use dimension=3, raw polynomial terms=[[18, [2, 1, 0]], [19, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[18/1, 19/2, 20/3]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [19, [0, 1, 1]], [18, [2, 1, 0]]], [18, 19/2, 20/3], [[[36, [1, 1, 0]]], [[19, [0, 0, 1]], [18, [2, 0, 0]]], [[19, [0, 1, 0]]]], [6156, 17876/3, 361/2], exact_rational_coordinate_evaluation)","explanation":"Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [19, [0, 1, 1]], [18, [2, 1, 0]]], [18, 19/2, 20/3], [[[36, [1, 1, 0]]], [[19, [0, 0, 1]], [18, [2, 0, 0]]], [[19, [0, 1, 0]]]], [6156, 17876/3, 361/2], exact_rational_coordinate_evaluation)."},"skills":["Evaluate Every Coordinate Derivative at the Same Exact Rational Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0044","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50880659ba093390e563d6becfbc284f25c0f0940a79fc5dbbc9c0cf4c067359","family_id":"qf-mod44-gradient-rational-point","seed":202611136044,"semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":2,\"point\":[[19,1],[10,1]],\"terms\":[[[1,1],[0,0]],[[19,1],[2,1]],[[20,1],[0,1]]]}"},"fields":{"title":"Evaluate Every Coordinate Derivative at the Same Exact Rational Point","stem":"Use dimension=2, raw polynomial terms=[[19, [2, 1]], [20, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[19/1, 20/2]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [20, [0, 1]], [19, [2, 1]]], [19, 10], [[[38, [1, 1]]], [[20, [0, 0]], [19, [2, 0]]]], [7220, 6879], exact_rational_coordinate_evaluation)","explanation":"Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [20, [0, 1]], [19, [2, 1]]], [19, 10], [[[38, [1, 1]]], [[20, [0, 0]], [19, [2, 0]]]], [7220, 6879], exact_rational_coordinate_evaluation)."},"skills":["Evaluate Every Coordinate Derivative at the Same Exact Rational Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0045","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4485bd5fb3f516e726d7dde06be6974cd85cfae1e76965a5b86299e569a03e2e","family_id":"qf-mod44-gradient-rational-point","seed":202611136045,"semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":3,\"point\":[[20,1],[21,2],[22,3]],\"terms\":[[[1,1],[0,0,0]],[[20,1],[2,1,0]],[[21,1],[0,1,1]]]}"},"fields":{"title":"Evaluate Every Coordinate Derivative at the Same Exact Rational Point","stem":"Use dimension=3, raw polynomial terms=[[20, [2, 1, 0]], [21, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[20/1, 21/2, 22/3]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [21, [0, 1, 1]], [20, [2, 1, 0]]], [20, 21/2, 22/3], [[[40, [1, 1, 0]]], [[21, [0, 0, 1]], [20, [2, 0, 0]]], [[21, [0, 1, 0]]]], [8400, 8154, 441/2], exact_rational_coordinate_evaluation)","explanation":"Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [21, [0, 1, 1]], [20, [2, 1, 0]]], [20, 21/2, 22/3], [[[40, [1, 1, 0]]], [[21, [0, 0, 1]], [20, [2, 0, 0]]], [[21, [0, 1, 0]]]], [8400, 8154, 441/2], exact_rational_coordinate_evaluation)."},"skills":["Evaluate Every Coordinate Derivative at the Same Exact Rational Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0046","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00e30467af06d2640122247e26924dee9166ae0fe48245f2130e16c8c6415224","family_id":"qf-mod44-gradient-rational-point","seed":202611136046,"semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":2,\"point\":[[21,1],[11,1]],\"terms\":[[[1,1],[0,0]],[[21,1],[2,1]],[[22,1],[0,1]]]}"},"fields":{"title":"Evaluate Every Coordinate Derivative at the Same Exact Rational Point","stem":"Use dimension=2, raw polynomial terms=[[21, [2, 1]], [22, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[21/1, 22/2]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [22, [0, 1]], [21, [2, 1]]], [21, 11], [[[42, [1, 1]]], [[22, [0, 0]], [21, [2, 0]]]], [9702, 9283], exact_rational_coordinate_evaluation)","explanation":"Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [22, [0, 1]], [21, [2, 1]]], [21, 11], [[[42, [1, 1]]], [[22, [0, 0]], [21, [2, 0]]]], [9702, 9283], exact_rational_coordinate_evaluation)."},"skills":["Evaluate Every Coordinate Derivative at the Same Exact Rational Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0047","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17e750417ad2afd831154488f4bc96af967d21daa06fd80e9a735780ae04a0aa","family_id":"qf-mod44-gradient-rational-point","seed":202611136047,"semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":3,\"point\":[[22,1],[23,2],[8,1]],\"terms\":[[[1,1],[0,0,0]],[[22,1],[2,1,0]],[[23,1],[0,1,1]]]}"},"fields":{"title":"Evaluate Every Coordinate Derivative at the Same Exact Rational Point","stem":"Use dimension=3, raw polynomial terms=[[22, [2, 1, 0]], [23, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[22/1, 23/2, 24/3]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [23, [0, 1, 1]], [22, [2, 1, 0]]], [22, 23/2, 8], [[[44, [1, 1, 0]]], [[23, [0, 0, 1]], [22, [2, 0, 0]]], [[23, [0, 1, 0]]]], [11132, 10832, 529/2], exact_rational_coordinate_evaluation)","explanation":"Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [23, [0, 1, 1]], [22, [2, 1, 0]]], [22, 23/2, 8], [[[44, [1, 1, 0]]], [[23, [0, 0, 1]], [22, [2, 0, 0]]], [[23, [0, 1, 0]]]], [11132, 10832, 529/2], exact_rational_coordinate_evaluation)."},"skills":["Evaluate Every Coordinate Derivative at the Same Exact Rational Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0048","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69204437945e54e31db7b4c4477c207d6db364b571a085bc6bbc97879b3161d1","family_id":"qf-mod44-gradient-rational-point","seed":202611136048,"semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":2,\"point\":[[23,1],[12,1]],\"terms\":[[[1,1],[0,0]],[[23,1],[2,1]],[[24,1],[0,1]]]}"},"fields":{"title":"Evaluate Every Coordinate Derivative at the Same Exact Rational Point","stem":"Use dimension=2, raw polynomial terms=[[23, [2, 1]], [24, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[23/1, 24/2]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [24, [0, 1]], [23, [2, 1]]], [23, 12], [[[46, [1, 1]]], [[24, [0, 0]], [23, [2, 0]]]], [12696, 12191], exact_rational_coordinate_evaluation)","explanation":"Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [24, [0, 1]], [23, [2, 1]]], [23, 12], [[[46, [1, 1]]], [[24, [0, 0]], [23, [2, 0]]]], [12696, 12191], exact_rational_coordinate_evaluation)."},"skills":["Evaluate Every Coordinate Derivative at the Same Exact Rational Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0049","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b16ee6c90271008daf21bcca36bb8f96cfe04a3d073c85e11c145417b4eac004","family_id":"qf-mod44-gradient-rational-point","seed":202611136049,"semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":3,\"point\":[[24,1],[25,2],[26,3]],\"terms\":[[[1,1],[0,0,0]],[[24,1],[2,1,0]],[[25,1],[0,1,1]]]}"},"fields":{"title":"Evaluate Every Coordinate Derivative at the Same Exact Rational Point","stem":"Use dimension=3, raw polynomial terms=[[24, [2, 1, 0]], [25, [0, 1, 1]], [1, [0, 0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[24/1, 25/2, 26/3]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).","answer":{"kind":"expression","value":"(3, [[1, [0, 0, 0]], [25, [0, 1, 1]], [24, [2, 1, 0]]], [24, 25/2, 26/3], [[[48, [1, 1, 0]]], [[25, [0, 0, 1]], [24, [2, 0, 0]]], [[25, [0, 1, 0]]]], [14400, 42122/3, 625/2], exact_rational_coordinate_evaluation)","explanation":"Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (3, [[1, [0, 0, 0]], [25, [0, 1, 1]], [24, [2, 1, 0]]], [24, 25/2, 26/3], [[[48, [1, 1, 0]]], [[25, [0, 0, 1]], [24, [2, 0, 0]]], [[25, [0, 1, 0]]]], [14400, 42122/3, 625/2], exact_rational_coordinate_evaluation)."},"skills":["Evaluate Every Coordinate Derivative at the Same Exact Rational Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0050","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d7644ddcb746d7d874205c37b75c8d9b9e01d9ce77d751a3dddd1c7ff19a20d","family_id":"qf-mod44-gradient-rational-point","seed":202611136050,"semantic_template":"qf-mod44-gradient-rational-point/{\"dimension\":2,\"point\":[[25,1],[13,1]],\"terms\":[[[1,1],[0,0]],[[25,1],[2,1]],[[26,1],[0,1]]]}"},"fields":{"title":"Evaluate Every Coordinate Derivative at the Same Exact Rational Point","stem":"Use dimension=2, raw polynomial terms=[[25, [2, 1]], [26, [0, 1]], [1, [0, 0]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[25/1, 26/2]. Give (dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate).","answer":{"kind":"expression","value":"(2, [[1, [0, 0]], [26, [0, 1]], [25, [2, 1]]], [25, 13], [[[50, [1, 1]]], [[26, [0, 0]], [25, [2, 0]]]], [16250, 15651], exact_rational_coordinate_evaluation)","explanation":"Evaluate each symbolic partial at this same point with exact rational arithmetic. Keep the component order and dimension; do not replace the ordered Euclidean gradient by a sampled directional derivative. The exact result, retaining the requested order, is (2, [[1, [0, 0]], [26, [0, 1]], [25, [2, 1]]], [25, 13], [[[50, [1, 1]]], [[26, [0, 0]], [25, [2, 0]]]], [16250, 15651], exact_rational_coordinate_evaluation)."},"skills":["Evaluate Every Coordinate Derivative at the Same Exact Rational Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0051","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"308a98326ac29518fa2f6f2ef35a9fe2d1e87016a2f5c83e8a316817bb5fc06e","family_id":"qf-mod44-gradient-stationary-audit","seed":202611136051,"semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[1,1],[2,0]],[[2,1],[0,2]]]}"},"fields":{"title":"A Zero Gradient Certifies Stationarity, Not an Automatic Extremum","stem":"Use dimension=2, raw polynomial terms=[[1, [2, 0]], [2, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)","explanation":"Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)."},"skills":["A Zero Gradient Certifies Stationarity, Not an Automatic Extremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0052","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4a6540ef5155cef1d1d67d9b080ee0649190caa53d3de8abe2613643b023445","family_id":"qf-mod44-gradient-stationary-audit","seed":202611136052,"semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[2,1],[2,0]],[[3,1],[0,2]]]}"},"fields":{"title":"A Zero Gradient Certifies Stationarity, Not an Automatic Extremum","stem":"Use dimension=2, raw polynomial terms=[[2, [2, 0]], [3, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)","explanation":"Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)."},"skills":["A Zero Gradient Certifies Stationarity, Not an Automatic Extremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0053","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5061c6c12a8cac6b51276d1c4df4e8fb0505a0b267e85cc1e49387d9d9fc3f8","family_id":"qf-mod44-gradient-stationary-audit","seed":202611136053,"semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[3,1],[2,0]],[[4,1],[0,2]]]}"},"fields":{"title":"A Zero Gradient Certifies Stationarity, Not an Automatic Extremum","stem":"Use dimension=2, raw polynomial terms=[[3, [2, 0]], [4, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)","explanation":"Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)."},"skills":["A Zero Gradient Certifies Stationarity, Not an Automatic Extremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0054","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f62dcc9699f95a01b57729f64a0267192472121a00697ac8ff43445289b027fb","family_id":"qf-mod44-gradient-stationary-audit","seed":202611136054,"semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[4,1],[2,0]],[[5,1],[0,2]]]}"},"fields":{"title":"A Zero Gradient Certifies Stationarity, Not an Automatic Extremum","stem":"Use dimension=2, raw polynomial terms=[[4, [2, 0]], [5, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)","explanation":"Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)."},"skills":["A Zero Gradient Certifies Stationarity, Not an Automatic Extremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0055","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"304c2101a097d0d75d3fec865c33f7d5faba5eac5a3c9383a3e08d201ded352e","family_id":"qf-mod44-gradient-stationary-audit","seed":202611136055,"semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[5,1],[2,0]],[[6,1],[0,2]]]}"},"fields":{"title":"A Zero Gradient Certifies Stationarity, Not an Automatic Extremum","stem":"Use dimension=2, raw polynomial terms=[[5, [2, 0]], [6, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)","explanation":"Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)."},"skills":["A Zero Gradient Certifies Stationarity, Not an Automatic Extremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0056","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"475a23fbff3c96b90edb9ae68ce28e2cef1a50d000fce7b0794cd5ad5aefd735","family_id":"qf-mod44-gradient-stationary-audit","seed":202611136056,"semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[6,1],[2,0]],[[7,1],[0,2]]]}"},"fields":{"title":"A Zero Gradient Certifies Stationarity, Not an Automatic Extremum","stem":"Use dimension=2, raw polynomial terms=[[6, [2, 0]], [7, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)","explanation":"Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)."},"skills":["A Zero Gradient Certifies Stationarity, Not an Automatic Extremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0057","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5443936179c436473ce4af9be88a0964373cda89a123ae2fffbbb5ce851f6d5d","family_id":"qf-mod44-gradient-stationary-audit","seed":202611136057,"semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[7,1],[2,0]],[[8,1],[0,2]]]}"},"fields":{"title":"A Zero Gradient Certifies Stationarity, Not an Automatic Extremum","stem":"Use dimension=2, raw polynomial terms=[[7, [2, 0]], [8, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)","explanation":"Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)."},"skills":["A Zero Gradient Certifies Stationarity, Not an Automatic Extremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0058","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e487d6b3458199a26a07f4ca9b409c943426399f2d59da83e3716e0d489428c","family_id":"qf-mod44-gradient-stationary-audit","seed":202611136058,"semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[8,1],[2,0]],[[9,1],[0,2]]]}"},"fields":{"title":"A Zero Gradient Certifies Stationarity, Not an Automatic Extremum","stem":"Use dimension=2, raw polynomial terms=[[8, [2, 0]], [9, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)","explanation":"Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)."},"skills":["A Zero Gradient Certifies Stationarity, Not an Automatic Extremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0059","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b47744f23fdc5e3f3d419e4892127eceaa8c2fbccaf51899c901ee8bcd40433","family_id":"qf-mod44-gradient-stationary-audit","seed":202611136059,"semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[10,1],[0,2]],[[9,1],[2,0]]]}"},"fields":{"title":"A Zero Gradient Certifies Stationarity, Not an Automatic Extremum","stem":"Use dimension=2, raw polynomial terms=[[9, [2, 0]], [10, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)","explanation":"Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)."},"skills":["A Zero Gradient Certifies Stationarity, Not an Automatic Extremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0060","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"175e357d908783dad2e0af3ed83411bac7173a182687f4a891499cb784c5838b","family_id":"qf-mod44-gradient-stationary-audit","seed":202611136060,"semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[10,1],[2,0]],[[11,1],[0,2]]]}"},"fields":{"title":"A Zero Gradient Certifies Stationarity, Not an Automatic Extremum","stem":"Use dimension=2, raw polynomial terms=[[10, [2, 0]], [11, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)","explanation":"Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)."},"skills":["A Zero Gradient Certifies Stationarity, Not an Automatic Extremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0061","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe2f9cb7741422081b5c145a23c6747a94483aa6a15bb8a3e78327299f986b53","family_id":"qf-mod44-gradient-stationary-audit","seed":202611136061,"semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[11,1],[2,0]],[[12,1],[0,2]]]}"},"fields":{"title":"A Zero Gradient Certifies Stationarity, Not an Automatic Extremum","stem":"Use dimension=2, raw polynomial terms=[[11, [2, 0]], [12, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)","explanation":"Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)."},"skills":["A Zero Gradient Certifies Stationarity, Not an Automatic Extremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0062","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe033a3acef7380b96f460069a2ca8f5b86aba86d7513548bd762606cf5a5e45","family_id":"qf-mod44-gradient-stationary-audit","seed":202611136062,"semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[12,1],[2,0]],[[13,1],[0,2]]]}"},"fields":{"title":"A Zero Gradient Certifies Stationarity, Not an Automatic Extremum","stem":"Use dimension=2, raw polynomial terms=[[12, [2, 0]], [13, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)","explanation":"Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)."},"skills":["A Zero Gradient Certifies Stationarity, Not an Automatic Extremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0063","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f77e25bd946e8dfc5a363b3a31860a1775b5bd361c0b4e9c84d6aa490306ac8","family_id":"qf-mod44-gradient-stationary-audit","seed":202611136063,"semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[13,1],[2,0]],[[14,1],[0,2]]]}"},"fields":{"title":"A Zero Gradient Certifies Stationarity, Not an Automatic Extremum","stem":"Use dimension=2, raw polynomial terms=[[13, [2, 0]], [14, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)","explanation":"Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)."},"skills":["A Zero Gradient Certifies Stationarity, Not an Automatic Extremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0064","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77987c832ee5053e66838f30006c59299a9355e0a515c0a141402372687754ec","family_id":"qf-mod44-gradient-stationary-audit","seed":202611136064,"semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[14,1],[2,0]],[[15,1],[0,2]]]}"},"fields":{"title":"A Zero Gradient Certifies Stationarity, Not an Automatic Extremum","stem":"Use dimension=2, raw polynomial terms=[[14, [2, 0]], [15, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)","explanation":"Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)."},"skills":["A Zero Gradient Certifies Stationarity, Not an Automatic Extremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0065","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57002f8ecdce024b3cfe9eb624156a803f9b7c50807cef8d0c3772c884b3df52","family_id":"qf-mod44-gradient-stationary-audit","seed":202611136065,"semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[15,1],[2,0]],[[16,1],[0,2]]]}"},"fields":{"title":"A Zero Gradient Certifies Stationarity, Not an Automatic Extremum","stem":"Use dimension=2, raw polynomial terms=[[15, [2, 0]], [16, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)","explanation":"Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)."},"skills":["A Zero Gradient Certifies Stationarity, Not an Automatic Extremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0066","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f50cff16e68be47fdfbc21e59bf99cc19cb395f4c1e25859ea1190d7c70f9c67","family_id":"qf-mod44-gradient-stationary-audit","seed":202611136066,"semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[16,1],[2,0]],[[17,1],[0,2]]]}"},"fields":{"title":"A Zero Gradient Certifies Stationarity, Not an Automatic Extremum","stem":"Use dimension=2, raw polynomial terms=[[16, [2, 0]], [17, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)","explanation":"Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)."},"skills":["A Zero Gradient Certifies Stationarity, Not an Automatic Extremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0067","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e05a7566fbb6197917aa86224649a57b3088e73b3da76e9857c30affc487494","family_id":"qf-mod44-gradient-stationary-audit","seed":202611136067,"semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[17,1],[2,0]],[[18,1],[0,2]]]}"},"fields":{"title":"A Zero Gradient Certifies Stationarity, Not an Automatic Extremum","stem":"Use dimension=2, raw polynomial terms=[[17, [2, 0]], [18, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)","explanation":"Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)."},"skills":["A Zero Gradient Certifies Stationarity, Not an Automatic Extremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0068","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1137e29405c64933b0481702cd7276130d60662d7c752a08c4d49188263bea46","family_id":"qf-mod44-gradient-stationary-audit","seed":202611136068,"semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[18,1],[2,0]],[[19,1],[0,2]]]}"},"fields":{"title":"A Zero Gradient Certifies Stationarity, Not an Automatic Extremum","stem":"Use dimension=2, raw polynomial terms=[[18, [2, 0]], [19, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)","explanation":"Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)."},"skills":["A Zero Gradient Certifies Stationarity, Not an Automatic Extremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0069","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30ffbc399942d3de082be30d1113b75c98eb835c183e212f9d6a7a57b0edade4","family_id":"qf-mod44-gradient-stationary-audit","seed":202611136069,"semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[19,1],[2,0]],[[20,1],[0,2]]]}"},"fields":{"title":"A Zero Gradient Certifies Stationarity, Not an Automatic Extremum","stem":"Use dimension=2, raw polynomial terms=[[19, [2, 0]], [20, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)","explanation":"Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)."},"skills":["A Zero Gradient Certifies Stationarity, Not an Automatic Extremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0070","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"98532c710fe96cdbc49004f71bc4e5e8c469448a2ff0905befc2f224f96c4cdc","family_id":"qf-mod44-gradient-stationary-audit","seed":202611136070,"semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[20,1],[2,0]],[[21,1],[0,2]]]}"},"fields":{"title":"A Zero Gradient Certifies Stationarity, Not an Automatic Extremum","stem":"Use dimension=2, raw polynomial terms=[[20, [2, 0]], [21, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)","explanation":"Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)."},"skills":["A Zero Gradient Certifies Stationarity, Not an Automatic Extremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0071","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b6df62b64ce43f1aa8344b4134714d181cd2c0f94785505d47cd55d6524d2c84","family_id":"qf-mod44-gradient-stationary-audit","seed":202611136071,"semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[21,1],[2,0]],[[22,1],[0,2]]]}"},"fields":{"title":"A Zero Gradient Certifies Stationarity, Not an Automatic Extremum","stem":"Use dimension=2, raw polynomial terms=[[21, [2, 0]], [22, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)","explanation":"Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)."},"skills":["A Zero Gradient Certifies Stationarity, Not an Automatic Extremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0072","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bcf536261ec3948e4a2f33a62e26db9620bf05f892876c6d7a1cd5b8192d9270","family_id":"qf-mod44-gradient-stationary-audit","seed":202611136072,"semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[22,1],[2,0]],[[23,1],[0,2]]]}"},"fields":{"title":"A Zero Gradient Certifies Stationarity, Not an Automatic Extremum","stem":"Use dimension=2, raw polynomial terms=[[22, [2, 0]], [23, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)","explanation":"Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)."},"skills":["A Zero Gradient Certifies Stationarity, Not an Automatic Extremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0073","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f84c32e71028cdd9b48d186bfdf7b65747eb8b9ae4d1efad51021e2c0a90cae4","family_id":"qf-mod44-gradient-stationary-audit","seed":202611136073,"semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[23,1],[2,0]],[[24,1],[0,2]]]}"},"fields":{"title":"A Zero Gradient Certifies Stationarity, Not an Automatic Extremum","stem":"Use dimension=2, raw polynomial terms=[[23, [2, 0]], [24, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)","explanation":"Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)."},"skills":["A Zero Gradient Certifies Stationarity, Not an Automatic Extremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0074","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"585337d292ce7efd5f250081aeec5b07b8b5777d39fc389ade2b760270a0ba83","family_id":"qf-mod44-gradient-stationary-audit","seed":202611136074,"semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[24,1],[2,0]],[[25,1],[0,2]]]}"},"fields":{"title":"A Zero Gradient Certifies Stationarity, Not an Automatic Extremum","stem":"Use dimension=2, raw polynomial terms=[[24, [2, 0]], [25, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)","explanation":"Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)."},"skills":["A Zero Gradient Certifies Stationarity, Not an Automatic Extremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0075","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8451fc2e179b84a13a5c06a35bf9d0d495e23d2269877417e92e69c036029965","family_id":"qf-mod44-gradient-stationary-audit","seed":202611136075,"semantic_template":"qf-mod44-gradient-stationary-audit/{\"dimension\":2,\"point\":[[0,1],[0,1]],\"terms\":[[[25,1],[2,0]],[[26,1],[0,2]]]}"},"fields":{"title":"A Zero Gradient Certifies Stationarity, Not an Automatic Extremum","stem":"Use dimension=2, raw polynomial terms=[[25, [2, 0]], [26, [0, 2]]], each [rational coefficient,ordered exponent vector]. Coefficients may be integers or [numerator,denominator]. Exponent/point arities equal the dimension. Combine equal monomials and remove zero coefficients before the canonical symbolic output. Coordinate order and the standard Euclidean basis/metric are fixed. Point=[0, 0]. Give (dimension, point, gradient_values, stationary, extremum_certificate, boundary).","answer":{"kind":"expression","value":"(2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)","explanation":"Report all gradient components and whether all are0. The fixed No withholds an extremum certificate; it does not deny extrema in special cases such as a constant function. Nonzero gradient means not stationary here, and no descent/convergence/global-optimum claim is added. The exact result, retaining the requested order, is (2, [0, 0], [0, 0], Yes, No, gradient_zero_means_stationary_not_necessarily_extremum)."},"skills":["A Zero Gradient Certifies Stationarity, Not an Automatic Extremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0076","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07d4d7bb38620eec1355e304b49a3e39c17e09c3e81cdc762f4ec37c43259a65","family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136076,"semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"incomplete_coordinate_list\",\"scale\":[1,1]}"},"fields":{"title":"Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts","stem":"Use mode=incomplete_coordinate_list, positive scale=1, with the source-declared explicit counterexample for that mode. Give (mode, required_dimension, provided_component_descriptions, provided_count, complete_gradient, boundary).","answer":{"kind":"expression","value":"(incomplete_coordinate_list, 3, [1*partial_x, 1*partial_y], 2, No, missing_coordinate_component_is_not_a_gradient_in_R3)","explanation":"incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (incomplete_coordinate_list, 3, [1*partial_x, 1*partial_y], 2, No, missing_coordinate_component_is_not_a_gradient_in_R3)."},"skills":["Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0077","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da611211d394609e3595be260d936ed80ee1ae9baabac6e0966ede47181c3513","family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136077,"semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"nondifferentiable_norm\",\"scale\":[2,1]}"},"fields":{"title":"Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts","stem":"Use mode=nondifferentiable_norm, positive scale=2, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, coordinate_partials_exist, gradient_defined, boundary).","answer":{"kind":"expression","value":"(nondifferentiable_norm, f(x,y)=2*sqrt(x^2+y^2), [0, 0], No, No, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)","explanation":"incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (nondifferentiable_norm, f(x,y)=2*sqrt(x^2+y^2), [0, 0], No, No, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)."},"skills":["Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0078","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4c4c3380a000924292c5fcef23561f2b5d6e7cd9018be9da81ddd8b8eaeb061","family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136078,"semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"partials_exist_not_differentiable\",\"scale\":[3,1]}"},"fields":{"title":"Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts","stem":"Use mode=partials_exist_not_differentiable, positive scale=3, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, coordinate_partial_values, coordinate_partials_exist, diagonal_comparison, differentiable, boundary).","answer":{"kind":"expression","value":"(partials_exist_not_differentiable, f(x,y)=3*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [3, 0], Yes, along_y=x:f(x,x)=3/2*x_but_linear_candidate=3*x, No, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)","explanation":"incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (partials_exist_not_differentiable, f(x,y)=3*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [3, 0], Yes, along_y=x:f(x,x)=3/2*x_but_linear_candidate=3*x, No, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)."},"skills":["Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0079","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8dc78eb099843bece319efb61f9f2881a7323cfe70115dae1bb3f68fa26963d9","family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136079,"semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"saddle_stationary\",\"scale\":[4,1]}"},"fields":{"title":"Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts","stem":"Use mode=saddle_stationary, positive scale=4, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, gradient_values, stationary, extremum_certificate, boundary).","answer":{"kind":"expression","value":"(saddle_stationary, f(x,y)=4*(x^2-y^2), [0, 0], [0, 0], Yes, No, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)","explanation":"incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (saddle_stationary, f(x,y)=4*(x^2-y^2), [0, 0], [0, 0], Yes, No, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)."},"skills":["Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0080","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79cbf7703c2833e0fa8ee8c8992e092a9f9d165d0fbf9f97bacf94647cd48b0b","family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136080,"semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"incomplete_coordinate_list\",\"scale\":[5,1]}"},"fields":{"title":"Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts","stem":"Use mode=incomplete_coordinate_list, positive scale=5, with the source-declared explicit counterexample for that mode. Give (mode, required_dimension, provided_component_descriptions, provided_count, complete_gradient, boundary).","answer":{"kind":"expression","value":"(incomplete_coordinate_list, 3, [5*partial_x, 5*partial_y], 2, No, missing_coordinate_component_is_not_a_gradient_in_R3)","explanation":"incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (incomplete_coordinate_list, 3, [5*partial_x, 5*partial_y], 2, No, missing_coordinate_component_is_not_a_gradient_in_R3)."},"skills":["Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0081","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59374393c1d8fd3f29aab61765cef8f9e0e699ae2f22b844a05184d307fb6d57","family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136081,"semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"nondifferentiable_norm\",\"scale\":[6,1]}"},"fields":{"title":"Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts","stem":"Use mode=nondifferentiable_norm, positive scale=6, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, coordinate_partials_exist, gradient_defined, boundary).","answer":{"kind":"expression","value":"(nondifferentiable_norm, f(x,y)=6*sqrt(x^2+y^2), [0, 0], No, No, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)","explanation":"incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (nondifferentiable_norm, f(x,y)=6*sqrt(x^2+y^2), [0, 0], No, No, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)."},"skills":["Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0082","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5993e727dbe8150aebd143db81aba0edb4711a3511afe3d9ca1a3a6ad7d3b107","family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136082,"semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"partials_exist_not_differentiable\",\"scale\":[7,1]}"},"fields":{"title":"Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts","stem":"Use mode=partials_exist_not_differentiable, positive scale=7, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, coordinate_partial_values, coordinate_partials_exist, diagonal_comparison, differentiable, boundary).","answer":{"kind":"expression","value":"(partials_exist_not_differentiable, f(x,y)=7*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [7, 0], Yes, along_y=x:f(x,x)=7/2*x_but_linear_candidate=7*x, No, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)","explanation":"incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (partials_exist_not_differentiable, f(x,y)=7*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [7, 0], Yes, along_y=x:f(x,x)=7/2*x_but_linear_candidate=7*x, No, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)."},"skills":["Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0083","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4fa325f679adfa8b9a60780982ae2582ff935af94c4c50cf6371440304c410b0","family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136083,"semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"saddle_stationary\",\"scale\":[8,1]}"},"fields":{"title":"Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts","stem":"Use mode=saddle_stationary, positive scale=8, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, gradient_values, stationary, extremum_certificate, boundary).","answer":{"kind":"expression","value":"(saddle_stationary, f(x,y)=8*(x^2-y^2), [0, 0], [0, 0], Yes, No, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)","explanation":"incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (saddle_stationary, f(x,y)=8*(x^2-y^2), [0, 0], [0, 0], Yes, No, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)."},"skills":["Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0084","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7beefda0f7f354c3ede7f9fa1f139c0c2f5de9063c7d550c384ea102e7f73958","family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136084,"semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"incomplete_coordinate_list\",\"scale\":[9,1]}"},"fields":{"title":"Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts","stem":"Use mode=incomplete_coordinate_list, positive scale=9, with the source-declared explicit counterexample for that mode. Give (mode, required_dimension, provided_component_descriptions, provided_count, complete_gradient, boundary).","answer":{"kind":"expression","value":"(incomplete_coordinate_list, 3, [9*partial_x, 9*partial_y], 2, No, missing_coordinate_component_is_not_a_gradient_in_R3)","explanation":"incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (incomplete_coordinate_list, 3, [9*partial_x, 9*partial_y], 2, No, missing_coordinate_component_is_not_a_gradient_in_R3)."},"skills":["Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0085","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb071b111b6d7bd0a3d838043a634d47bb48f69da5b0cf2b45abf82d73e28f0f","family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136085,"semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"nondifferentiable_norm\",\"scale\":[10,1]}"},"fields":{"title":"Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts","stem":"Use mode=nondifferentiable_norm, positive scale=10, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, coordinate_partials_exist, gradient_defined, boundary).","answer":{"kind":"expression","value":"(nondifferentiable_norm, f(x,y)=10*sqrt(x^2+y^2), [0, 0], No, No, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)","explanation":"incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (nondifferentiable_norm, f(x,y)=10*sqrt(x^2+y^2), [0, 0], No, No, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)."},"skills":["Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0086","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed0a73461716b7697c07d3fd29e1cf500985cf026dcb094d036ea3b6c04bd765","family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136086,"semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"partials_exist_not_differentiable\",\"scale\":[11,1]}"},"fields":{"title":"Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts","stem":"Use mode=partials_exist_not_differentiable, positive scale=11, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, coordinate_partial_values, coordinate_partials_exist, diagonal_comparison, differentiable, boundary).","answer":{"kind":"expression","value":"(partials_exist_not_differentiable, f(x,y)=11*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [11, 0], Yes, along_y=x:f(x,x)=11/2*x_but_linear_candidate=11*x, No, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)","explanation":"incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (partials_exist_not_differentiable, f(x,y)=11*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [11, 0], Yes, along_y=x:f(x,x)=11/2*x_but_linear_candidate=11*x, No, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)."},"skills":["Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0087","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3dd900d0146260a7e1de959803cf91d34193f294a22effe3a2480f72c88b8f69","family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136087,"semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"saddle_stationary\",\"scale\":[12,1]}"},"fields":{"title":"Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts","stem":"Use mode=saddle_stationary, positive scale=12, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, gradient_values, stationary, extremum_certificate, boundary).","answer":{"kind":"expression","value":"(saddle_stationary, f(x,y)=12*(x^2-y^2), [0, 0], [0, 0], Yes, No, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)","explanation":"incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (saddle_stationary, f(x,y)=12*(x^2-y^2), [0, 0], [0, 0], Yes, No, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)."},"skills":["Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0088","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0b4a2f0658a36b7b808d2bc3344d145da1ba7db8823dbabb1b2f7121ee60c34","family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136088,"semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"incomplete_coordinate_list\",\"scale\":[13,1]}"},"fields":{"title":"Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts","stem":"Use mode=incomplete_coordinate_list, positive scale=13, with the source-declared explicit counterexample for that mode. Give (mode, required_dimension, provided_component_descriptions, provided_count, complete_gradient, boundary).","answer":{"kind":"expression","value":"(incomplete_coordinate_list, 3, [13*partial_x, 13*partial_y], 2, No, missing_coordinate_component_is_not_a_gradient_in_R3)","explanation":"incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (incomplete_coordinate_list, 3, [13*partial_x, 13*partial_y], 2, No, missing_coordinate_component_is_not_a_gradient_in_R3)."},"skills":["Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0089","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"564236b3298ae3e910fd75aa4f6f6d35f33c228a5d08f7cd0449798e5b7d32ce","family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136089,"semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"nondifferentiable_norm\",\"scale\":[14,1]}"},"fields":{"title":"Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts","stem":"Use mode=nondifferentiable_norm, positive scale=14, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, coordinate_partials_exist, gradient_defined, boundary).","answer":{"kind":"expression","value":"(nondifferentiable_norm, f(x,y)=14*sqrt(x^2+y^2), [0, 0], No, No, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)","explanation":"incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (nondifferentiable_norm, f(x,y)=14*sqrt(x^2+y^2), [0, 0], No, No, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)."},"skills":["Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0090","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b598575e0dcef9cc9ca037e2edf155ed6dff737718d1eddcb34a4253d037233f","family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136090,"semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"partials_exist_not_differentiable\",\"scale\":[15,1]}"},"fields":{"title":"Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts","stem":"Use mode=partials_exist_not_differentiable, positive scale=15, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, coordinate_partial_values, coordinate_partials_exist, diagonal_comparison, differentiable, boundary).","answer":{"kind":"expression","value":"(partials_exist_not_differentiable, f(x,y)=15*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [15, 0], Yes, along_y=x:f(x,x)=15/2*x_but_linear_candidate=15*x, No, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)","explanation":"incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (partials_exist_not_differentiable, f(x,y)=15*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [15, 0], Yes, along_y=x:f(x,x)=15/2*x_but_linear_candidate=15*x, No, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)."},"skills":["Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0091","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9eecc50f2c7b6c1907222dc05809d469511bebf34ca75fbeb64381381378a3dd","family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136091,"semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"saddle_stationary\",\"scale\":[16,1]}"},"fields":{"title":"Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts","stem":"Use mode=saddle_stationary, positive scale=16, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, gradient_values, stationary, extremum_certificate, boundary).","answer":{"kind":"expression","value":"(saddle_stationary, f(x,y)=16*(x^2-y^2), [0, 0], [0, 0], Yes, No, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)","explanation":"incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (saddle_stationary, f(x,y)=16*(x^2-y^2), [0, 0], [0, 0], Yes, No, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)."},"skills":["Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0092","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9e61172a18c498404cf85f3d5eda9140d2a81b40d16b4384b462850c93c1736","family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136092,"semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"incomplete_coordinate_list\",\"scale\":[17,1]}"},"fields":{"title":"Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts","stem":"Use mode=incomplete_coordinate_list, positive scale=17, with the source-declared explicit counterexample for that mode. Give (mode, required_dimension, provided_component_descriptions, provided_count, complete_gradient, boundary).","answer":{"kind":"expression","value":"(incomplete_coordinate_list, 3, [17*partial_x, 17*partial_y], 2, No, missing_coordinate_component_is_not_a_gradient_in_R3)","explanation":"incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (incomplete_coordinate_list, 3, [17*partial_x, 17*partial_y], 2, No, missing_coordinate_component_is_not_a_gradient_in_R3)."},"skills":["Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0093","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3f55c5d0a345162212f9abdf2008dd898dafa00c471730ed7a0c46bdcd18a0a","family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136093,"semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"nondifferentiable_norm\",\"scale\":[18,1]}"},"fields":{"title":"Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts","stem":"Use mode=nondifferentiable_norm, positive scale=18, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, coordinate_partials_exist, gradient_defined, boundary).","answer":{"kind":"expression","value":"(nondifferentiable_norm, f(x,y)=18*sqrt(x^2+y^2), [0, 0], No, No, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)","explanation":"incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (nondifferentiable_norm, f(x,y)=18*sqrt(x^2+y^2), [0, 0], No, No, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)."},"skills":["Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0094","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eee410783af7a95ec0a0f7b303de07acc3cdfe0154e94944822551fc93b4b29f","family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136094,"semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"partials_exist_not_differentiable\",\"scale\":[19,1]}"},"fields":{"title":"Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts","stem":"Use mode=partials_exist_not_differentiable, positive scale=19, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, coordinate_partial_values, coordinate_partials_exist, diagonal_comparison, differentiable, boundary).","answer":{"kind":"expression","value":"(partials_exist_not_differentiable, f(x,y)=19*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [19, 0], Yes, along_y=x:f(x,x)=19/2*x_but_linear_candidate=19*x, No, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)","explanation":"incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (partials_exist_not_differentiable, f(x,y)=19*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [19, 0], Yes, along_y=x:f(x,x)=19/2*x_but_linear_candidate=19*x, No, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)."},"skills":["Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0095","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d7ad6dc3d8871fe7009ad96859a2f0eaedad61878e6df72949f5433eac9298e","family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136095,"semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"saddle_stationary\",\"scale\":[20,1]}"},"fields":{"title":"Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts","stem":"Use mode=saddle_stationary, positive scale=20, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, gradient_values, stationary, extremum_certificate, boundary).","answer":{"kind":"expression","value":"(saddle_stationary, f(x,y)=20*(x^2-y^2), [0, 0], [0, 0], Yes, No, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)","explanation":"incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (saddle_stationary, f(x,y)=20*(x^2-y^2), [0, 0], [0, 0], Yes, No, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)."},"skills":["Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0096","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38fd4e9d5d07445f5e15ad84ed24f903f103acde84eb8fe57e939ee1d8d99398","family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136096,"semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"incomplete_coordinate_list\",\"scale\":[21,1]}"},"fields":{"title":"Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts","stem":"Use mode=incomplete_coordinate_list, positive scale=21, with the source-declared explicit counterexample for that mode. Give (mode, required_dimension, provided_component_descriptions, provided_count, complete_gradient, boundary).","answer":{"kind":"expression","value":"(incomplete_coordinate_list, 3, [21*partial_x, 21*partial_y], 2, No, missing_coordinate_component_is_not_a_gradient_in_R3)","explanation":"incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (incomplete_coordinate_list, 3, [21*partial_x, 21*partial_y], 2, No, missing_coordinate_component_is_not_a_gradient_in_R3)."},"skills":["Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0097","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0adf3e70064001d282ddcaf4790e64c235b1873803fdf1aee595ef7b37c4337b","family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136097,"semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"nondifferentiable_norm\",\"scale\":[22,1]}"},"fields":{"title":"Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts","stem":"Use mode=nondifferentiable_norm, positive scale=22, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, coordinate_partials_exist, gradient_defined, boundary).","answer":{"kind":"expression","value":"(nondifferentiable_norm, f(x,y)=22*sqrt(x^2+y^2), [0, 0], No, No, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)","explanation":"incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (nondifferentiable_norm, f(x,y)=22*sqrt(x^2+y^2), [0, 0], No, No, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)."},"skills":["Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0098","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c73df35688faaf5776694ceae62c7a331b8fedeba4f9dd574748cc8855f40b10","family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136098,"semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"partials_exist_not_differentiable\",\"scale\":[23,1]}"},"fields":{"title":"Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts","stem":"Use mode=partials_exist_not_differentiable, positive scale=23, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, coordinate_partial_values, coordinate_partials_exist, diagonal_comparison, differentiable, boundary).","answer":{"kind":"expression","value":"(partials_exist_not_differentiable, f(x,y)=23*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [23, 0], Yes, along_y=x:f(x,x)=23/2*x_but_linear_candidate=23*x, No, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)","explanation":"incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (partials_exist_not_differentiable, f(x,y)=23*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [23, 0], Yes, along_y=x:f(x,x)=23/2*x_but_linear_candidate=23*x, No, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)."},"skills":["Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0099","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5972108849cb3567fe6a89b76f8f4ec8869577aa5808e7be8258d8721e95e25c","family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136099,"semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"saddle_stationary\",\"scale\":[24,1]}"},"fields":{"title":"Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts","stem":"Use mode=saddle_stationary, positive scale=24, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, gradient_values, stationary, extremum_certificate, boundary).","answer":{"kind":"expression","value":"(saddle_stationary, f(x,y)=24*(x^2-y^2), [0, 0], [0, 0], Yes, No, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)","explanation":"incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (saddle_stationary, f(x,y)=24*(x^2-y^2), [0, 0], [0, 0], Yes, No, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)."},"skills":["Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261113-r06-gradient-vector-0100","batch_id":"batch-20261015-advance-20261113-r06-gradient-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22a857e900efe7dad80121574bc3365f7c4fc0b9416d61c09992ef4b6a759cba","family_id":"qf-mod44-gradient-boundary-counterexample","seed":202611136100,"semantic_template":"qf-mod44-gradient-boundary-counterexample/{\"mode\":\"incomplete_coordinate_list\",\"scale\":[25,1]}"},"fields":{"title":"Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts","stem":"Use mode=incomplete_coordinate_list, positive scale=25, with the source-declared explicit counterexample for that mode. Give (mode, required_dimension, provided_component_descriptions, provided_count, complete_gradient, boundary).","answer":{"kind":"expression","value":"(incomplete_coordinate_list, 3, [25*partial_x, 25*partial_y], 2, No, missing_coordinate_component_is_not_a_gradient_in_R3)","explanation":"incomplete_coordinate_list supplies only two descriptions for required R3. nondifferentiable_norm uses scale*sqrt(x^2+y^2) at0, where two-sided coordinate partials fail. partials_exist_not_differentiable uses scale*x^3/(x^2+y^2), extended by0 at the origin: partial values are [scale,0], but along y=x the linear remainder is not little-o of the norm. saddle_stationary uses scale*(x^2−y^2), whose zero gradient is not an extremum. Use explicit mode-selected6/8/7 fields without padding or answer-derived guessing. The exact result, retaining the requested order, is (incomplete_coordinate_list, 3, [25*partial_x, 25*partial_y], 2, No, missing_coordinate_component_is_not_a_gradient_in_R3)."},"skills":["Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}