{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r03-0001","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5e6b49367667bcf33830e3074ca1219f2146f7b3a7de5cdba04e88d7b9c01bf","family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193001,"semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[1,1],[2,1]],\"linear\":[[-1,1],[-2,1]],\"point\":[[-2,1],[-3,1]]}"},"fields":{"title":"A Positive Diagonal Quadratic Has an Exact Ordered Gradient","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 2], linear b=[-1, -2], and point x=[-2, -3] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).","answer":{"kind":"expression","value":"([1, 2], [-1, -2], [-2, -3], [-3, -8], 19, fixed_standard_Euclidean_coordinate_order)","explanation":"Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([1, 2], [-1, -2], [-2, -3], [-3, -8], 19, fixed_standard_Euclidean_coordinate_order)."},"skills":["A Positive Diagonal Quadratic Has an Exact Ordered Gradient"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0002","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f259a9797ae881c931a1fa5a0806465e5de9b1a7855e64f3f379652a10c6acd","family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193002,"semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[2,1],[3,1]],\"linear\":[[0,1],[-1,1]],\"point\":[[-1,1],[-2,1]]}"},"fields":{"title":"A Positive Diagonal Quadratic Has an Exact Ordered Gradient","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 3], linear b=[0, -1], and point x=[-1, -2] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).","answer":{"kind":"expression","value":"([2, 3], [0, -1], [-1, -2], [-2, -7], 9, fixed_standard_Euclidean_coordinate_order)","explanation":"Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([2, 3], [0, -1], [-1, -2], [-2, -7], 9, fixed_standard_Euclidean_coordinate_order)."},"skills":["A Positive Diagonal Quadratic Has an Exact Ordered Gradient"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0003","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a68c5677519df14a66cb9e988aea9facf052140fed50f172dc7f962fa9629d0","family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193003,"semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[3,1],[4,1]],\"linear\":[[1,1],[0,1]],\"point\":[[0,1],[-1,1]]}"},"fields":{"title":"A Positive Diagonal Quadratic Has an Exact Ordered Gradient","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 4], linear b=[1, 0], and point x=[0, -1] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).","answer":{"kind":"expression","value":"([3, 4], [1, 0], [0, -1], [1, -4], 2, fixed_standard_Euclidean_coordinate_order)","explanation":"Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([3, 4], [1, 0], [0, -1], [1, -4], 2, fixed_standard_Euclidean_coordinate_order)."},"skills":["A Positive Diagonal Quadratic Has an Exact Ordered Gradient"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0004","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a7607cfa57d758a98b5b6359bb78583d58158af6d67af7eda7e046e117af675","family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193004,"semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[1,1],[5,1]],\"linear\":[[2,1],[1,1]],\"point\":[[1,1],[0,1]]}"},"fields":{"title":"A Positive Diagonal Quadratic Has an Exact Ordered Gradient","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 5], linear b=[2, 1], and point x=[1, 0] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).","answer":{"kind":"expression","value":"([1, 5], [2, 1], [1, 0], [3, 1], 5/2, fixed_standard_Euclidean_coordinate_order)","explanation":"Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([1, 5], [2, 1], [1, 0], [3, 1], 5/2, fixed_standard_Euclidean_coordinate_order)."},"skills":["A Positive Diagonal Quadratic Has an Exact Ordered Gradient"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0005","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b77af2de2156f9d57423322a3607451894af14f71a85af8cf6885a649709c8b2","family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193005,"semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[2,1],[2,1]],\"linear\":[[-2,1],[2,1]],\"point\":[[2,1],[1,1]]}"},"fields":{"title":"A Positive Diagonal Quadratic Has an Exact Ordered Gradient","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 2], linear b=[-2, 2], and point x=[2, 1] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).","answer":{"kind":"expression","value":"([2, 2], [-2, 2], [2, 1], [2, 4], 3, fixed_standard_Euclidean_coordinate_order)","explanation":"Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([2, 2], [-2, 2], [2, 1], [2, 4], 3, fixed_standard_Euclidean_coordinate_order)."},"skills":["A Positive Diagonal Quadratic Has an Exact Ordered Gradient"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0006","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93f8e126a2158da43ef6b083256e1005a6dbbb97e3ba8e820b66314a9dcb6a6d","family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193006,"semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[3,1],[3,1]],\"linear\":[[-1,1],[3,1]],\"point\":[[-2,1],[2,1]]}"},"fields":{"title":"A Positive Diagonal Quadratic Has an Exact Ordered Gradient","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 3], linear b=[-1, 3], and point x=[-2, 2] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).","answer":{"kind":"expression","value":"([3, 3], [-1, 3], [-2, 2], [-7, 9], 20, fixed_standard_Euclidean_coordinate_order)","explanation":"Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([3, 3], [-1, 3], [-2, 2], [-7, 9], 20, fixed_standard_Euclidean_coordinate_order)."},"skills":["A Positive Diagonal Quadratic Has an Exact Ordered Gradient"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0007","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d4dd4ede62e064fe0568ac2b05bef4468efd833868a25e7a3e73ed2e37dee81","family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193007,"semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[1,1],[4,1]],\"linear\":[[0,1],[-3,1]],\"point\":[[-1,1],[3,1]]}"},"fields":{"title":"A Positive Diagonal Quadratic Has an Exact Ordered Gradient","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 4], linear b=[0, -3], and point x=[-1, 3] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).","answer":{"kind":"expression","value":"([1, 4], [0, -3], [-1, 3], [-1, 9], 19/2, fixed_standard_Euclidean_coordinate_order)","explanation":"Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([1, 4], [0, -3], [-1, 3], [-1, 9], 19/2, fixed_standard_Euclidean_coordinate_order)."},"skills":["A Positive Diagonal Quadratic Has an Exact Ordered Gradient"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0008","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"825aad6c8fbf3f466d2ba4724860ccf45c169833f409d1eb5e70f67f4fa4b6fc","family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193008,"semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[2,1],[5,1]],\"linear\":[[1,1],[-2,1]],\"point\":[[0,1],[-3,1]]}"},"fields":{"title":"A Positive Diagonal Quadratic Has an Exact Ordered Gradient","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 5], linear b=[1, -2], and point x=[0, -3] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).","answer":{"kind":"expression","value":"([2, 5], [1, -2], [0, -3], [1, -17], 57/2, fixed_standard_Euclidean_coordinate_order)","explanation":"Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([2, 5], [1, -2], [0, -3], [1, -17], 57/2, fixed_standard_Euclidean_coordinate_order)."},"skills":["A Positive Diagonal Quadratic Has an Exact Ordered Gradient"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0009","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9649a4c8eb2a36797d32f9f419ae6f41dc68653079ab33df7844a7e229cacd31","family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193009,"semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[3,1],[2,1]],\"linear\":[[2,1],[-1,1]],\"point\":[[1,1],[-2,1]]}"},"fields":{"title":"A Positive Diagonal Quadratic Has an Exact Ordered Gradient","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 2], linear b=[2, -1], and point x=[1, -2] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).","answer":{"kind":"expression","value":"([3, 2], [2, -1], [1, -2], [5, -5], 19/2, fixed_standard_Euclidean_coordinate_order)","explanation":"Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([3, 2], [2, -1], [1, -2], [5, -5], 19/2, fixed_standard_Euclidean_coordinate_order)."},"skills":["A Positive Diagonal Quadratic Has an Exact Ordered Gradient"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0010","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c6fe703adcf1dfb6cd2961af3ed920cb5469769637f47d3d00fff0b2d47ec2fc","family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193010,"semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[1,1],[3,1]],\"linear\":[[-2,1],[0,1]],\"point\":[[2,1],[-1,1]]}"},"fields":{"title":"A Positive Diagonal Quadratic Has an Exact Ordered Gradient","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 3], linear b=[-2, 0], and point x=[2, -1] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).","answer":{"kind":"expression","value":"([1, 3], [-2, 0], [2, -1], [0, -3], -1/2, fixed_standard_Euclidean_coordinate_order)","explanation":"Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([1, 3], [-2, 0], [2, -1], [0, -3], -1/2, fixed_standard_Euclidean_coordinate_order)."},"skills":["A Positive Diagonal Quadratic Has an Exact Ordered Gradient"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0011","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0c873c266f8042879c7de8d62259a3d6b4174c4a80362b2871377dad29bb25a","family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193011,"semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[2,1],[4,1]],\"linear\":[[-1,1],[1,1]],\"point\":[[-2,1],[0,1]]}"},"fields":{"title":"A Positive Diagonal Quadratic Has an Exact Ordered Gradient","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 4], linear b=[-1, 1], and point x=[-2, 0] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).","answer":{"kind":"expression","value":"([2, 4], [-1, 1], [-2, 0], [-5, 1], 6, fixed_standard_Euclidean_coordinate_order)","explanation":"Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([2, 4], [-1, 1], [-2, 0], [-5, 1], 6, fixed_standard_Euclidean_coordinate_order)."},"skills":["A Positive Diagonal Quadratic Has an Exact Ordered Gradient"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0012","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc9f6357419744186469060f7c5544e2389fec884907817f7322dea2eab9917d","family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193012,"semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[3,1],[5,1]],\"linear\":[[0,1],[2,1]],\"point\":[[-1,1],[1,1]]}"},"fields":{"title":"A Positive Diagonal Quadratic Has an Exact Ordered Gradient","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 5], linear b=[0, 2], and point x=[-1, 1] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).","answer":{"kind":"expression","value":"([3, 5], [0, 2], [-1, 1], [-3, 7], 6, fixed_standard_Euclidean_coordinate_order)","explanation":"Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([3, 5], [0, 2], [-1, 1], [-3, 7], 6, fixed_standard_Euclidean_coordinate_order)."},"skills":["A Positive Diagonal Quadratic Has an Exact Ordered Gradient"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0013","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1088bfafd9b3a00f8c91019607271935fa3daca8781e887b44561973b239e225","family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193013,"semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[1,1],[2,1]],\"linear\":[[1,1],[3,1]],\"point\":[[0,1],[2,1]]}"},"fields":{"title":"A Positive Diagonal Quadratic Has an Exact Ordered Gradient","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 2], linear b=[1, 3], and point x=[0, 2] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).","answer":{"kind":"expression","value":"([1, 2], [1, 3], [0, 2], [1, 7], 10, fixed_standard_Euclidean_coordinate_order)","explanation":"Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([1, 2], [1, 3], [0, 2], [1, 7], 10, fixed_standard_Euclidean_coordinate_order)."},"skills":["A Positive Diagonal Quadratic Has an Exact Ordered Gradient"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0014","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3117b1d87f01ce3cd60b6fd5a9f30d3f67be58aad251cde530c310b6a1f0d0a2","family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193014,"semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[2,1],[3,1]],\"linear\":[[2,1],[-3,1]],\"point\":[[1,1],[3,1]]}"},"fields":{"title":"A Positive Diagonal Quadratic Has an Exact Ordered Gradient","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 3], linear b=[2, -3], and point x=[1, 3] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).","answer":{"kind":"expression","value":"([2, 3], [2, -3], [1, 3], [4, 6], 15/2, fixed_standard_Euclidean_coordinate_order)","explanation":"Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([2, 3], [2, -3], [1, 3], [4, 6], 15/2, fixed_standard_Euclidean_coordinate_order)."},"skills":["A Positive Diagonal Quadratic Has an Exact Ordered Gradient"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0015","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99d69e7da1f571b01be9c502210e46457751aa2affbfcc1c98a1ca3e984eb552","family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193015,"semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[3,1],[4,1]],\"linear\":[[-2,1],[-2,1]],\"point\":[[2,1],[-3,1]]}"},"fields":{"title":"A Positive Diagonal Quadratic Has an Exact Ordered Gradient","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 4], linear b=[-2, -2], and point x=[2, -3] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).","answer":{"kind":"expression","value":"([3, 4], [-2, -2], [2, -3], [4, -14], 26, fixed_standard_Euclidean_coordinate_order)","explanation":"Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([3, 4], [-2, -2], [2, -3], [4, -14], 26, fixed_standard_Euclidean_coordinate_order)."},"skills":["A Positive Diagonal Quadratic Has an Exact Ordered Gradient"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0016","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c45cb32cbda8148d97ddada0b14c09a787e2c38efb795fe70ae4c793c9aa3037","family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193016,"semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[1,1],[5,1]],\"linear\":[[-1,1],[-1,1]],\"point\":[[-2,1],[-2,1]]}"},"fields":{"title":"A Positive Diagonal Quadratic Has an Exact Ordered Gradient","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 5], linear b=[-1, -1], and point x=[-2, -2] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).","answer":{"kind":"expression","value":"([1, 5], [-1, -1], [-2, -2], [-3, -11], 16, fixed_standard_Euclidean_coordinate_order)","explanation":"Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([1, 5], [-1, -1], [-2, -2], [-3, -11], 16, fixed_standard_Euclidean_coordinate_order)."},"skills":["A Positive Diagonal Quadratic Has an Exact Ordered Gradient"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0017","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d55f4b6b3862cb903169e3e597ad9377e83a317b676f0b37053c2b2dfc063d87","family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193017,"semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[2,1],[2,1]],\"linear\":[[0,1],[0,1]],\"point\":[[-1,1],[-1,1]]}"},"fields":{"title":"A Positive Diagonal Quadratic Has an Exact Ordered Gradient","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 2], linear b=[0, 0], and point x=[-1, -1] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).","answer":{"kind":"expression","value":"([2, 2], [0, 0], [-1, -1], [-2, -2], 2, fixed_standard_Euclidean_coordinate_order)","explanation":"Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([2, 2], [0, 0], [-1, -1], [-2, -2], 2, fixed_standard_Euclidean_coordinate_order)."},"skills":["A Positive Diagonal Quadratic Has an Exact Ordered Gradient"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0018","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4cfce7b3adfc8e8a373b3740b01ffb6f442a07114ff684347e91766acdeae335","family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193018,"semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[3,1],[3,1]],\"linear\":[[1,1],[1,1]],\"point\":[[0,1],[0,1]]}"},"fields":{"title":"A Positive Diagonal Quadratic Has an Exact Ordered Gradient","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 3], linear b=[1, 1], and point x=[0, 0] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).","answer":{"kind":"expression","value":"([3, 3], [1, 1], [0, 0], [1, 1], 0, fixed_standard_Euclidean_coordinate_order)","explanation":"Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([3, 3], [1, 1], [0, 0], [1, 1], 0, fixed_standard_Euclidean_coordinate_order)."},"skills":["A Positive Diagonal Quadratic Has an Exact Ordered Gradient"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0019","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e9c63362110a5ae01afdc97cdcffc8e59289287c44d7b1400c2c40ef42c9d6e","family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193019,"semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[1,1],[4,1]],\"linear\":[[2,1],[2,1]],\"point\":[[1,1],[1,1]]}"},"fields":{"title":"A Positive Diagonal Quadratic Has an Exact Ordered Gradient","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 4], linear b=[2, 2], and point x=[1, 1] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).","answer":{"kind":"expression","value":"([1, 4], [2, 2], [1, 1], [3, 6], 13/2, fixed_standard_Euclidean_coordinate_order)","explanation":"Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([1, 4], [2, 2], [1, 1], [3, 6], 13/2, fixed_standard_Euclidean_coordinate_order)."},"skills":["A Positive Diagonal Quadratic Has an Exact Ordered Gradient"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0020","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9220ea9783cbc95391be76da15045230ad7485d22d273fa18e8827192bf1467e","family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193020,"semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[2,1],[5,1]],\"linear\":[[-2,1],[3,1]],\"point\":[[2,1],[2,1]]}"},"fields":{"title":"A Positive Diagonal Quadratic Has an Exact Ordered Gradient","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 5], linear b=[-2, 3], and point x=[2, 2] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).","answer":{"kind":"expression","value":"([2, 5], [-2, 3], [2, 2], [2, 13], 16, fixed_standard_Euclidean_coordinate_order)","explanation":"Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([2, 5], [-2, 3], [2, 2], [2, 13], 16, fixed_standard_Euclidean_coordinate_order)."},"skills":["A Positive Diagonal Quadratic Has an Exact Ordered Gradient"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0021","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9736755c9ba5190f44b28443d1144f60b8670ca3605eb13199f8a4b365d04941","family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193021,"semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[3,1],[2,1]],\"linear\":[[-1,1],[-3,1]],\"point\":[[-2,1],[3,1]]}"},"fields":{"title":"A Positive Diagonal Quadratic Has an Exact Ordered Gradient","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 2], linear b=[-1, -3], and point x=[-2, 3] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).","answer":{"kind":"expression","value":"([3, 2], [-1, -3], [-2, 3], [-7, 3], 8, fixed_standard_Euclidean_coordinate_order)","explanation":"Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([3, 2], [-1, -3], [-2, 3], [-7, 3], 8, fixed_standard_Euclidean_coordinate_order)."},"skills":["A Positive Diagonal Quadratic Has an Exact Ordered Gradient"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0022","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e4f13a44532ac38710ed6b5213c95c2d66ee41ce36aab334f9e5fd60c2c3fe0","family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193022,"semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[1,1],[3,1]],\"linear\":[[0,1],[-2,1]],\"point\":[[-1,1],[-3,1]]}"},"fields":{"title":"A Positive Diagonal Quadratic Has an Exact Ordered Gradient","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 3], linear b=[0, -2], and point x=[-1, -3] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).","answer":{"kind":"expression","value":"([1, 3], [0, -2], [-1, -3], [-1, -11], 20, fixed_standard_Euclidean_coordinate_order)","explanation":"Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([1, 3], [0, -2], [-1, -3], [-1, -11], 20, fixed_standard_Euclidean_coordinate_order)."},"skills":["A Positive Diagonal Quadratic Has an Exact Ordered Gradient"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0023","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da04596ff0287561c45d6dafbb95d4672b12bf1afced313fd4a973ed75dd325b","family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193023,"semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[2,1],[4,1]],\"linear\":[[1,1],[-1,1]],\"point\":[[0,1],[-2,1]]}"},"fields":{"title":"A Positive Diagonal Quadratic Has an Exact Ordered Gradient","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 4], linear b=[1, -1], and point x=[0, -2] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).","answer":{"kind":"expression","value":"([2, 4], [1, -1], [0, -2], [1, -9], 10, fixed_standard_Euclidean_coordinate_order)","explanation":"Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([2, 4], [1, -1], [0, -2], [1, -9], 10, fixed_standard_Euclidean_coordinate_order)."},"skills":["A Positive Diagonal Quadratic Has an Exact Ordered Gradient"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0024","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf315deacf1b51621ef1029cabc88f15ab864faeb2b07ce8ab29b8aae38cdf80","family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193024,"semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[3,1],[5,1]],\"linear\":[[2,1],[0,1]],\"point\":[[1,1],[-1,1]]}"},"fields":{"title":"A Positive Diagonal Quadratic Has an Exact Ordered Gradient","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 5], linear b=[2, 0], and point x=[1, -1] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).","answer":{"kind":"expression","value":"([3, 5], [2, 0], [1, -1], [5, -5], 6, fixed_standard_Euclidean_coordinate_order)","explanation":"Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([3, 5], [2, 0], [1, -1], [5, -5], 6, fixed_standard_Euclidean_coordinate_order)."},"skills":["A Positive Diagonal Quadratic Has an Exact Ordered Gradient"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0025","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c15ddff91e044a6b4ca10bac49517e456002bab3a70c95b9580dcfb40cb9a21","family_id":"qf-mod50-gd-quadratic-gradient","seed":202611193025,"semantic_template":"gd-quadratic-gradient|{\"diagonal\":[[1,1],[2,1]],\"linear\":[[-2,1],[1,1]],\"point\":[[2,1],[0,1]]}"},"fields":{"title":"A Positive Diagonal Quadratic Has an Exact Ordered Gradient","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 2], linear b=[-2, 1], and point x=[2, 0] in fixed standard Euclidean coordinates. Give (positive_diagonal, linear_coefficients, point, gradient, objective_value, coordinate_convention).","answer":{"kind":"expression","value":"([1, 2], [-2, 1], [2, 0], [0, 1], -2, fixed_standard_Euclidean_coordinate_order)","explanation":"Calculate gradient_i=d_i*x_i+b_i and the complete objective value. Keep coordinate order and all vector dimensions. This is evaluation of the specified quadratic, not a convergence result or a classification of arbitrary stationary points. The exact result, retaining the requested order, is ([1, 2], [-2, 1], [2, 0], [0, 1], -2, fixed_standard_Euclidean_coordinate_order)."},"skills":["A Positive Diagonal Quadratic Has an Exact Ordered Gradient"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0026","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"583202c7ec9e04ca8f9264f579d875eb565f46b4c27f499d0809000f012d4987","family_id":"qf-mod50-gd-one-step","seed":202611193026,"semantic_template":"gd-one-step|{\"diagonal\":[[1,1],[2,1]],\"linear\":[[-1,1],[-2,1]],\"point\":[[-2,1],[-3,1]],\"step_size\":[1,4]}"},"fields":{"title":"Positive Step Size Alone Does Not Guarantee Objective Descent","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 2], linear b=[-1, -2], and point x=[-2, -3] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).","answer":{"kind":"expression","value":"([-2, -3], [-3, -8], 1/4, [-5/4, -1], 19, 161/32, Yes, positive_step_alone_does_not_guarantee_descent)","explanation":"Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([-2, -3], [-3, -8], 1/4, [-5/4, -1], 19, 161/32, Yes, positive_step_alone_does_not_guarantee_descent)."},"skills":["Positive Step Size Alone Does Not Guarantee Objective Descent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0027","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14f4535169e087bc0ed4e8cdc2317568c0b849fb982d53f54d9578a4a6b5ffc5","family_id":"qf-mod50-gd-one-step","seed":202611193027,"semantic_template":"gd-one-step|{\"diagonal\":[[2,1],[3,1]],\"linear\":[[0,1],[-1,1]],\"point\":[[-1,1],[-2,1]],\"step_size\":[1,4]}"},"fields":{"title":"Positive Step Size Alone Does Not Guarantee Objective Descent","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 3], linear b=[0, -1], and point x=[-1, -2] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).","answer":{"kind":"expression","value":"([-1, -2], [-2, -7], 1/4, [-1/2, -1/4], 9, 19/32, Yes, positive_step_alone_does_not_guarantee_descent)","explanation":"Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([-1, -2], [-2, -7], 1/4, [-1/2, -1/4], 9, 19/32, Yes, positive_step_alone_does_not_guarantee_descent)."},"skills":["Positive Step Size Alone Does Not Guarantee Objective Descent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0028","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16b5d0e5a10483eb71781bebb7845197c11dfaea27b6e1c1de84ace8781b7f2e","family_id":"qf-mod50-gd-one-step","seed":202611193028,"semantic_template":"gd-one-step|{\"diagonal\":[[3,1],[4,1]],\"linear\":[[1,1],[0,1]],\"point\":[[0,1],[-1,1]],\"step_size\":[1,4]}"},"fields":{"title":"Positive Step Size Alone Does Not Guarantee Objective Descent","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 4], linear b=[1, 0], and point x=[0, -1] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).","answer":{"kind":"expression","value":"([0, -1], [1, -4], 1/4, [-1/4, 0], 2, -5/32, Yes, positive_step_alone_does_not_guarantee_descent)","explanation":"Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([0, -1], [1, -4], 1/4, [-1/4, 0], 2, -5/32, Yes, positive_step_alone_does_not_guarantee_descent)."},"skills":["Positive Step Size Alone Does Not Guarantee Objective Descent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0029","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ebfbb083afe4fc31c484324053e5f15d2cebbe1a04e7cdf45269503321ba58f","family_id":"qf-mod50-gd-one-step","seed":202611193029,"semantic_template":"gd-one-step|{\"diagonal\":[[1,1],[5,1]],\"linear\":[[2,1],[1,1]],\"point\":[[1,1],[0,1]],\"step_size\":[1,4]}"},"fields":{"title":"Positive Step Size Alone Does Not Guarantee Objective Descent","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 5], linear b=[2, 1], and point x=[1, 0] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).","answer":{"kind":"expression","value":"([1, 0], [3, 1], 1/4, [1/4, -1/4], 5/2, 7/16, Yes, positive_step_alone_does_not_guarantee_descent)","explanation":"Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([1, 0], [3, 1], 1/4, [1/4, -1/4], 5/2, 7/16, Yes, positive_step_alone_does_not_guarantee_descent)."},"skills":["Positive Step Size Alone Does Not Guarantee Objective Descent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0030","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1073ebbb65827d600a90c14eca601ffd5a1c9fad585347cd7c5ed74a92b993c0","family_id":"qf-mod50-gd-one-step","seed":202611193030,"semantic_template":"gd-one-step|{\"diagonal\":[[2,1],[2,1]],\"linear\":[[-2,1],[2,1]],\"point\":[[2,1],[1,1]],\"step_size\":[1,4]}"},"fields":{"title":"Positive Step Size Alone Does Not Guarantee Objective Descent","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 2], linear b=[-2, 2], and point x=[2, 1] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).","answer":{"kind":"expression","value":"([2, 1], [2, 4], 1/4, [3/2, 0], 3, -3/4, Yes, positive_step_alone_does_not_guarantee_descent)","explanation":"Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([2, 1], [2, 4], 1/4, [3/2, 0], 3, -3/4, Yes, positive_step_alone_does_not_guarantee_descent)."},"skills":["Positive Step Size Alone Does Not Guarantee Objective Descent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0031","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a54f5839e0680508e22922294b29012199b6241f11f2b46ae72108fb3b939bf0","family_id":"qf-mod50-gd-one-step","seed":202611193031,"semantic_template":"gd-one-step|{\"diagonal\":[[3,1],[3,1]],\"linear\":[[-1,1],[3,1]],\"point\":[[-2,1],[2,1]],\"step_size\":[1,4]}"},"fields":{"title":"Positive Step Size Alone Does Not Guarantee Objective Descent","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 3], linear b=[-1, 3], and point x=[-2, 2] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).","answer":{"kind":"expression","value":"([-2, 2], [-7, 9], 1/4, [-1/4, -1/4], 20, -5/16, Yes, positive_step_alone_does_not_guarantee_descent)","explanation":"Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([-2, 2], [-7, 9], 1/4, [-1/4, -1/4], 20, -5/16, Yes, positive_step_alone_does_not_guarantee_descent)."},"skills":["Positive Step Size Alone Does Not Guarantee Objective Descent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0032","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c9a80b7af03b6c087e1ff44408e14f38059967e72f950257e494636210330d5","family_id":"qf-mod50-gd-one-step","seed":202611193032,"semantic_template":"gd-one-step|{\"diagonal\":[[1,1],[4,1]],\"linear\":[[0,1],[-3,1]],\"point\":[[-1,1],[3,1]],\"step_size\":[1,4]}"},"fields":{"title":"Positive Step Size Alone Does Not Guarantee Objective Descent","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 4], linear b=[0, -3], and point x=[-1, 3] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).","answer":{"kind":"expression","value":"([-1, 3], [-1, 9], 1/4, [-3/4, 3/4], 19/2, -27/32, Yes, positive_step_alone_does_not_guarantee_descent)","explanation":"Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([-1, 3], [-1, 9], 1/4, [-3/4, 3/4], 19/2, -27/32, Yes, positive_step_alone_does_not_guarantee_descent)."},"skills":["Positive Step Size Alone Does Not Guarantee Objective Descent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0033","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49633ed1861bc1baa4acc5a746a9775e99b2e014479ef667eafb58ff0d70993a","family_id":"qf-mod50-gd-one-step","seed":202611193033,"semantic_template":"gd-one-step|{\"diagonal\":[[2,1],[5,1]],\"linear\":[[1,1],[-2,1]],\"point\":[[0,1],[-3,1]],\"step_size\":[1,4]}"},"fields":{"title":"Positive Step Size Alone Does Not Guarantee Objective Descent","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 5], linear b=[1, -2], and point x=[0, -3] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).","answer":{"kind":"expression","value":"([0, -3], [1, -17], 1/4, [-1/4, 5/4], 57/2, 39/32, Yes, positive_step_alone_does_not_guarantee_descent)","explanation":"Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([0, -3], [1, -17], 1/4, [-1/4, 5/4], 57/2, 39/32, Yes, positive_step_alone_does_not_guarantee_descent)."},"skills":["Positive Step Size Alone Does Not Guarantee Objective Descent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0034","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cbfd94cffc641a56f483b11c2791902bbb64cb06d165a466b8c349b1043d7b3f","family_id":"qf-mod50-gd-one-step","seed":202611193034,"semantic_template":"gd-one-step|{\"diagonal\":[[3,1],[2,1]],\"linear\":[[2,1],[-1,1]],\"point\":[[1,1],[-2,1]],\"step_size\":[1,4]}"},"fields":{"title":"Positive Step Size Alone Does Not Guarantee Objective Descent","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 2], linear b=[2, -1], and point x=[1, -2] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).","answer":{"kind":"expression","value":"([1, -2], [5, -5], 1/4, [-1/4, -3/4], 19/2, 29/32, Yes, positive_step_alone_does_not_guarantee_descent)","explanation":"Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([1, -2], [5, -5], 1/4, [-1/4, -3/4], 19/2, 29/32, Yes, positive_step_alone_does_not_guarantee_descent)."},"skills":["Positive Step Size Alone Does Not Guarantee Objective Descent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0035","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08b86a335ba1f4daa4aba8a38132fafaf545a3ded32b3d5854e55b7c0c40fd98","family_id":"qf-mod50-gd-one-step","seed":202611193035,"semantic_template":"gd-one-step|{\"diagonal\":[[1,1],[3,1]],\"linear\":[[-2,1],[0,1]],\"point\":[[2,1],[-1,1]],\"step_size\":[1,4]}"},"fields":{"title":"Positive Step Size Alone Does Not Guarantee Objective Descent","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 3], linear b=[-2, 0], and point x=[2, -1] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).","answer":{"kind":"expression","value":"([2, -1], [0, -3], 1/4, [2, -1/4], -1/2, -61/32, Yes, positive_step_alone_does_not_guarantee_descent)","explanation":"Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([2, -1], [0, -3], 1/4, [2, -1/4], -1/2, -61/32, Yes, positive_step_alone_does_not_guarantee_descent)."},"skills":["Positive Step Size Alone Does Not Guarantee Objective Descent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0036","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83d2d8b6830272adf9e1eb2c8677fc88e34fdf0648a8d1526128ecf860be1914","family_id":"qf-mod50-gd-one-step","seed":202611193036,"semantic_template":"gd-one-step|{\"diagonal\":[[2,1],[4,1]],\"linear\":[[-1,1],[1,1]],\"point\":[[-2,1],[0,1]],\"step_size\":[1,4]}"},"fields":{"title":"Positive Step Size Alone Does Not Guarantee Objective Descent","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 4], linear b=[-1, 1], and point x=[-2, 0] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).","answer":{"kind":"expression","value":"([-2, 0], [-5, 1], 1/4, [-3/4, -1/4], 6, 19/16, Yes, positive_step_alone_does_not_guarantee_descent)","explanation":"Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([-2, 0], [-5, 1], 1/4, [-3/4, -1/4], 6, 19/16, Yes, positive_step_alone_does_not_guarantee_descent)."},"skills":["Positive Step Size Alone Does Not Guarantee Objective Descent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0037","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d68b783c0e128fa91e06628eb30d89e3d73a6ec14bc789d6fb47ae41aba1a4b","family_id":"qf-mod50-gd-one-step","seed":202611193037,"semantic_template":"gd-one-step|{\"diagonal\":[[3,1],[5,1]],\"linear\":[[0,1],[2,1]],\"point\":[[-1,1],[1,1]],\"step_size\":[1,4]}"},"fields":{"title":"Positive Step Size Alone Does Not Guarantee Objective Descent","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 5], linear b=[0, 2], and point x=[-1, 1] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).","answer":{"kind":"expression","value":"([-1, 1], [-3, 7], 1/4, [-1/4, -3/4], 6, 0, Yes, positive_step_alone_does_not_guarantee_descent)","explanation":"Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([-1, 1], [-3, 7], 1/4, [-1/4, -3/4], 6, 0, Yes, positive_step_alone_does_not_guarantee_descent)."},"skills":["Positive Step Size Alone Does Not Guarantee Objective Descent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0038","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2553deeaae87e422f05936ed85edd976c61ccfa3cb115e2902748f3a2e7e4ff1","family_id":"qf-mod50-gd-one-step","seed":202611193038,"semantic_template":"gd-one-step|{\"diagonal\":[[1,1],[2,1]],\"linear\":[[1,1],[3,1]],\"point\":[[0,1],[2,1]],\"step_size\":[1,4]}"},"fields":{"title":"Positive Step Size Alone Does Not Guarantee Objective Descent","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 2], linear b=[1, 3], and point x=[0, 2] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).","answer":{"kind":"expression","value":"([0, 2], [1, 7], 1/4, [-1/4, 1/4], 10, 19/32, Yes, positive_step_alone_does_not_guarantee_descent)","explanation":"Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([0, 2], [1, 7], 1/4, [-1/4, 1/4], 10, 19/32, Yes, positive_step_alone_does_not_guarantee_descent)."},"skills":["Positive Step Size Alone Does Not Guarantee Objective Descent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0039","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2977dfa7d72fe857adf8a71956b5ae91037cd2888f71e37dc5afec3f1fa019e","family_id":"qf-mod50-gd-one-step","seed":202611193039,"semantic_template":"gd-one-step|{\"diagonal\":[[2,1],[3,1]],\"linear\":[[2,1],[-3,1]],\"point\":[[1,1],[3,1]],\"step_size\":[1,4]}"},"fields":{"title":"Positive Step Size Alone Does Not Guarantee Objective Descent","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 3], linear b=[2, -3], and point x=[1, 3] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).","answer":{"kind":"expression","value":"([1, 3], [4, 6], 1/4, [0, 3/2], 15/2, -9/8, Yes, positive_step_alone_does_not_guarantee_descent)","explanation":"Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([1, 3], [4, 6], 1/4, [0, 3/2], 15/2, -9/8, Yes, positive_step_alone_does_not_guarantee_descent)."},"skills":["Positive Step Size Alone Does Not Guarantee Objective Descent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0040","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"200201e4e45fedf1122005124e79277df906089284fec0a3dfc7486940fdbd07","family_id":"qf-mod50-gd-one-step","seed":202611193040,"semantic_template":"gd-one-step|{\"diagonal\":[[3,1],[4,1]],\"linear\":[[-2,1],[-2,1]],\"point\":[[2,1],[-3,1]],\"step_size\":[1,4]}"},"fields":{"title":"Positive Step Size Alone Does Not Guarantee Objective Descent","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 4], linear b=[-2, -2], and point x=[2, -3] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).","answer":{"kind":"expression","value":"([2, -3], [4, -14], 1/4, [1, 1/2], 26, -1, Yes, positive_step_alone_does_not_guarantee_descent)","explanation":"Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([2, -3], [4, -14], 1/4, [1, 1/2], 26, -1, Yes, positive_step_alone_does_not_guarantee_descent)."},"skills":["Positive Step Size Alone Does Not Guarantee Objective Descent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0041","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"837ae2913870a83d6a1124e39b47e53e82541648626b03e51d833a866c8fd6db","family_id":"qf-mod50-gd-one-step","seed":202611193041,"semantic_template":"gd-one-step|{\"diagonal\":[[1,1],[5,1]],\"linear\":[[-1,1],[-1,1]],\"point\":[[-2,1],[-2,1]],\"step_size\":[1,4]}"},"fields":{"title":"Positive Step Size Alone Does Not Guarantee Objective Descent","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 5], linear b=[-1, -1], and point x=[-2, -2] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).","answer":{"kind":"expression","value":"([-2, -2], [-3, -11], 1/4, [-5/4, 3/4], 16, 43/16, Yes, positive_step_alone_does_not_guarantee_descent)","explanation":"Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([-2, -2], [-3, -11], 1/4, [-5/4, 3/4], 16, 43/16, Yes, positive_step_alone_does_not_guarantee_descent)."},"skills":["Positive Step Size Alone Does Not Guarantee Objective Descent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0042","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6308f0f90fa29fedfe2ed42adca7f47365f007b36a96472d40bf27041f48bc27","family_id":"qf-mod50-gd-one-step","seed":202611193042,"semantic_template":"gd-one-step|{\"diagonal\":[[2,1],[2,1]],\"linear\":[[0,1],[0,1]],\"point\":[[-1,1],[-1,1]],\"step_size\":[1,4]}"},"fields":{"title":"Positive Step Size Alone Does Not Guarantee Objective Descent","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 2], linear b=[0, 0], and point x=[-1, -1] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).","answer":{"kind":"expression","value":"([-1, -1], [-2, -2], 1/4, [-1/2, -1/2], 2, 1/2, Yes, positive_step_alone_does_not_guarantee_descent)","explanation":"Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([-1, -1], [-2, -2], 1/4, [-1/2, -1/2], 2, 1/2, Yes, positive_step_alone_does_not_guarantee_descent)."},"skills":["Positive Step Size Alone Does Not Guarantee Objective Descent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0043","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4abeff22bfb58eae724ec5ac210f1f1c942e6cef1e8bffc6a0623e740308d078","family_id":"qf-mod50-gd-one-step","seed":202611193043,"semantic_template":"gd-one-step|{\"diagonal\":[[3,1],[3,1]],\"linear\":[[1,1],[1,1]],\"point\":[[0,1],[0,1]],\"step_size\":[1,4]}"},"fields":{"title":"Positive Step Size Alone Does Not Guarantee Objective Descent","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 3], linear b=[1, 1], and point x=[0, 0] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).","answer":{"kind":"expression","value":"([0, 0], [1, 1], 1/4, [-1/4, -1/4], 0, -5/16, Yes, positive_step_alone_does_not_guarantee_descent)","explanation":"Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([0, 0], [1, 1], 1/4, [-1/4, -1/4], 0, -5/16, Yes, positive_step_alone_does_not_guarantee_descent)."},"skills":["Positive Step Size Alone Does Not Guarantee Objective Descent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0044","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2af2840ecfc3b2c19f60843d894e971d222190f7d2f2546c24e1c63e4b6cf99b","family_id":"qf-mod50-gd-one-step","seed":202611193044,"semantic_template":"gd-one-step|{\"diagonal\":[[1,1],[4,1]],\"linear\":[[2,1],[2,1]],\"point\":[[1,1],[1,1]],\"step_size\":[1,4]}"},"fields":{"title":"Positive Step Size Alone Does Not Guarantee Objective Descent","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 4], linear b=[2, 2], and point x=[1, 1] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).","answer":{"kind":"expression","value":"([1, 1], [3, 6], 1/4, [1/4, -1/2], 13/2, 1/32, Yes, positive_step_alone_does_not_guarantee_descent)","explanation":"Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([1, 1], [3, 6], 1/4, [1/4, -1/2], 13/2, 1/32, Yes, positive_step_alone_does_not_guarantee_descent)."},"skills":["Positive Step Size Alone Does Not Guarantee Objective Descent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0045","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d4442a03c8b6289c20939a9445a5da39884a2da2e640c2d7e61bfc031dcc709","family_id":"qf-mod50-gd-one-step","seed":202611193045,"semantic_template":"gd-one-step|{\"diagonal\":[[2,1],[5,1]],\"linear\":[[-2,1],[3,1]],\"point\":[[2,1],[2,1]],\"step_size\":[1,4]}"},"fields":{"title":"Positive Step Size Alone Does Not Guarantee Objective Descent","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 5], linear b=[-2, 3], and point x=[2, 2] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).","answer":{"kind":"expression","value":"([2, 2], [2, 13], 1/4, [3/2, -5/4], 16, -19/32, Yes, positive_step_alone_does_not_guarantee_descent)","explanation":"Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([2, 2], [2, 13], 1/4, [3/2, -5/4], 16, -19/32, Yes, positive_step_alone_does_not_guarantee_descent)."},"skills":["Positive Step Size Alone Does Not Guarantee Objective Descent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0046","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af44e2a5072e85843426ae1f64001618405f609600f4d1e22c36db1fdb048de3","family_id":"qf-mod50-gd-one-step","seed":202611193046,"semantic_template":"gd-one-step|{\"diagonal\":[[3,1],[2,1]],\"linear\":[[-1,1],[-3,1]],\"point\":[[-2,1],[3,1]],\"step_size\":[1,4]}"},"fields":{"title":"Positive Step Size Alone Does Not Guarantee Objective Descent","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 2], linear b=[-1, -3], and point x=[-2, 3] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).","answer":{"kind":"expression","value":"([-2, 3], [-7, 3], 1/4, [-1/4, 9/4], 8, -43/32, Yes, positive_step_alone_does_not_guarantee_descent)","explanation":"Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([-2, 3], [-7, 3], 1/4, [-1/4, 9/4], 8, -43/32, Yes, positive_step_alone_does_not_guarantee_descent)."},"skills":["Positive Step Size Alone Does Not Guarantee Objective Descent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0047","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5e92175bbec7dcfc6340ade1c8396001ed6542f1647f497f4bd2af83fe81059","family_id":"qf-mod50-gd-one-step","seed":202611193047,"semantic_template":"gd-one-step|{\"diagonal\":[[1,1],[3,1]],\"linear\":[[0,1],[-2,1]],\"point\":[[-1,1],[-3,1]],\"step_size\":[1,4]}"},"fields":{"title":"Positive Step Size Alone Does Not Guarantee Objective Descent","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 3], linear b=[0, -2], and point x=[-1, -3] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).","answer":{"kind":"expression","value":"([-1, -3], [-1, -11], 1/4, [-3/4, -1/4], 20, 7/8, Yes, positive_step_alone_does_not_guarantee_descent)","explanation":"Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([-1, -3], [-1, -11], 1/4, [-3/4, -1/4], 20, 7/8, Yes, positive_step_alone_does_not_guarantee_descent)."},"skills":["Positive Step Size Alone Does Not Guarantee Objective Descent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0048","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69197c79cdcfcf35ce05d640606d6f606df963db23d349d65329b1904f48e9d1","family_id":"qf-mod50-gd-one-step","seed":202611193048,"semantic_template":"gd-one-step|{\"diagonal\":[[2,1],[4,1]],\"linear\":[[1,1],[-1,1]],\"point\":[[0,1],[-2,1]],\"step_size\":[1,4]}"},"fields":{"title":"Positive Step Size Alone Does Not Guarantee Objective Descent","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 4], linear b=[1, -1], and point x=[0, -2] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).","answer":{"kind":"expression","value":"([0, -2], [1, -9], 1/4, [-1/4, 1/4], 10, -5/16, Yes, positive_step_alone_does_not_guarantee_descent)","explanation":"Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([0, -2], [1, -9], 1/4, [-1/4, 1/4], 10, -5/16, Yes, positive_step_alone_does_not_guarantee_descent)."},"skills":["Positive Step Size Alone Does Not Guarantee Objective Descent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0049","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8912851d24f45f9fbe41a04c73c6d5eaebf5af640d8fffdfcad554126725b77","family_id":"qf-mod50-gd-one-step","seed":202611193049,"semantic_template":"gd-one-step|{\"diagonal\":[[3,1],[5,1]],\"linear\":[[2,1],[0,1]],\"point\":[[1,1],[-1,1]],\"step_size\":[1,4]}"},"fields":{"title":"Positive Step Size Alone Does Not Guarantee Objective Descent","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 5], linear b=[2, 0], and point x=[1, -1] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).","answer":{"kind":"expression","value":"([1, -1], [5, -5], 1/4, [-1/4, 1/4], 6, -1/4, Yes, positive_step_alone_does_not_guarantee_descent)","explanation":"Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([1, -1], [5, -5], 1/4, [-1/4, 1/4], 6, -1/4, Yes, positive_step_alone_does_not_guarantee_descent)."},"skills":["Positive Step Size Alone Does Not Guarantee Objective Descent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0050","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7bbc7b334258d19ea60c84f4a272e092e3f1a568e2f1d27e3a30648e90f01c3a","family_id":"qf-mod50-gd-one-step","seed":202611193050,"semantic_template":"gd-one-step|{\"diagonal\":[[1,1],[2,1]],\"linear\":[[-2,1],[1,1]],\"point\":[[2,1],[0,1]],\"step_size\":[1,4]}"},"fields":{"title":"Positive Step Size Alone Does Not Guarantee Objective Descent","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 2], linear b=[-2, 1], and point x=[2, 0] in fixed standard Euclidean coordinates. Use positive step size alpha=1/4. Give (point, gradient, step_size, next_point, objective_before, objective_after, nonincreasing, step_scope).","answer":{"kind":"expression","value":"([2, 0], [0, 1], 1/4, [2, -1/4], -2, -35/16, Yes, positive_step_alone_does_not_guarantee_descent)","explanation":"Update every coordinate by x_next=x−alpha*gradient and evaluate the complete objective again. The nonincreasing flag is an actual before/after comparison and may be No. Do not silently impose the stronger safe-step range on this one-step family or infer descent from alpha>0 alone. The exact result, retaining the requested order, is ([2, 0], [0, 1], 1/4, [2, -1/4], -2, -35/16, Yes, positive_step_alone_does_not_guarantee_descent)."},"skills":["Positive Step Size Alone Does Not Guarantee Objective Descent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0051","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae057ab74cd0c53784154d2ebe45cc494f3e9082deac9637104edb83ded93e06","family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193051,"semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[1,1],[2,1]],\"linear\":[[-1,1],[-2,1]],\"point\":[[-2,1],[-3,1]],\"step_size\":[1,4],\"steps\":1}"},"fields":{"title":"A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 2], linear b=[-1, -2], and point x=[-2, -3] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).","answer":{"kind":"expression","value":"(2, 1/4, 1, 1, [[0, [-2, -3], 19, [-3, -8]], [1, [-5/4, -1], 161/32, [-9/4, -4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation":"Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (2, 1/4, 1, 1, [[0, [-2, -3], 19, [-3, -8]], [1, [-5/4, -1], 161/32, [-9/4, -4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)."},"skills":["A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0052","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6134f1f71eb5577066e84a8e2fb248b8eda60a0423bab1932e2264e4a7f32fce","family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193052,"semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[2,1],[3,1]],\"linear\":[[0,1],[-1,1]],\"point\":[[-1,1],[-2,1]],\"step_size\":[1,4],\"steps\":1}"},"fields":{"title":"A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 3], linear b=[0, -1], and point x=[-1, -2] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).","answer":{"kind":"expression","value":"(3, 1/4, 2/3, 1, [[0, [-1, -2], 9, [-2, -7]], [1, [-1/2, -1/4], 19/32, [-1, -7/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation":"Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (3, 1/4, 2/3, 1, [[0, [-1, -2], 9, [-2, -7]], [1, [-1/2, -1/4], 19/32, [-1, -7/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)."},"skills":["A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0053","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3664d333711285313ab8be7b4ecb7757eb386a33c1ebb5f039ecf2b87a6c0a12","family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193053,"semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[3,1],[4,1]],\"linear\":[[1,1],[0,1]],\"point\":[[0,1],[-1,1]],\"step_size\":[1,4],\"steps\":1}"},"fields":{"title":"A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 4], linear b=[1, 0], and point x=[0, -1] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).","answer":{"kind":"expression","value":"(4, 1/4, 1/2, 1, [[0, [0, -1], 2, [1, -4]], [1, [-1/4, 0], -5/32, [1/4, 0]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation":"Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (4, 1/4, 1/2, 1, [[0, [0, -1], 2, [1, -4]], [1, [-1/4, 0], -5/32, [1/4, 0]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)."},"skills":["A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0054","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0297bd0ab25b6b6216867d65bc37ba0eac3cfa2456a1d9f98c9f2b8db3424223","family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193054,"semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[1,1],[5,1]],\"linear\":[[2,1],[1,1]],\"point\":[[1,1],[0,1]],\"step_size\":[1,4],\"steps\":1}"},"fields":{"title":"A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 5], linear b=[2, 1], and point x=[1, 0] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).","answer":{"kind":"expression","value":"(5, 1/4, 2/5, 1, [[0, [1, 0], 5/2, [3, 1]], [1, [1/4, -1/4], 7/16, [9/4, -1/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation":"Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (5, 1/4, 2/5, 1, [[0, [1, 0], 5/2, [3, 1]], [1, [1/4, -1/4], 7/16, [9/4, -1/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)."},"skills":["A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0055","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d5405cb453d9b6ec7ae3f49bdad5907fd4b99b1e36180d3950c79d9ed9eddd0","family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193055,"semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[2,1],[2,1]],\"linear\":[[-2,1],[2,1]],\"point\":[[2,1],[1,1]],\"step_size\":[1,4],\"steps\":1}"},"fields":{"title":"A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 2], linear b=[-2, 2], and point x=[2, 1] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).","answer":{"kind":"expression","value":"(2, 1/4, 1, 1, [[0, [2, 1], 3, [2, 4]], [1, [3/2, 0], -3/4, [1, 2]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation":"Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (2, 1/4, 1, 1, [[0, [2, 1], 3, [2, 4]], [1, [3/2, 0], -3/4, [1, 2]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)."},"skills":["A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0056","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b6a9032ad8bb095ee636d395e752bfbbefae48463963a4920a8fe406d85aed46","family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193056,"semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[3,1],[3,1]],\"linear\":[[-1,1],[3,1]],\"point\":[[-2,1],[2,1]],\"step_size\":[1,4],\"steps\":1}"},"fields":{"title":"A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 3], linear b=[-1, 3], and point x=[-2, 2] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).","answer":{"kind":"expression","value":"(3, 1/4, 2/3, 1, [[0, [-2, 2], 20, [-7, 9]], [1, [-1/4, -1/4], -5/16, [-7/4, 9/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation":"Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (3, 1/4, 2/3, 1, [[0, [-2, 2], 20, [-7, 9]], [1, [-1/4, -1/4], -5/16, [-7/4, 9/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)."},"skills":["A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0057","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"965f9a5af58bcbbf61ae86d403a6a3f1ba99fb43f2c24a991d9077e8fc73962e","family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193057,"semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[1,1],[4,1]],\"linear\":[[0,1],[-3,1]],\"point\":[[-1,1],[3,1]],\"step_size\":[1,4],\"steps\":1}"},"fields":{"title":"A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 4], linear b=[0, -3], and point x=[-1, 3] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).","answer":{"kind":"expression","value":"(4, 1/4, 1/2, 1, [[0, [-1, 3], 19/2, [-1, 9]], [1, [-3/4, 3/4], -27/32, [-3/4, 0]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation":"Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (4, 1/4, 1/2, 1, [[0, [-1, 3], 19/2, [-1, 9]], [1, [-3/4, 3/4], -27/32, [-3/4, 0]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)."},"skills":["A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0058","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b8157ec51ab1c118a4caeaf61e3f990941b755111472e591a9e75b0f9ca92c4","family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193058,"semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[2,1],[5,1]],\"linear\":[[1,1],[-2,1]],\"point\":[[0,1],[-3,1]],\"step_size\":[1,4],\"steps\":1}"},"fields":{"title":"A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 5], linear b=[1, -2], and point x=[0, -3] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).","answer":{"kind":"expression","value":"(5, 1/4, 2/5, 1, [[0, [0, -3], 57/2, [1, -17]], [1, [-1/4, 5/4], 39/32, [1/2, 17/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation":"Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (5, 1/4, 2/5, 1, [[0, [0, -3], 57/2, [1, -17]], [1, [-1/4, 5/4], 39/32, [1/2, 17/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)."},"skills":["A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0059","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d14fac3672df0e68ebc91ec9b1a376646d78d71f51409879733b4f945237a05","family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193059,"semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[3,1],[2,1]],\"linear\":[[2,1],[-1,1]],\"point\":[[1,1],[-2,1]],\"step_size\":[1,4],\"steps\":1}"},"fields":{"title":"A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 2], linear b=[2, -1], and point x=[1, -2] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).","answer":{"kind":"expression","value":"(3, 1/4, 2/3, 1, [[0, [1, -2], 19/2, [5, -5]], [1, [-1/4, -3/4], 29/32, [5/4, -5/2]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation":"Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (3, 1/4, 2/3, 1, [[0, [1, -2], 19/2, [5, -5]], [1, [-1/4, -3/4], 29/32, [5/4, -5/2]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)."},"skills":["A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0060","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1358da639e2f138ab4375507ddd7241e65b8e403f1e0f8c010582c2406979a01","family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193060,"semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[1,1],[3,1]],\"linear\":[[-2,1],[0,1]],\"point\":[[2,1],[-1,1]],\"step_size\":[1,4],\"steps\":1}"},"fields":{"title":"A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 3], linear b=[-2, 0], and point x=[2, -1] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).","answer":{"kind":"expression","value":"(3, 1/4, 2/3, 1, [[0, [2, -1], -1/2, [0, -3]], [1, [2, -1/4], -61/32, [0, -3/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation":"Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (3, 1/4, 2/3, 1, [[0, [2, -1], -1/2, [0, -3]], [1, [2, -1/4], -61/32, [0, -3/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)."},"skills":["A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0061","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a43bc865f0f791c634510e5c8132d95c0bb8461eee97462eec989911bb92e4d","family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193061,"semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[2,1],[4,1]],\"linear\":[[-1,1],[1,1]],\"point\":[[-2,1],[0,1]],\"step_size\":[1,4],\"steps\":1}"},"fields":{"title":"A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 4], linear b=[-1, 1], and point x=[-2, 0] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).","answer":{"kind":"expression","value":"(4, 1/4, 1/2, 1, [[0, [-2, 0], 6, [-5, 1]], [1, [-3/4, -1/4], 19/16, [-5/2, 0]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation":"Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (4, 1/4, 1/2, 1, [[0, [-2, 0], 6, [-5, 1]], [1, [-3/4, -1/4], 19/16, [-5/2, 0]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)."},"skills":["A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0062","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6550026463eab2e5ebe4d99bfdd1a0b532d0cdf5037d1901e6456f6437a11e1","family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193062,"semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[3,1],[5,1]],\"linear\":[[0,1],[2,1]],\"point\":[[-1,1],[1,1]],\"step_size\":[1,4],\"steps\":1}"},"fields":{"title":"A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 5], linear b=[0, 2], and point x=[-1, 1] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).","answer":{"kind":"expression","value":"(5, 1/4, 2/5, 1, [[0, [-1, 1], 6, [-3, 7]], [1, [-1/4, -3/4], 0, [-3/4, -7/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation":"Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (5, 1/4, 2/5, 1, [[0, [-1, 1], 6, [-3, 7]], [1, [-1/4, -3/4], 0, [-3/4, -7/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)."},"skills":["A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0063","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b05e8167735eefed84502d06e66b8e9aa6ded5f2f9a05ffa4eb441af8ccaab9","family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193063,"semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[1,1],[2,1]],\"linear\":[[1,1],[3,1]],\"point\":[[0,1],[2,1]],\"step_size\":[1,4],\"steps\":1}"},"fields":{"title":"A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 2], linear b=[1, 3], and point x=[0, 2] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).","answer":{"kind":"expression","value":"(2, 1/4, 1, 1, [[0, [0, 2], 10, [1, 7]], [1, [-1/4, 1/4], 19/32, [3/4, 7/2]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation":"Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (2, 1/4, 1, 1, [[0, [0, 2], 10, [1, 7]], [1, [-1/4, 1/4], 19/32, [3/4, 7/2]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)."},"skills":["A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0064","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"933cc97756a893c5d0ac2f98a1793d8f250847d0bfb80675c0914999ae2e33a7","family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193064,"semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[2,1],[3,1]],\"linear\":[[2,1],[-3,1]],\"point\":[[1,1],[3,1]],\"step_size\":[1,4],\"steps\":1}"},"fields":{"title":"A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 3], linear b=[2, -3], and point x=[1, 3] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).","answer":{"kind":"expression","value":"(3, 1/4, 2/3, 1, [[0, [1, 3], 15/2, [4, 6]], [1, [0, 3/2], -9/8, [2, 3/2]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation":"Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (3, 1/4, 2/3, 1, [[0, [1, 3], 15/2, [4, 6]], [1, [0, 3/2], -9/8, [2, 3/2]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)."},"skills":["A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0065","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5f7822f0855d7451f0cad9d1d0d92f0b6eb2227f13c4d488527a6343434044d","family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193065,"semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[3,1],[4,1]],\"linear\":[[-2,1],[-2,1]],\"point\":[[2,1],[-3,1]],\"step_size\":[1,4],\"steps\":1}"},"fields":{"title":"A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 4], linear b=[-2, -2], and point x=[2, -3] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).","answer":{"kind":"expression","value":"(4, 1/4, 1/2, 1, [[0, [2, -3], 26, [4, -14]], [1, [1, 1/2], -1, [1, 0]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation":"Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (4, 1/4, 1/2, 1, [[0, [2, -3], 26, [4, -14]], [1, [1, 1/2], -1, [1, 0]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)."},"skills":["A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0066","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a5ee0a22eebd3ca442d9851b9d30f16c5e1ebd92c0a4d6f3070fbc984b6b180","family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193066,"semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[1,1],[5,1]],\"linear\":[[-1,1],[-1,1]],\"point\":[[-2,1],[-2,1]],\"step_size\":[1,4],\"steps\":1}"},"fields":{"title":"A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 5], linear b=[-1, -1], and point x=[-2, -2] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).","answer":{"kind":"expression","value":"(5, 1/4, 2/5, 1, [[0, [-2, -2], 16, [-3, -11]], [1, [-5/4, 3/4], 43/16, [-9/4, 11/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation":"Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (5, 1/4, 2/5, 1, [[0, [-2, -2], 16, [-3, -11]], [1, [-5/4, 3/4], 43/16, [-9/4, 11/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)."},"skills":["A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0067","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a42bc5677df121f5198b95b2afceba5f8ca0ab5c08c2ad8acd70c2b81763d50c","family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193067,"semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[2,1],[2,1]],\"linear\":[[0,1],[0,1]],\"point\":[[-1,1],[-1,1]],\"step_size\":[1,4],\"steps\":1}"},"fields":{"title":"A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 2], linear b=[0, 0], and point x=[-1, -1] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).","answer":{"kind":"expression","value":"(2, 1/4, 1, 1, [[0, [-1, -1], 2, [-2, -2]], [1, [-1/2, -1/2], 1/2, [-1, -1]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation":"Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (2, 1/4, 1, 1, [[0, [-1, -1], 2, [-2, -2]], [1, [-1/2, -1/2], 1/2, [-1, -1]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)."},"skills":["A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0068","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6fa901ab273f8d1ddd8382f828104cebeda42b5ab48162c7a241e33a9a99cbd","family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193068,"semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[3,1],[3,1]],\"linear\":[[1,1],[1,1]],\"point\":[[0,1],[0,1]],\"step_size\":[1,4],\"steps\":1}"},"fields":{"title":"A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 3], linear b=[1, 1], and point x=[0, 0] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).","answer":{"kind":"expression","value":"(3, 1/4, 2/3, 1, [[0, [0, 0], 0, [1, 1]], [1, [-1/4, -1/4], -5/16, [1/4, 1/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation":"Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (3, 1/4, 2/3, 1, [[0, [0, 0], 0, [1, 1]], [1, [-1/4, -1/4], -5/16, [1/4, 1/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)."},"skills":["A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0069","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6bf03e9e211dd6bd0dd8f67b4155e5d734704eda4583b041f751cb709ec4e5b2","family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193069,"semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[1,1],[4,1]],\"linear\":[[2,1],[2,1]],\"point\":[[1,1],[1,1]],\"step_size\":[1,4],\"steps\":1}"},"fields":{"title":"A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 4], linear b=[2, 2], and point x=[1, 1] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).","answer":{"kind":"expression","value":"(4, 1/4, 1/2, 1, [[0, [1, 1], 13/2, [3, 6]], [1, [1/4, -1/2], 1/32, [9/4, 0]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation":"Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (4, 1/4, 1/2, 1, [[0, [1, 1], 13/2, [3, 6]], [1, [1/4, -1/2], 1/32, [9/4, 0]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)."},"skills":["A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0070","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"313cbe99cd57e6dc85661006a157d84df9fd61f738862439701d3f5657c9b9e1","family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193070,"semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[2,1],[5,1]],\"linear\":[[-2,1],[3,1]],\"point\":[[2,1],[2,1]],\"step_size\":[1,4],\"steps\":1}"},"fields":{"title":"A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 5], linear b=[-2, 3], and point x=[2, 2] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).","answer":{"kind":"expression","value":"(5, 1/4, 2/5, 1, [[0, [2, 2], 16, [2, 13]], [1, [3/2, -5/4], -19/32, [1, -13/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation":"Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (5, 1/4, 2/5, 1, [[0, [2, 2], 16, [2, 13]], [1, [3/2, -5/4], -19/32, [1, -13/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)."},"skills":["A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0071","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b27c6f5f83c647ff4a39c18da21a8ab8561b7a38936de3b00078052eb932c07a","family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193071,"semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[3,1],[2,1]],\"linear\":[[-1,1],[-3,1]],\"point\":[[-2,1],[3,1]],\"step_size\":[1,4],\"steps\":1}"},"fields":{"title":"A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 2], linear b=[-1, -3], and point x=[-2, 3] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).","answer":{"kind":"expression","value":"(3, 1/4, 2/3, 1, [[0, [-2, 3], 8, [-7, 3]], [1, [-1/4, 9/4], -43/32, [-7/4, 3/2]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation":"Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (3, 1/4, 2/3, 1, [[0, [-2, 3], 8, [-7, 3]], [1, [-1/4, 9/4], -43/32, [-7/4, 3/2]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)."},"skills":["A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0072","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7cc2d13df9e4f2795ff32495700563aa7d3e6474cb851658447b913a11502d85","family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193072,"semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[1,1],[3,1]],\"linear\":[[0,1],[-2,1]],\"point\":[[-1,1],[-3,1]],\"step_size\":[1,4],\"steps\":1}"},"fields":{"title":"A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 3], linear b=[0, -2], and point x=[-1, -3] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).","answer":{"kind":"expression","value":"(3, 1/4, 2/3, 1, [[0, [-1, -3], 20, [-1, -11]], [1, [-3/4, -1/4], 7/8, [-3/4, -11/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation":"Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (3, 1/4, 2/3, 1, [[0, [-1, -3], 20, [-1, -11]], [1, [-3/4, -1/4], 7/8, [-3/4, -11/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)."},"skills":["A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0073","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b4324a67dc1ad3f11cdfaa603d43b038a10d7ea65f70e3f65f00518ba519cc5","family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193073,"semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[2,1],[4,1]],\"linear\":[[1,1],[-1,1]],\"point\":[[0,1],[-2,1]],\"step_size\":[1,4],\"steps\":1}"},"fields":{"title":"A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[2, 4], linear b=[1, -1], and point x=[0, -2] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).","answer":{"kind":"expression","value":"(4, 1/4, 1/2, 1, [[0, [0, -2], 10, [1, -9]], [1, [-1/4, 1/4], -5/16, [1/2, 0]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation":"Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (4, 1/4, 1/2, 1, [[0, [0, -2], 10, [1, -9]], [1, [-1/4, 1/4], -5/16, [1/2, 0]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)."},"skills":["A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0074","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e21e2203e5db8577f42ca2dbb18996bcc028c89b995908ca34a0a69760592585","family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193074,"semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[3,1],[5,1]],\"linear\":[[2,1],[0,1]],\"point\":[[1,1],[-1,1]],\"step_size\":[1,4],\"steps\":1}"},"fields":{"title":"A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[3, 5], linear b=[2, 0], and point x=[1, -1] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).","answer":{"kind":"expression","value":"(5, 1/4, 2/5, 1, [[0, [1, -1], 6, [5, -5]], [1, [-1/4, 1/4], -1/4, [5/4, 5/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation":"Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (5, 1/4, 2/5, 1, [[0, [1, -1], 6, [5, -5]], [1, [-1/4, 1/4], -1/4, [5/4, 5/4]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)."},"skills":["A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0075","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"321276bff86ceca955f6e7092a6f9e481260866e75945928f634a17fc3425290","family_id":"qf-mod50-gd-safe-quadratic-trace","seed":202611193075,"semantic_template":"gd-safe-quadratic-trace|{\"diagonal\":[[1,1],[2,1]],\"linear\":[[-2,1],[1,1]],\"point\":[[2,1],[0,1]],\"step_size\":[1,4],\"steps\":1}"},"fields":{"title":"A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace","stem":"Use f(x)=1/2*sum_i d_i*x_i^2+sum_i b_i*x_i, with positive diagonal d=[1, 2], linear b=[-2, 1], and point x=[2, 0] in fixed standard Euclidean coordinates. Use constant alpha=1/4, step count=1. Give (largest_diagonal_L, step_size, strict_step_upper_bound, step_count, iterate_objective_gradient_records, finite_trace_nonincreasing, unbounded_convergence_claim, finite_scope).","answer":{"kind":"expression","value":"(2, 1/4, 1, 1, [[0, [2, 0], -2, [0, 1]], [1, [2, -1/4], -35/16, [0, 1/2]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)","explanation":"Compute L=max_i d_i and require strictly0<alpha<2/L. Retain the initial state and every iterate, objective and gradient after each update. Check finite nonincrease separately from any unbounded convergence or general convex/nonconvex rate claim, which this trace does not prove. The exact result, retaining the requested order, is (2, 1/4, 1, 1, [[0, [2, 0], -2, [0, 1]], [1, [2, -1/4], -35/16, [0, 1/2]]], Yes, No, verified_finite_safe_trace_is_not_an_unbounded_convergence_certificate)."},"skills":["A Strict Safe Step Gives a Checked Finite Diagonal-Quadratic Trace"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0076","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b437d970f7971396655f7808cbe60796ec41de0e35736fbfeb7d7434fe343070","family_id":"qf-mod50-gd-boundary","seed":202611193076,"semantic_template":"gd-boundary|{\"mode\":\"oversized_step\",\"model\":{\"curvature\":[1,1],\"point\":[1,1],\"step_size\":[3,1]}}"},"fields":{"title":"Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences","stem":"Use mode=oversized_step, complete model={\"curvature\":1,\"point\":1,\"step_size\":3}. Give (mode, curvature, point, step_size, next_point, objective_before, objective_after, automatic_descent_claim, boundary).","answer":{"kind":"expression","value":"(oversized_step, 1, 1, 3, -2, 1/2, 2, No, positive_but_oversized_step_can_increase_the_objective)","explanation":"A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (oversized_step, 1, 1, 3, -2, 1/2, 2, No, positive_but_oversized_step_can_increase_the_objective)."},"skills":["Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0077","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5af61c9bd84fc8f51c37c5ef0f0f33bae8dba8931badb3c3456ac12a5c28b767","family_id":"qf-mod50-gd-boundary","seed":202611193077,"semantic_template":"gd-boundary|{\"mode\":\"zero_gradient_nonminimum\",\"model\":{\"scale\":[2,1]}}"},"fields":{"title":"Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences","stem":"Use mode=zero_gradient_nonminimum, complete model={\"scale\":2}. Give (mode, objective_description, stationary_point, gradient_statement, actual_point_status, minimum_claim, boundary).","answer":{"kind":"expression","value":"(zero_gradient_nonminimum, f(x)=-2*x^2, x=0, gradient=0, strict_local_maximum, No, zero_gradient_is_stationary_not_a_minimum_certificate)","explanation":"A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (zero_gradient_nonminimum, f(x)=-2*x^2, x=0, gradient=0, strict_local_maximum, No, zero_gradient_is_stationary_not_a_minimum_certificate)."},"skills":["Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0078","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2bc07e63595e2be41a4c5bebd8c14b3c2417130eda805e57f4881d0485d9040","family_id":"qf-mod50-gd-boundary","seed":202611193078,"semantic_template":"gd-boundary|{\"mode\":\"finite_trace_only\",\"model\":{\"diagonal\":[[1,1],[2,1]],\"linear\":[[2,1],[-2,1]],\"point\":[[3,1],[1,1]],\"step_size\":[1,4],\"steps\":1}}"},"fields":{"title":"Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences","stem":"Use mode=finite_trace_only, complete model={\"diagonal\":[1,2],\"linear\":[2,-2],\"point\":[3,1],\"step_size\":[1,4],\"steps\":1}. Give (mode, step_count, iterate_objective_records, convergence_or_global_optimality_claim, boundary).","answer":{"kind":"expression","value":"(finite_trace_only, 1, [[0, [3, 1], 19/2], [1, [7/4, 1], 129/32]], No, finite_trace_does_not_prove_convergence_or_global_optimality)","explanation":"A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (finite_trace_only, 1, [[0, [3, 1], 19/2], [1, [7/4, 1], 129/32]], No, finite_trace_does_not_prove_convergence_or_global_optimality)."},"skills":["Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0079","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93ac6538b87d5d5768b84d1db1696cba66dc46eb28b76edc78a943107fa0de77","family_id":"qf-mod50-gd-boundary","seed":202611193079,"semantic_template":"gd-boundary|{\"mode\":\"constrained_confusion\",\"model\":{\"gradient\":[4,1],\"lower\":[-1,1],\"point\":[0,1],\"step_size\":[2,1],\"upper\":[1,1]}}"},"fields":{"title":"Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences","stem":"Use mode=constrained_confusion, complete model={\"point\":0,\"gradient\":4,\"step_size\":2,\"lower\":-1,\"upper\":1}. Give (mode, lower_bound, upper_bound, feasible_point, unconstrained_next_point, constrained_descent_claim, boundary).","answer":{"kind":"expression","value":"(constrained_confusion, -1, 1, 0, -8, No, unconstrained_gradient_step_is_not_projected_or_constrained_descent)","explanation":"A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (constrained_confusion, -1, 1, 0, -8, No, unconstrained_gradient_step_is_not_projected_or_constrained_descent)."},"skills":["Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0080","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb479f6c155531ba1cf4b5020937f6b35e33b83208293a3b0fd5c6702b014662","family_id":"qf-mod50-gd-boundary","seed":202611193080,"semantic_template":"gd-boundary|{\"mode\":\"oversized_step\",\"model\":{\"curvature\":[5,1],\"point\":[5,1],\"step_size\":[3,1]}}"},"fields":{"title":"Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences","stem":"Use mode=oversized_step, complete model={\"curvature\":5,\"point\":5,\"step_size\":3}. Give (mode, curvature, point, step_size, next_point, objective_before, objective_after, automatic_descent_claim, boundary).","answer":{"kind":"expression","value":"(oversized_step, 5, 5, 3, -70, 125/2, 12250, No, positive_but_oversized_step_can_increase_the_objective)","explanation":"A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (oversized_step, 5, 5, 3, -70, 125/2, 12250, No, positive_but_oversized_step_can_increase_the_objective)."},"skills":["Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0081","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e055c98d52571412cc174262ecd86198e8f2905864f067a8b2b046585918f6ea","family_id":"qf-mod50-gd-boundary","seed":202611193081,"semantic_template":"gd-boundary|{\"mode\":\"zero_gradient_nonminimum\",\"model\":{\"scale\":[6,1]}}"},"fields":{"title":"Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences","stem":"Use mode=zero_gradient_nonminimum, complete model={\"scale\":6}. Give (mode, objective_description, stationary_point, gradient_statement, actual_point_status, minimum_claim, boundary).","answer":{"kind":"expression","value":"(zero_gradient_nonminimum, f(x)=-6*x^2, x=0, gradient=0, strict_local_maximum, No, zero_gradient_is_stationary_not_a_minimum_certificate)","explanation":"A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (zero_gradient_nonminimum, f(x)=-6*x^2, x=0, gradient=0, strict_local_maximum, No, zero_gradient_is_stationary_not_a_minimum_certificate)."},"skills":["Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0082","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f93fb03af1cfee291bedb71220ef150041092bb46dd926574554af06b8c4861b","family_id":"qf-mod50-gd-boundary","seed":202611193082,"semantic_template":"gd-boundary|{\"mode\":\"finite_trace_only\",\"model\":{\"diagonal\":[[1,1],[2,1]],\"linear\":[[6,1],[-6,1]],\"point\":[[7,1],[1,1]],\"step_size\":[1,4],\"steps\":1}}"},"fields":{"title":"Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences","stem":"Use mode=finite_trace_only, complete model={\"diagonal\":[1,2],\"linear\":[6,-6],\"point\":[7,1],\"step_size\":[1,4],\"steps\":1}. Give (mode, step_count, iterate_objective_records, convergence_or_global_optimality_claim, boundary).","answer":{"kind":"expression","value":"(finite_trace_only, 1, [[0, [7, 1], 123/2], [1, [15/4, 2], 689/32]], No, finite_trace_does_not_prove_convergence_or_global_optimality)","explanation":"A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (finite_trace_only, 1, [[0, [7, 1], 123/2], [1, [15/4, 2], 689/32]], No, finite_trace_does_not_prove_convergence_or_global_optimality)."},"skills":["Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0083","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef178341a4e9afb2fee345209fbe35cb347c279c179af977a180158538e0b1c4","family_id":"qf-mod50-gd-boundary","seed":202611193083,"semantic_template":"gd-boundary|{\"mode\":\"constrained_confusion\",\"model\":{\"gradient\":[8,1],\"lower\":[-1,1],\"point\":[0,1],\"step_size\":[2,1],\"upper\":[1,1]}}"},"fields":{"title":"Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences","stem":"Use mode=constrained_confusion, complete model={\"point\":0,\"gradient\":8,\"step_size\":2,\"lower\":-1,\"upper\":1}. Give (mode, lower_bound, upper_bound, feasible_point, unconstrained_next_point, constrained_descent_claim, boundary).","answer":{"kind":"expression","value":"(constrained_confusion, -1, 1, 0, -16, No, unconstrained_gradient_step_is_not_projected_or_constrained_descent)","explanation":"A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (constrained_confusion, -1, 1, 0, -16, No, unconstrained_gradient_step_is_not_projected_or_constrained_descent)."},"skills":["Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0084","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b330f1da90cd915c9e634515e72f633aea2a98aae2610cdd2fafe610f1344ac","family_id":"qf-mod50-gd-boundary","seed":202611193084,"semantic_template":"gd-boundary|{\"mode\":\"oversized_step\",\"model\":{\"curvature\":[4,1],\"point\":[9,1],\"step_size\":[3,1]}}"},"fields":{"title":"Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences","stem":"Use mode=oversized_step, complete model={\"curvature\":4,\"point\":9,\"step_size\":3}. Give (mode, curvature, point, step_size, next_point, objective_before, objective_after, automatic_descent_claim, boundary).","answer":{"kind":"expression","value":"(oversized_step, 4, 9, 3, -99, 162, 19602, No, positive_but_oversized_step_can_increase_the_objective)","explanation":"A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (oversized_step, 4, 9, 3, -99, 162, 19602, No, positive_but_oversized_step_can_increase_the_objective)."},"skills":["Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0085","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e54485959e65533398fe2adafe21c08680309f96c8ba1b2bf5378d7b9b6eda5","family_id":"qf-mod50-gd-boundary","seed":202611193085,"semantic_template":"gd-boundary|{\"mode\":\"zero_gradient_nonminimum\",\"model\":{\"scale\":[10,1]}}"},"fields":{"title":"Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences","stem":"Use mode=zero_gradient_nonminimum, complete model={\"scale\":10}. Give (mode, objective_description, stationary_point, gradient_statement, actual_point_status, minimum_claim, boundary).","answer":{"kind":"expression","value":"(zero_gradient_nonminimum, f(x)=-10*x^2, x=0, gradient=0, strict_local_maximum, No, zero_gradient_is_stationary_not_a_minimum_certificate)","explanation":"A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (zero_gradient_nonminimum, f(x)=-10*x^2, x=0, gradient=0, strict_local_maximum, No, zero_gradient_is_stationary_not_a_minimum_certificate)."},"skills":["Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0086","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"239e23119cb7061854b8a54e954467a6dd776cd63e2d074b419a0bd0d190b02b","family_id":"qf-mod50-gd-boundary","seed":202611193086,"semantic_template":"gd-boundary|{\"mode\":\"finite_trace_only\",\"model\":{\"diagonal\":[[1,1],[2,1]],\"linear\":[[10,1],[-10,1]],\"point\":[[11,1],[1,1]],\"step_size\":[1,4],\"steps\":1}}"},"fields":{"title":"Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences","stem":"Use mode=finite_trace_only, complete model={\"diagonal\":[1,2],\"linear\":[10,-10],\"point\":[11,1],\"step_size\":[1,4],\"steps\":1}. Give (mode, step_count, iterate_objective_records, convergence_or_global_optimality_claim, boundary).","answer":{"kind":"expression","value":"(finite_trace_only, 1, [[0, [11, 1], 323/2], [1, [23/4, 3], 1697/32]], No, finite_trace_does_not_prove_convergence_or_global_optimality)","explanation":"A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (finite_trace_only, 1, [[0, [11, 1], 323/2], [1, [23/4, 3], 1697/32]], No, finite_trace_does_not_prove_convergence_or_global_optimality)."},"skills":["Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0087","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cbf83be21dd01ae634ca34c6ccf85c7d5b4091e3f9eafac73627b9ca1c5f6023","family_id":"qf-mod50-gd-boundary","seed":202611193087,"semantic_template":"gd-boundary|{\"mode\":\"constrained_confusion\",\"model\":{\"gradient\":[12,1],\"lower\":[-1,1],\"point\":[0,1],\"step_size\":[2,1],\"upper\":[1,1]}}"},"fields":{"title":"Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences","stem":"Use mode=constrained_confusion, complete model={\"point\":0,\"gradient\":12,\"step_size\":2,\"lower\":-1,\"upper\":1}. Give (mode, lower_bound, upper_bound, feasible_point, unconstrained_next_point, constrained_descent_claim, boundary).","answer":{"kind":"expression","value":"(constrained_confusion, -1, 1, 0, -24, No, unconstrained_gradient_step_is_not_projected_or_constrained_descent)","explanation":"A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (constrained_confusion, -1, 1, 0, -24, No, unconstrained_gradient_step_is_not_projected_or_constrained_descent)."},"skills":["Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0088","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eec6ab51dcfcc31b42a95c777fa874a2e14d7369e5afdc52feac4c43f0577c56","family_id":"qf-mod50-gd-boundary","seed":202611193088,"semantic_template":"gd-boundary|{\"mode\":\"oversized_step\",\"model\":{\"curvature\":[3,1],\"point\":[13,1],\"step_size\":[3,1]}}"},"fields":{"title":"Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences","stem":"Use mode=oversized_step, complete model={\"curvature\":3,\"point\":13,\"step_size\":3}. Give (mode, curvature, point, step_size, next_point, objective_before, objective_after, automatic_descent_claim, boundary).","answer":{"kind":"expression","value":"(oversized_step, 3, 13, 3, -104, 507/2, 16224, No, positive_but_oversized_step_can_increase_the_objective)","explanation":"A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (oversized_step, 3, 13, 3, -104, 507/2, 16224, No, positive_but_oversized_step_can_increase_the_objective)."},"skills":["Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0089","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16d71488db3a5cf0a18b2e6793e5f372ffe48f59f521a43fb05e0f774a27b409","family_id":"qf-mod50-gd-boundary","seed":202611193089,"semantic_template":"gd-boundary|{\"mode\":\"zero_gradient_nonminimum\",\"model\":{\"scale\":[14,1]}}"},"fields":{"title":"Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences","stem":"Use mode=zero_gradient_nonminimum, complete model={\"scale\":14}. Give (mode, objective_description, stationary_point, gradient_statement, actual_point_status, minimum_claim, boundary).","answer":{"kind":"expression","value":"(zero_gradient_nonminimum, f(x)=-14*x^2, x=0, gradient=0, strict_local_maximum, No, zero_gradient_is_stationary_not_a_minimum_certificate)","explanation":"A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (zero_gradient_nonminimum, f(x)=-14*x^2, x=0, gradient=0, strict_local_maximum, No, zero_gradient_is_stationary_not_a_minimum_certificate)."},"skills":["Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0090","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f63019781111f8a82499ed0cbbf000df8a6af959ce9616b6e414426b7ceb7686","family_id":"qf-mod50-gd-boundary","seed":202611193090,"semantic_template":"gd-boundary|{\"mode\":\"finite_trace_only\",\"model\":{\"diagonal\":[[1,1],[2,1]],\"linear\":[[14,1],[-14,1]],\"point\":[[15,1],[1,1]],\"step_size\":[1,4],\"steps\":1}}"},"fields":{"title":"Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences","stem":"Use mode=finite_trace_only, complete model={\"diagonal\":[1,2],\"linear\":[14,-14],\"point\":[15,1],\"step_size\":[1,4],\"steps\":1}. Give (mode, step_count, iterate_objective_records, convergence_or_global_optimality_claim, boundary).","answer":{"kind":"expression","value":"(finite_trace_only, 1, [[0, [15, 1], 619/2], [1, [31/4, 4], 3153/32]], No, finite_trace_does_not_prove_convergence_or_global_optimality)","explanation":"A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (finite_trace_only, 1, [[0, [15, 1], 619/2], [1, [31/4, 4], 3153/32]], No, finite_trace_does_not_prove_convergence_or_global_optimality)."},"skills":["Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0091","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e834aebfea5720b2517c94f4b614ec7bb9305608350c72135d9f3d99cbb15b22","family_id":"qf-mod50-gd-boundary","seed":202611193091,"semantic_template":"gd-boundary|{\"mode\":\"constrained_confusion\",\"model\":{\"gradient\":[16,1],\"lower\":[-1,1],\"point\":[0,1],\"step_size\":[2,1],\"upper\":[1,1]}}"},"fields":{"title":"Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences","stem":"Use mode=constrained_confusion, complete model={\"point\":0,\"gradient\":16,\"step_size\":2,\"lower\":-1,\"upper\":1}. Give (mode, lower_bound, upper_bound, feasible_point, unconstrained_next_point, constrained_descent_claim, boundary).","answer":{"kind":"expression","value":"(constrained_confusion, -1, 1, 0, -32, No, unconstrained_gradient_step_is_not_projected_or_constrained_descent)","explanation":"A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (constrained_confusion, -1, 1, 0, -32, No, unconstrained_gradient_step_is_not_projected_or_constrained_descent)."},"skills":["Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0092","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95fd5b957260fd2c034d3f607a7307e39a587351a0a6e49e5e72e95eba5de1db","family_id":"qf-mod50-gd-boundary","seed":202611193092,"semantic_template":"gd-boundary|{\"mode\":\"oversized_step\",\"model\":{\"curvature\":[2,1],\"point\":[17,1],\"step_size\":[3,1]}}"},"fields":{"title":"Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences","stem":"Use mode=oversized_step, complete model={\"curvature\":2,\"point\":17,\"step_size\":3}. Give (mode, curvature, point, step_size, next_point, objective_before, objective_after, automatic_descent_claim, boundary).","answer":{"kind":"expression","value":"(oversized_step, 2, 17, 3, -85, 289, 7225, No, positive_but_oversized_step_can_increase_the_objective)","explanation":"A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (oversized_step, 2, 17, 3, -85, 289, 7225, No, positive_but_oversized_step_can_increase_the_objective)."},"skills":["Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0093","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"44725d681de63677d738f0fb89c123109136cd64f315791d97e06d9e3139a400","family_id":"qf-mod50-gd-boundary","seed":202611193093,"semantic_template":"gd-boundary|{\"mode\":\"zero_gradient_nonminimum\",\"model\":{\"scale\":[18,1]}}"},"fields":{"title":"Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences","stem":"Use mode=zero_gradient_nonminimum, complete model={\"scale\":18}. Give (mode, objective_description, stationary_point, gradient_statement, actual_point_status, minimum_claim, boundary).","answer":{"kind":"expression","value":"(zero_gradient_nonminimum, f(x)=-18*x^2, x=0, gradient=0, strict_local_maximum, No, zero_gradient_is_stationary_not_a_minimum_certificate)","explanation":"A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (zero_gradient_nonminimum, f(x)=-18*x^2, x=0, gradient=0, strict_local_maximum, No, zero_gradient_is_stationary_not_a_minimum_certificate)."},"skills":["Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0094","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e58d2e96b579ade8cf75dddc921e9cf2f04344f7cd748bab9756d4942a81873","family_id":"qf-mod50-gd-boundary","seed":202611193094,"semantic_template":"gd-boundary|{\"mode\":\"finite_trace_only\",\"model\":{\"diagonal\":[[1,1],[2,1]],\"linear\":[[18,1],[-18,1]],\"point\":[[19,1],[1,1]],\"step_size\":[1,4],\"steps\":1}}"},"fields":{"title":"Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences","stem":"Use mode=finite_trace_only, complete model={\"diagonal\":[1,2],\"linear\":[18,-18],\"point\":[19,1],\"step_size\":[1,4],\"steps\":1}. Give (mode, step_count, iterate_objective_records, convergence_or_global_optimality_claim, boundary).","answer":{"kind":"expression","value":"(finite_trace_only, 1, [[0, [19, 1], 1011/2], [1, [39/4, 5], 5057/32]], No, finite_trace_does_not_prove_convergence_or_global_optimality)","explanation":"A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (finite_trace_only, 1, [[0, [19, 1], 1011/2], [1, [39/4, 5], 5057/32]], No, finite_trace_does_not_prove_convergence_or_global_optimality)."},"skills":["Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0095","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45ab72e8354f040a1031b98f06c4b991fb12875b6e4a15a5c283d33f46d831e9","family_id":"qf-mod50-gd-boundary","seed":202611193095,"semantic_template":"gd-boundary|{\"mode\":\"constrained_confusion\",\"model\":{\"gradient\":[20,1],\"lower\":[-1,1],\"point\":[0,1],\"step_size\":[2,1],\"upper\":[1,1]}}"},"fields":{"title":"Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences","stem":"Use mode=constrained_confusion, complete model={\"point\":0,\"gradient\":20,\"step_size\":2,\"lower\":-1,\"upper\":1}. Give (mode, lower_bound, upper_bound, feasible_point, unconstrained_next_point, constrained_descent_claim, boundary).","answer":{"kind":"expression","value":"(constrained_confusion, -1, 1, 0, -40, No, unconstrained_gradient_step_is_not_projected_or_constrained_descent)","explanation":"A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (constrained_confusion, -1, 1, 0, -40, No, unconstrained_gradient_step_is_not_projected_or_constrained_descent)."},"skills":["Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0096","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"618588a2881d69c256464d14590c7ee24b9ff47628b58283b8f13f6967e74fed","family_id":"qf-mod50-gd-boundary","seed":202611193096,"semantic_template":"gd-boundary|{\"mode\":\"oversized_step\",\"model\":{\"curvature\":[1,1],\"point\":[21,1],\"step_size\":[3,1]}}"},"fields":{"title":"Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences","stem":"Use mode=oversized_step, complete model={\"curvature\":1,\"point\":21,\"step_size\":3}. Give (mode, curvature, point, step_size, next_point, objective_before, objective_after, automatic_descent_claim, boundary).","answer":{"kind":"expression","value":"(oversized_step, 1, 21, 3, -42, 441/2, 882, No, positive_but_oversized_step_can_increase_the_objective)","explanation":"A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (oversized_step, 1, 21, 3, -42, 441/2, 882, No, positive_but_oversized_step_can_increase_the_objective)."},"skills":["Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0097","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c14b24e02b4affe7c7cd0ef43b269532dfcee856ea52382879a441c9eed729b2","family_id":"qf-mod50-gd-boundary","seed":202611193097,"semantic_template":"gd-boundary|{\"mode\":\"zero_gradient_nonminimum\",\"model\":{\"scale\":[22,1]}}"},"fields":{"title":"Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences","stem":"Use mode=zero_gradient_nonminimum, complete model={\"scale\":22}. Give (mode, objective_description, stationary_point, gradient_statement, actual_point_status, minimum_claim, boundary).","answer":{"kind":"expression","value":"(zero_gradient_nonminimum, f(x)=-22*x^2, x=0, gradient=0, strict_local_maximum, No, zero_gradient_is_stationary_not_a_minimum_certificate)","explanation":"A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (zero_gradient_nonminimum, f(x)=-22*x^2, x=0, gradient=0, strict_local_maximum, No, zero_gradient_is_stationary_not_a_minimum_certificate)."},"skills":["Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0098","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ec680728ed6d19ab07389a8f2af8ed385204f544eb663cc252af876da963f22","family_id":"qf-mod50-gd-boundary","seed":202611193098,"semantic_template":"gd-boundary|{\"mode\":\"finite_trace_only\",\"model\":{\"diagonal\":[[1,1],[2,1]],\"linear\":[[22,1],[-22,1]],\"point\":[[23,1],[1,1]],\"step_size\":[1,4],\"steps\":1}}"},"fields":{"title":"Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences","stem":"Use mode=finite_trace_only, complete model={\"diagonal\":[1,2],\"linear\":[22,-22],\"point\":[23,1],\"step_size\":[1,4],\"steps\":1}. Give (mode, step_count, iterate_objective_records, convergence_or_global_optimality_claim, boundary).","answer":{"kind":"expression","value":"(finite_trace_only, 1, [[0, [23, 1], 1499/2], [1, [47/4, 6], 7409/32]], No, finite_trace_does_not_prove_convergence_or_global_optimality)","explanation":"A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (finite_trace_only, 1, [[0, [23, 1], 1499/2], [1, [47/4, 6], 7409/32]], No, finite_trace_does_not_prove_convergence_or_global_optimality)."},"skills":["Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0099","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a97e490134d6475096ea34bd4b63cec816ba86b1f07b3e77a44df0724d36010e","family_id":"qf-mod50-gd-boundary","seed":202611193099,"semantic_template":"gd-boundary|{\"mode\":\"constrained_confusion\",\"model\":{\"gradient\":[24,1],\"lower\":[-1,1],\"point\":[0,1],\"step_size\":[2,1],\"upper\":[1,1]}}"},"fields":{"title":"Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences","stem":"Use mode=constrained_confusion, complete model={\"point\":0,\"gradient\":24,\"step_size\":2,\"lower\":-1,\"upper\":1}. Give (mode, lower_bound, upper_bound, feasible_point, unconstrained_next_point, constrained_descent_claim, boundary).","answer":{"kind":"expression","value":"(constrained_confusion, -1, 1, 0, -48, No, unconstrained_gradient_step_is_not_projected_or_constrained_descent)","explanation":"A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (constrained_confusion, -1, 1, 0, -48, No, unconstrained_gradient_step_is_not_projected_or_constrained_descent)."},"skills":["Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-pulse-20261119-r03-0100","batch_id":"batch-20261015-advance-20261119-r03-gradient-descent-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4570291b8dbe66f1dce10050ea98844801e60a19d6394dc4450f89268f9a57ba","family_id":"qf-mod50-gd-boundary","seed":202611193100,"semantic_template":"gd-boundary|{\"mode\":\"oversized_step\",\"model\":{\"curvature\":[5,1],\"point\":[25,1],\"step_size\":[3,1]}}"},"fields":{"title":"Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences","stem":"Use mode=oversized_step, complete model={\"curvature\":5,\"point\":25,\"step_size\":3}. Give (mode, curvature, point, step_size, next_point, objective_before, objective_after, automatic_descent_claim, boundary).","answer":{"kind":"expression","value":"(oversized_step, 5, 25, 3, -350, 3125/2, 306250, No, positive_but_oversized_step_can_increase_the_objective)","explanation":"A strictly oversized positive step can increase a scalar quadratic objective. A zero gradient may be a strict maximum. A finite trajectory does not certify a limit or global optimum. An unconstrained step leaving the feasible interval is not a projected or constrained gradient update. Keep each actual witness and branch-specific field order. The exact result, retaining the requested order, is (oversized_step, 5, 25, 3, -350, 3125/2, 306250, No, positive_but_oversized_step_can_increase_the_objective)."},"skills":["Oversized Steps, Stationary Maxima and Lost Feasibility Block Different Inferences"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}