{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r04-0001","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5a818e0121efa702f09856ad936dccbb135b82fbf1c84ad0cb2f7fd4c499d43","family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194001,"semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]]}"},"fields":{"title":"Full Row Rank Means the Declared Constraint Gradients Are Independent","stem":"Use constraint gradients as ordered rows=[[1, 0], [0, 1]]. Give (constraint_gradient_rows, constraint_count, dimension, rank, full_row_rank, regularity_definition).","answer":{"kind":"expression","value":"([[1, 0], [0, 1]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation":"Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 1]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)."},"skills":["Full Row Rank Means the Declared Constraint Gradients Are Independent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0002","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"993e0af37b788acd13b234fa755939975d9599c2f5c88e36e8232cd654241d3f","family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194002,"semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[2,1]],[[1,1],[0,1]]]}"},"fields":{"title":"Full Row Rank Means the Declared Constraint Gradients Are Independent","stem":"Use constraint gradients as ordered rows=[[1, 0], [0, 2]]. Give (constraint_gradient_rows, constraint_count, dimension, rank, full_row_rank, regularity_definition).","answer":{"kind":"expression","value":"([[1, 0], [0, 2]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation":"Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 2]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)."},"skills":["Full Row Rank Means the Declared Constraint Gradients Are Independent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0003","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2208ef839eeb10cc73d17a8269f6e5383b59f27ea77ba0c5e6b969bb56f224fa","family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194003,"semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[3,1]],[[1,1],[0,1]]]}"},"fields":{"title":"Full Row Rank Means the Declared Constraint Gradients Are Independent","stem":"Use constraint gradients as ordered rows=[[1, 0], [0, 3]]. Give (constraint_gradient_rows, constraint_count, dimension, rank, full_row_rank, regularity_definition).","answer":{"kind":"expression","value":"([[1, 0], [0, 3]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation":"Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 3]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)."},"skills":["Full Row Rank Means the Declared Constraint Gradients Are Independent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0004","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad9f26615987889a90bcb181906f9dae1f11d48ac3b9b91ba26029ef6657357f","family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194004,"semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[4,1]],[[1,1],[0,1]]]}"},"fields":{"title":"Full Row Rank Means the Declared Constraint Gradients Are Independent","stem":"Use constraint gradients as ordered rows=[[1, 0], [0, 4]]. Give (constraint_gradient_rows, constraint_count, dimension, rank, full_row_rank, regularity_definition).","answer":{"kind":"expression","value":"([[1, 0], [0, 4]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation":"Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 4]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)."},"skills":["Full Row Rank Means the Declared Constraint Gradients Are Independent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0005","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc9c539239ab74eb7a40768460ce4e475fa063336554953e777c26d722141f32","family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194005,"semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[5,1]],[[1,1],[0,1]]]}"},"fields":{"title":"Full Row Rank Means the Declared Constraint Gradients Are Independent","stem":"Use constraint gradients as ordered rows=[[1, 0], [0, 5]]. Give (constraint_gradient_rows, constraint_count, dimension, rank, full_row_rank, regularity_definition).","answer":{"kind":"expression","value":"([[1, 0], [0, 5]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation":"Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 5]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)."},"skills":["Full Row Rank Means the Declared Constraint Gradients Are Independent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0006","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4115e4dabe91fdc5fb94b3b69493c907e3139fb06ffc627834d28c6051cd474d","family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194006,"semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[6,1]],[[1,1],[0,1]]]}"},"fields":{"title":"Full Row Rank Means the Declared Constraint Gradients Are Independent","stem":"Use constraint gradients as ordered rows=[[1, 0], [0, 6]]. Give (constraint_gradient_rows, constraint_count, dimension, rank, full_row_rank, regularity_definition).","answer":{"kind":"expression","value":"([[1, 0], [0, 6]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation":"Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 6]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)."},"skills":["Full Row Rank Means the Declared Constraint Gradients Are Independent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0007","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3c2ebc7ccc4cfbeac13be1a58fc6c5a3860f37286bde2fbfeb74c82404b9ab3","family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194007,"semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[7,1]],[[1,1],[0,1]]]}"},"fields":{"title":"Full Row Rank Means the Declared Constraint Gradients Are Independent","stem":"Use constraint gradients as ordered rows=[[1, 0], [0, 7]]. Give (constraint_gradient_rows, constraint_count, dimension, rank, full_row_rank, regularity_definition).","answer":{"kind":"expression","value":"([[1, 0], [0, 7]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation":"Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 7]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)."},"skills":["Full Row Rank Means the Declared Constraint Gradients Are Independent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0008","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9cbd1256e09522426c4ee2cf3018b94d7f63703c4172f49f20d7cc50498aa056","family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194008,"semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[8,1]],[[1,1],[0,1]]]}"},"fields":{"title":"Full Row Rank Means the Declared Constraint Gradients Are Independent","stem":"Use constraint gradients as ordered rows=[[1, 0], [0, 8]]. Give (constraint_gradient_rows, constraint_count, dimension, rank, full_row_rank, regularity_definition).","answer":{"kind":"expression","value":"([[1, 0], [0, 8]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation":"Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 8]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)."},"skills":["Full Row Rank Means the Declared Constraint Gradients Are Independent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0009","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d876c92b09691826a2aacb9a95cfde385b1cae5f9b9788bdad9311137280d05","family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194009,"semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[9,1]],[[1,1],[0,1]]]}"},"fields":{"title":"Full Row Rank Means the Declared Constraint Gradients Are Independent","stem":"Use constraint gradients as ordered rows=[[1, 0], [0, 9]]. Give (constraint_gradient_rows, constraint_count, dimension, rank, full_row_rank, regularity_definition).","answer":{"kind":"expression","value":"([[1, 0], [0, 9]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation":"Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 9]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)."},"skills":["Full Row Rank Means the Declared Constraint Gradients Are Independent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0010","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f299934c9e7fa013534ab79f508a70e4c448642112cbfda94a0760e34eb3b49","family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194010,"semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[10,1]],[[1,1],[0,1]]]}"},"fields":{"title":"Full Row Rank Means the Declared Constraint Gradients Are Independent","stem":"Use constraint gradients as ordered rows=[[1, 0], [0, 10]]. Give (constraint_gradient_rows, constraint_count, dimension, rank, full_row_rank, regularity_definition).","answer":{"kind":"expression","value":"([[1, 0], [0, 10]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation":"Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 10]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)."},"skills":["Full Row Rank Means the Declared Constraint Gradients Are Independent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0011","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d96e2b6e776a6fb5f551d5c84c858e1a591672c9033390b365627cddfed590ea","family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194011,"semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[11,1]],[[1,1],[0,1]]]}"},"fields":{"title":"Full Row Rank Means the Declared Constraint Gradients Are Independent","stem":"Use constraint gradients as ordered rows=[[1, 0], [0, 11]]. Give (constraint_gradient_rows, constraint_count, dimension, rank, full_row_rank, regularity_definition).","answer":{"kind":"expression","value":"([[1, 0], [0, 11]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation":"Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 11]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)."},"skills":["Full Row Rank Means the Declared Constraint Gradients Are Independent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0012","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fcd8982380383b09856bdcb09740152477231abc545c16713277cf2f28fd4d82","family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194012,"semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[12,1]],[[1,1],[0,1]]]}"},"fields":{"title":"Full Row Rank Means the Declared Constraint Gradients Are Independent","stem":"Use constraint gradients as ordered rows=[[1, 0], [0, 12]]. Give (constraint_gradient_rows, constraint_count, dimension, rank, full_row_rank, regularity_definition).","answer":{"kind":"expression","value":"([[1, 0], [0, 12]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation":"Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 12]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)."},"skills":["Full Row Rank Means the Declared Constraint Gradients Are Independent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0013","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9256f6bde6e16d912257d1cc73f715d85cb7cd45ffd4775618e6eca2b0806a4","family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194013,"semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[13,1]],[[1,1],[0,1]]]}"},"fields":{"title":"Full Row Rank Means the Declared Constraint Gradients Are Independent","stem":"Use constraint gradients as ordered rows=[[1, 0], [0, 13]]. Give (constraint_gradient_rows, constraint_count, dimension, rank, full_row_rank, regularity_definition).","answer":{"kind":"expression","value":"([[1, 0], [0, 13]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation":"Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 13]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)."},"skills":["Full Row Rank Means the Declared Constraint Gradients Are Independent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0014","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90e09d00261f10e607531af1381b981a4fa699fe2d851c8fe095cf47a36cc1f1","family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194014,"semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[14,1]],[[1,1],[0,1]]]}"},"fields":{"title":"Full Row Rank Means the Declared Constraint Gradients Are Independent","stem":"Use constraint gradients as ordered rows=[[1, 0], [0, 14]]. Give (constraint_gradient_rows, constraint_count, dimension, rank, full_row_rank, regularity_definition).","answer":{"kind":"expression","value":"([[1, 0], [0, 14]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation":"Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 14]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)."},"skills":["Full Row Rank Means the Declared Constraint Gradients Are Independent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0015","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01613791be58dd2b5aa2b81c418278668cdb3a8005c48630ce4636ea4939a126","family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194015,"semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[15,1]],[[1,1],[0,1]]]}"},"fields":{"title":"Full Row Rank Means the Declared Constraint Gradients Are Independent","stem":"Use constraint gradients as ordered rows=[[1, 0], [0, 15]]. Give (constraint_gradient_rows, constraint_count, dimension, rank, full_row_rank, regularity_definition).","answer":{"kind":"expression","value":"([[1, 0], [0, 15]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation":"Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 15]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)."},"skills":["Full Row Rank Means the Declared Constraint Gradients Are Independent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0016","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83729a471dc733f141e198f02f0f9fb192ddbef42117422700654fdf141c9684","family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194016,"semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[16,1]],[[1,1],[0,1]]]}"},"fields":{"title":"Full Row Rank Means the Declared Constraint Gradients Are Independent","stem":"Use constraint gradients as ordered rows=[[1, 0], [0, 16]]. Give (constraint_gradient_rows, constraint_count, dimension, rank, full_row_rank, regularity_definition).","answer":{"kind":"expression","value":"([[1, 0], [0, 16]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation":"Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 16]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)."},"skills":["Full Row Rank Means the Declared Constraint Gradients Are Independent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0017","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"419ce20b1dff4d19c691554ca48f8e3bfd4463c6253da82056dc94c3bb6dbbe8","family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194017,"semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[17,1]],[[1,1],[0,1]]]}"},"fields":{"title":"Full Row Rank Means the Declared Constraint Gradients Are Independent","stem":"Use constraint gradients as ordered rows=[[1, 0], [0, 17]]. Give (constraint_gradient_rows, constraint_count, dimension, rank, full_row_rank, regularity_definition).","answer":{"kind":"expression","value":"([[1, 0], [0, 17]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation":"Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 17]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)."},"skills":["Full Row Rank Means the Declared Constraint Gradients Are Independent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0018","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30f8d9789e7c89c6dc9e0842fbc8a3f2fc4e0c3938972daaf7661693a21d8924","family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194018,"semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[18,1]],[[1,1],[0,1]]]}"},"fields":{"title":"Full Row Rank Means the Declared Constraint Gradients Are Independent","stem":"Use constraint gradients as ordered rows=[[1, 0], [0, 18]]. Give (constraint_gradient_rows, constraint_count, dimension, rank, full_row_rank, regularity_definition).","answer":{"kind":"expression","value":"([[1, 0], [0, 18]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation":"Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 18]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)."},"skills":["Full Row Rank Means the Declared Constraint Gradients Are Independent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0019","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"108d9a6817a8cba60ee7ac493f6089c786526ac899cfa47e224ea2aab5c4d287","family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194019,"semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[19,1]],[[1,1],[0,1]]]}"},"fields":{"title":"Full Row Rank Means the Declared Constraint Gradients Are Independent","stem":"Use constraint gradients as ordered rows=[[1, 0], [0, 19]]. Give (constraint_gradient_rows, constraint_count, dimension, rank, full_row_rank, regularity_definition).","answer":{"kind":"expression","value":"([[1, 0], [0, 19]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation":"Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 19]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)."},"skills":["Full Row Rank Means the Declared Constraint Gradients Are Independent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0020","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"700b1fec8a97858c5d1c702f0da8ec07494643b815a14aeb9e7216064690de18","family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194020,"semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[20,1]],[[1,1],[0,1]]]}"},"fields":{"title":"Full Row Rank Means the Declared Constraint Gradients Are Independent","stem":"Use constraint gradients as ordered rows=[[1, 0], [0, 20]]. Give (constraint_gradient_rows, constraint_count, dimension, rank, full_row_rank, regularity_definition).","answer":{"kind":"expression","value":"([[1, 0], [0, 20]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation":"Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 20]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)."},"skills":["Full Row Rank Means the Declared Constraint Gradients Are Independent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0021","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0dd4fddcc3a7df986293d4a91fd5f645c237e0679adf1b2b636ee93a8b040c4","family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194021,"semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[21,1]],[[1,1],[0,1]]]}"},"fields":{"title":"Full Row Rank Means the Declared Constraint Gradients Are Independent","stem":"Use constraint gradients as ordered rows=[[1, 0], [0, 21]]. Give (constraint_gradient_rows, constraint_count, dimension, rank, full_row_rank, regularity_definition).","answer":{"kind":"expression","value":"([[1, 0], [0, 21]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation":"Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 21]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)."},"skills":["Full Row Rank Means the Declared Constraint Gradients Are Independent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0022","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8096b9709337b65cd3737877ebcb6aa561c35d9bca0cce458f44891f8481bdf7","family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194022,"semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[22,1]],[[1,1],[0,1]]]}"},"fields":{"title":"Full Row Rank Means the Declared Constraint Gradients Are Independent","stem":"Use constraint gradients as ordered rows=[[1, 0], [0, 22]]. Give (constraint_gradient_rows, constraint_count, dimension, rank, full_row_rank, regularity_definition).","answer":{"kind":"expression","value":"([[1, 0], [0, 22]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation":"Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 22]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)."},"skills":["Full Row Rank Means the Declared Constraint Gradients Are Independent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0023","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d191949b1dda6c8317e4ecde4eaa050c94c6adcefd9a7f45ee7b868eb196de8","family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194023,"semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[23,1]],[[1,1],[0,1]]]}"},"fields":{"title":"Full Row Rank Means the Declared Constraint Gradients Are Independent","stem":"Use constraint gradients as ordered rows=[[1, 0], [0, 23]]. Give (constraint_gradient_rows, constraint_count, dimension, rank, full_row_rank, regularity_definition).","answer":{"kind":"expression","value":"([[1, 0], [0, 23]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation":"Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 23]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)."},"skills":["Full Row Rank Means the Declared Constraint Gradients Are Independent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0024","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73392b710fbce39a62434b90a390e12ab2330b4a2eaad2caae2109942da72d2b","family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194024,"semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[24,1]],[[1,1],[0,1]]]}"},"fields":{"title":"Full Row Rank Means the Declared Constraint Gradients Are Independent","stem":"Use constraint gradients as ordered rows=[[1, 0], [0, 24]]. Give (constraint_gradient_rows, constraint_count, dimension, rank, full_row_rank, regularity_definition).","answer":{"kind":"expression","value":"([[1, 0], [0, 24]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation":"Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 24]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)."},"skills":["Full Row Rank Means the Declared Constraint Gradients Are Independent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0025","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e88bffa85da69b98e928a7cc7ef987ef5bb78a78e4c19408ef9ed03d05f48f5a","family_id":"qf-mod50-lagrange-regularity-audit","seed":202611194025,"semantic_template":"lagrange-regularity-audit|{\"constraint_gradients\":[[[0,1],[25,1]],[[1,1],[0,1]]]}"},"fields":{"title":"Full Row Rank Means the Declared Constraint Gradients Are Independent","stem":"Use constraint gradients as ordered rows=[[1, 0], [0, 25]]. Give (constraint_gradient_rows, constraint_count, dimension, rank, full_row_rank, regularity_definition).","answer":{"kind":"expression","value":"([[1, 0], [0, 25]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)","explanation":"Compute exact row rank and compare it with the number of constraints, retaining the ambient dimension. This is an algebraic qualification check on supplied vectors; it does not independently establish differentiability, feasibility or an extremum for functions not supplied here. The exact result, retaining the requested order, is ([[1, 0], [0, 25]], 2, 2, 2, Yes, full_row_rank_means_constraint_gradients_are_linearly_independent)."},"skills":["Full Row Rank Means the Declared Constraint Gradients Are Independent"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0026","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d6ef735d0770e7e0007443ee7d9c792a894d94dea88038e0a037d3375695389","family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194026,"semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[2,1],[1,1]],\"objective_gradient\":[[1,1],[2,1]]}"},"fields":{"title":"The Multiplier Equation Uses a Fixed Positive-Sign Convention","stem":"Use objective gradient=[1, 2], constraint-gradient rows=[[1, 0], [0, 1]], aligned multipliers=[1, 2]. Give (objective_gradient, constraint_gradient_rows, multipliers, gradient_combination, stationarity_residual, equation_holds, sign_convention).","answer":{"kind":"expression","value":"([1, 2], [[1, 0], [0, 1]], [1, 2], [1, 2], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)","explanation":"Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([1, 2], [[1, 0], [0, 1]], [1, 2], [1, 2], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)."},"skills":["The Multiplier Equation Uses a Fixed Positive-Sign Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0027","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75555bc94a9830f688767e0b84458327441e732e4e89ce3123831268f073518b","family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194027,"semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[4,1],[2,1]],\"objective_gradient\":[[2,1],[4,1]]}"},"fields":{"title":"The Multiplier Equation Uses a Fixed Positive-Sign Convention","stem":"Use objective gradient=[2, 4], constraint-gradient rows=[[1, 0], [0, 1]], aligned multipliers=[2, 4]. Give (objective_gradient, constraint_gradient_rows, multipliers, gradient_combination, stationarity_residual, equation_holds, sign_convention).","answer":{"kind":"expression","value":"([2, 4], [[1, 0], [0, 1]], [2, 4], [2, 4], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)","explanation":"Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([2, 4], [[1, 0], [0, 1]], [2, 4], [2, 4], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)."},"skills":["The Multiplier Equation Uses a Fixed Positive-Sign Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0028","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"efbbd31777d72eab0ffea9b5e94096af32e77920bd9c353e33626d3aad8cde69","family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194028,"semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[6,1],[3,1]],\"objective_gradient\":[[3,1],[6,1]]}"},"fields":{"title":"The Multiplier Equation Uses a Fixed Positive-Sign Convention","stem":"Use objective gradient=[3, 6], constraint-gradient rows=[[1, 0], [0, 1]], aligned multipliers=[3, 6]. Give (objective_gradient, constraint_gradient_rows, multipliers, gradient_combination, stationarity_residual, equation_holds, sign_convention).","answer":{"kind":"expression","value":"([3, 6], [[1, 0], [0, 1]], [3, 6], [3, 6], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)","explanation":"Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([3, 6], [[1, 0], [0, 1]], [3, 6], [3, 6], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)."},"skills":["The Multiplier Equation Uses a Fixed Positive-Sign Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0029","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa0834624d7024b2e6bb407a3ca585c8422e958a69cb5fd4decf765534a55fd0","family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194029,"semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[8,1],[4,1]],\"objective_gradient\":[[4,1],[8,1]]}"},"fields":{"title":"The Multiplier Equation Uses a Fixed Positive-Sign Convention","stem":"Use objective gradient=[4, 8], constraint-gradient rows=[[1, 0], [0, 1]], aligned multipliers=[4, 8]. Give (objective_gradient, constraint_gradient_rows, multipliers, gradient_combination, stationarity_residual, equation_holds, sign_convention).","answer":{"kind":"expression","value":"([4, 8], [[1, 0], [0, 1]], [4, 8], [4, 8], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)","explanation":"Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([4, 8], [[1, 0], [0, 1]], [4, 8], [4, 8], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)."},"skills":["The Multiplier Equation Uses a Fixed Positive-Sign Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0030","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"144c1cbfaecf85bbe71fe4ae7f515202677b2a22e5aa3fbb269f3adc3992df77","family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194030,"semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[10,1],[5,1]],\"objective_gradient\":[[5,1],[10,1]]}"},"fields":{"title":"The Multiplier Equation Uses a Fixed Positive-Sign Convention","stem":"Use objective gradient=[5, 10], constraint-gradient rows=[[1, 0], [0, 1]], aligned multipliers=[5, 10]. Give (objective_gradient, constraint_gradient_rows, multipliers, gradient_combination, stationarity_residual, equation_holds, sign_convention).","answer":{"kind":"expression","value":"([5, 10], [[1, 0], [0, 1]], [5, 10], [5, 10], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)","explanation":"Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([5, 10], [[1, 0], [0, 1]], [5, 10], [5, 10], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)."},"skills":["The Multiplier Equation Uses a Fixed Positive-Sign Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0031","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46d6301d2e817cae0624fcbcb07856e5cc8dd0c3a6d496b10e7964f43e174517","family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194031,"semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[12,1],[6,1]],\"objective_gradient\":[[6,1],[12,1]]}"},"fields":{"title":"The Multiplier Equation Uses a Fixed Positive-Sign Convention","stem":"Use objective gradient=[6, 12], constraint-gradient rows=[[1, 0], [0, 1]], aligned multipliers=[6, 12]. Give (objective_gradient, constraint_gradient_rows, multipliers, gradient_combination, stationarity_residual, equation_holds, sign_convention).","answer":{"kind":"expression","value":"([6, 12], [[1, 0], [0, 1]], [6, 12], [6, 12], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)","explanation":"Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([6, 12], [[1, 0], [0, 1]], [6, 12], [6, 12], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)."},"skills":["The Multiplier Equation Uses a Fixed Positive-Sign Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0032","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55692145e25c45aa214f31c394f94d65ba53e9058fc8f25dad4ed789ad74cc57","family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194032,"semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[14,1],[7,1]],\"objective_gradient\":[[7,1],[14,1]]}"},"fields":{"title":"The Multiplier Equation Uses a Fixed Positive-Sign Convention","stem":"Use objective gradient=[7, 14], constraint-gradient rows=[[1, 0], [0, 1]], aligned multipliers=[7, 14]. Give (objective_gradient, constraint_gradient_rows, multipliers, gradient_combination, stationarity_residual, equation_holds, sign_convention).","answer":{"kind":"expression","value":"([7, 14], [[1, 0], [0, 1]], [7, 14], [7, 14], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)","explanation":"Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([7, 14], [[1, 0], [0, 1]], [7, 14], [7, 14], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)."},"skills":["The Multiplier Equation Uses a Fixed Positive-Sign Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0033","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8eef280398fbeba47de924e1b968da9a2e288ad15c88e0e2b3a7f6094cab99e8","family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194033,"semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[16,1],[8,1]],\"objective_gradient\":[[8,1],[16,1]]}"},"fields":{"title":"The Multiplier Equation Uses a Fixed Positive-Sign Convention","stem":"Use objective gradient=[8, 16], constraint-gradient rows=[[1, 0], [0, 1]], aligned multipliers=[8, 16]. Give (objective_gradient, constraint_gradient_rows, multipliers, gradient_combination, stationarity_residual, equation_holds, sign_convention).","answer":{"kind":"expression","value":"([8, 16], [[1, 0], [0, 1]], [8, 16], [8, 16], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)","explanation":"Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([8, 16], [[1, 0], [0, 1]], [8, 16], [8, 16], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)."},"skills":["The Multiplier Equation Uses a Fixed Positive-Sign Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0034","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8bc1d4b90ea42f97dbd50b2015a80f4380383878c7509f0804cbef35bbf8f9d2","family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194034,"semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[18,1],[9,1]],\"objective_gradient\":[[9,1],[18,1]]}"},"fields":{"title":"The Multiplier Equation Uses a Fixed Positive-Sign Convention","stem":"Use objective gradient=[9, 18], constraint-gradient rows=[[1, 0], [0, 1]], aligned multipliers=[9, 18]. Give (objective_gradient, constraint_gradient_rows, multipliers, gradient_combination, stationarity_residual, equation_holds, sign_convention).","answer":{"kind":"expression","value":"([9, 18], [[1, 0], [0, 1]], [9, 18], [9, 18], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)","explanation":"Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([9, 18], [[1, 0], [0, 1]], [9, 18], [9, 18], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)."},"skills":["The Multiplier Equation Uses a Fixed Positive-Sign Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0035","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d1b2aaaf364037a493c2ed376658db45756ac85fedcd8250d318193c30d7afc","family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194035,"semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[20,1],[10,1]],\"objective_gradient\":[[10,1],[20,1]]}"},"fields":{"title":"The Multiplier Equation Uses a Fixed Positive-Sign Convention","stem":"Use objective gradient=[10, 20], constraint-gradient rows=[[1, 0], [0, 1]], aligned multipliers=[10, 20]. Give (objective_gradient, constraint_gradient_rows, multipliers, gradient_combination, stationarity_residual, equation_holds, sign_convention).","answer":{"kind":"expression","value":"([10, 20], [[1, 0], [0, 1]], [10, 20], [10, 20], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)","explanation":"Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([10, 20], [[1, 0], [0, 1]], [10, 20], [10, 20], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)."},"skills":["The Multiplier Equation Uses a Fixed Positive-Sign Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0036","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c11e532b8a73fbc2a0ea1d3ea9a6158ffb9ef7a369fd3b9d3ea2e84b5eccdff3","family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194036,"semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[22,1],[11,1]],\"objective_gradient\":[[11,1],[22,1]]}"},"fields":{"title":"The Multiplier Equation Uses a Fixed Positive-Sign Convention","stem":"Use objective gradient=[11, 22], constraint-gradient rows=[[1, 0], [0, 1]], aligned multipliers=[11, 22]. Give (objective_gradient, constraint_gradient_rows, multipliers, gradient_combination, stationarity_residual, equation_holds, sign_convention).","answer":{"kind":"expression","value":"([11, 22], [[1, 0], [0, 1]], [11, 22], [11, 22], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)","explanation":"Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([11, 22], [[1, 0], [0, 1]], [11, 22], [11, 22], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)."},"skills":["The Multiplier Equation Uses a Fixed Positive-Sign Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0037","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c77430b51b69a5c9262ee705cbef0f3a4851f5aabc3c3841e7626d37a83e2766","family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194037,"semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[24,1],[12,1]],\"objective_gradient\":[[12,1],[24,1]]}"},"fields":{"title":"The Multiplier Equation Uses a Fixed Positive-Sign Convention","stem":"Use objective gradient=[12, 24], constraint-gradient rows=[[1, 0], [0, 1]], aligned multipliers=[12, 24]. Give (objective_gradient, constraint_gradient_rows, multipliers, gradient_combination, stationarity_residual, equation_holds, sign_convention).","answer":{"kind":"expression","value":"([12, 24], [[1, 0], [0, 1]], [12, 24], [12, 24], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)","explanation":"Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([12, 24], [[1, 0], [0, 1]], [12, 24], [12, 24], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)."},"skills":["The Multiplier Equation Uses a Fixed Positive-Sign Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0038","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65dc27b55609fb68d66a8d172220743fbe907d1b59284a38e3b6decdaa7d7fd0","family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194038,"semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[26,1],[13,1]],\"objective_gradient\":[[13,1],[26,1]]}"},"fields":{"title":"The Multiplier Equation Uses a Fixed Positive-Sign Convention","stem":"Use objective gradient=[13, 26], constraint-gradient rows=[[1, 0], [0, 1]], aligned multipliers=[13, 26]. Give (objective_gradient, constraint_gradient_rows, multipliers, gradient_combination, stationarity_residual, equation_holds, sign_convention).","answer":{"kind":"expression","value":"([13, 26], [[1, 0], [0, 1]], [13, 26], [13, 26], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)","explanation":"Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([13, 26], [[1, 0], [0, 1]], [13, 26], [13, 26], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)."},"skills":["The Multiplier Equation Uses a Fixed Positive-Sign Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0039","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8a0593d9f60b09cc267420fe046c296bc33a8162fb1de5c77afae2f2c7974aa","family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194039,"semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[28,1],[14,1]],\"objective_gradient\":[[14,1],[28,1]]}"},"fields":{"title":"The Multiplier Equation Uses a Fixed Positive-Sign Convention","stem":"Use objective gradient=[14, 28], constraint-gradient rows=[[1, 0], [0, 1]], aligned multipliers=[14, 28]. Give (objective_gradient, constraint_gradient_rows, multipliers, gradient_combination, stationarity_residual, equation_holds, sign_convention).","answer":{"kind":"expression","value":"([14, 28], [[1, 0], [0, 1]], [14, 28], [14, 28], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)","explanation":"Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([14, 28], [[1, 0], [0, 1]], [14, 28], [14, 28], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)."},"skills":["The Multiplier Equation Uses a Fixed Positive-Sign Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0040","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d5630cf4cea5d3d7bfa834afda1ade4c8f9d8a34dbfdef3738b06510d78e098","family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194040,"semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[30,1],[15,1]],\"objective_gradient\":[[15,1],[30,1]]}"},"fields":{"title":"The Multiplier Equation Uses a Fixed Positive-Sign Convention","stem":"Use objective gradient=[15, 30], constraint-gradient rows=[[1, 0], [0, 1]], aligned multipliers=[15, 30]. Give (objective_gradient, constraint_gradient_rows, multipliers, gradient_combination, stationarity_residual, equation_holds, sign_convention).","answer":{"kind":"expression","value":"([15, 30], [[1, 0], [0, 1]], [15, 30], [15, 30], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)","explanation":"Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([15, 30], [[1, 0], [0, 1]], [15, 30], [15, 30], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)."},"skills":["The Multiplier Equation Uses a Fixed Positive-Sign Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0041","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cffa5933e835effbbef79b3173acb95adad584443b830e6958c1bc2d389e0e25","family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194041,"semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[32,1],[16,1]],\"objective_gradient\":[[16,1],[32,1]]}"},"fields":{"title":"The Multiplier Equation Uses a Fixed Positive-Sign Convention","stem":"Use objective gradient=[16, 32], constraint-gradient rows=[[1, 0], [0, 1]], aligned multipliers=[16, 32]. Give (objective_gradient, constraint_gradient_rows, multipliers, gradient_combination, stationarity_residual, equation_holds, sign_convention).","answer":{"kind":"expression","value":"([16, 32], [[1, 0], [0, 1]], [16, 32], [16, 32], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)","explanation":"Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([16, 32], [[1, 0], [0, 1]], [16, 32], [16, 32], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)."},"skills":["The Multiplier Equation Uses a Fixed Positive-Sign Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0042","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8018b06407196c3dd224ade0dfd0030358315038613ca73ba00a08142f5cf05b","family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194042,"semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[34,1],[17,1]],\"objective_gradient\":[[17,1],[34,1]]}"},"fields":{"title":"The Multiplier Equation Uses a Fixed Positive-Sign Convention","stem":"Use objective gradient=[17, 34], constraint-gradient rows=[[1, 0], [0, 1]], aligned multipliers=[17, 34]. Give (objective_gradient, constraint_gradient_rows, multipliers, gradient_combination, stationarity_residual, equation_holds, sign_convention).","answer":{"kind":"expression","value":"([17, 34], [[1, 0], [0, 1]], [17, 34], [17, 34], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)","explanation":"Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([17, 34], [[1, 0], [0, 1]], [17, 34], [17, 34], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)."},"skills":["The Multiplier Equation Uses a Fixed Positive-Sign Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0043","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8780d2bd65ac8f2caa9fd05d493d51817e26a5c9b2e2d3958e0b1a57b297226d","family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194043,"semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[36,1],[18,1]],\"objective_gradient\":[[18,1],[36,1]]}"},"fields":{"title":"The Multiplier Equation Uses a Fixed Positive-Sign Convention","stem":"Use objective gradient=[18, 36], constraint-gradient rows=[[1, 0], [0, 1]], aligned multipliers=[18, 36]. Give (objective_gradient, constraint_gradient_rows, multipliers, gradient_combination, stationarity_residual, equation_holds, sign_convention).","answer":{"kind":"expression","value":"([18, 36], [[1, 0], [0, 1]], [18, 36], [18, 36], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)","explanation":"Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([18, 36], [[1, 0], [0, 1]], [18, 36], [18, 36], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)."},"skills":["The Multiplier Equation Uses a Fixed Positive-Sign Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0044","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9aa4ff13648a6ff3d2674ee4208fb643a1141497f8764b01cd1cd66cd58bf1c3","family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194044,"semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[38,1],[19,1]],\"objective_gradient\":[[19,1],[38,1]]}"},"fields":{"title":"The Multiplier Equation Uses a Fixed Positive-Sign Convention","stem":"Use objective gradient=[19, 38], constraint-gradient rows=[[1, 0], [0, 1]], aligned multipliers=[19, 38]. Give (objective_gradient, constraint_gradient_rows, multipliers, gradient_combination, stationarity_residual, equation_holds, sign_convention).","answer":{"kind":"expression","value":"([19, 38], [[1, 0], [0, 1]], [19, 38], [19, 38], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)","explanation":"Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([19, 38], [[1, 0], [0, 1]], [19, 38], [19, 38], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)."},"skills":["The Multiplier Equation Uses a Fixed Positive-Sign Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0045","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c4f35a86b47fd1dcae6970c27e31dd1e788d285d0e137dfb559c7b1fed7d9c1","family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194045,"semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[40,1],[20,1]],\"objective_gradient\":[[20,1],[40,1]]}"},"fields":{"title":"The Multiplier Equation Uses a Fixed Positive-Sign Convention","stem":"Use objective gradient=[20, 40], constraint-gradient rows=[[1, 0], [0, 1]], aligned multipliers=[20, 40]. Give (objective_gradient, constraint_gradient_rows, multipliers, gradient_combination, stationarity_residual, equation_holds, sign_convention).","answer":{"kind":"expression","value":"([20, 40], [[1, 0], [0, 1]], [20, 40], [20, 40], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)","explanation":"Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([20, 40], [[1, 0], [0, 1]], [20, 40], [20, 40], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)."},"skills":["The Multiplier Equation Uses a Fixed Positive-Sign Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0046","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f28f76080cb3858b76f5dc14e31997f3db5d2632aa95c4c7f63ce536acd64c0","family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194046,"semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[42,1],[21,1]],\"objective_gradient\":[[21,1],[42,1]]}"},"fields":{"title":"The Multiplier Equation Uses a Fixed Positive-Sign Convention","stem":"Use objective gradient=[21, 42], constraint-gradient rows=[[1, 0], [0, 1]], aligned multipliers=[21, 42]. Give (objective_gradient, constraint_gradient_rows, multipliers, gradient_combination, stationarity_residual, equation_holds, sign_convention).","answer":{"kind":"expression","value":"([21, 42], [[1, 0], [0, 1]], [21, 42], [21, 42], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)","explanation":"Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([21, 42], [[1, 0], [0, 1]], [21, 42], [21, 42], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)."},"skills":["The Multiplier Equation Uses a Fixed Positive-Sign Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0047","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ee24259cb5b4d58f40ee3ce1cbbb9508253e1b4cca205db54b4c972a1a590d4","family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194047,"semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[44,1],[22,1]],\"objective_gradient\":[[22,1],[44,1]]}"},"fields":{"title":"The Multiplier Equation Uses a Fixed Positive-Sign Convention","stem":"Use objective gradient=[22, 44], constraint-gradient rows=[[1, 0], [0, 1]], aligned multipliers=[22, 44]. Give (objective_gradient, constraint_gradient_rows, multipliers, gradient_combination, stationarity_residual, equation_holds, sign_convention).","answer":{"kind":"expression","value":"([22, 44], [[1, 0], [0, 1]], [22, 44], [22, 44], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)","explanation":"Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([22, 44], [[1, 0], [0, 1]], [22, 44], [22, 44], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)."},"skills":["The Multiplier Equation Uses a Fixed Positive-Sign Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0048","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5f47bbf51d928439ab9f0569a738018cdf2154724922d435f5444aca0f4de37","family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194048,"semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[46,1],[23,1]],\"objective_gradient\":[[23,1],[46,1]]}"},"fields":{"title":"The Multiplier Equation Uses a Fixed Positive-Sign Convention","stem":"Use objective gradient=[23, 46], constraint-gradient rows=[[1, 0], [0, 1]], aligned multipliers=[23, 46]. Give (objective_gradient, constraint_gradient_rows, multipliers, gradient_combination, stationarity_residual, equation_holds, sign_convention).","answer":{"kind":"expression","value":"([23, 46], [[1, 0], [0, 1]], [23, 46], [23, 46], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)","explanation":"Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([23, 46], [[1, 0], [0, 1]], [23, 46], [23, 46], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)."},"skills":["The Multiplier Equation Uses a Fixed Positive-Sign Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0049","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a7a981245efe9c92ee9fd8f4090a10e6cef5dbf00deac1959311c3e388ca39a","family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194049,"semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[48,1],[24,1]],\"objective_gradient\":[[24,1],[48,1]]}"},"fields":{"title":"The Multiplier Equation Uses a Fixed Positive-Sign Convention","stem":"Use objective gradient=[24, 48], constraint-gradient rows=[[1, 0], [0, 1]], aligned multipliers=[24, 48]. Give (objective_gradient, constraint_gradient_rows, multipliers, gradient_combination, stationarity_residual, equation_holds, sign_convention).","answer":{"kind":"expression","value":"([24, 48], [[1, 0], [0, 1]], [24, 48], [24, 48], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)","explanation":"Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([24, 48], [[1, 0], [0, 1]], [24, 48], [24, 48], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)."},"skills":["The Multiplier Equation Uses a Fixed Positive-Sign Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0050","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e52e443776c7bff92304564330d51bac73ae72906548f46052f5fb98877bdb18","family_id":"qf-mod50-lagrange-multiplier-residual","seed":202611194050,"semantic_template":"lagrange-multiplier-residual|{\"constraint_gradients\":[[[0,1],[1,1]],[[1,1],[0,1]]],\"multipliers\":[[50,1],[25,1]],\"objective_gradient\":[[25,1],[50,1]]}"},"fields":{"title":"The Multiplier Equation Uses a Fixed Positive-Sign Convention","stem":"Use objective gradient=[25, 50], constraint-gradient rows=[[1, 0], [0, 1]], aligned multipliers=[25, 50]. Give (objective_gradient, constraint_gradient_rows, multipliers, gradient_combination, stationarity_residual, equation_holds, sign_convention).","answer":{"kind":"expression","value":"([25, 50], [[1, 0], [0, 1]], [25, 50], [25, 50], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)","explanation":"Require full row rank, compute J_g^T*lambda and residual grad_f−J_g^T*lambda in the declared coordinate order. The equation holds exactly when every residual component is0. These supplied vectors audit algebra only; separate function/point data must establish feasibility, differentiability and theorem applicability. Stationarity is not a minimum certificate. The exact result, retaining the requested order, is ([25, 50], [[1, 0], [0, 1]], [25, 50], [25, 50], [0, 0], Yes, convention_grad_f_equals_J_g_transpose_lambda)."},"skills":["The Multiplier Equation Uses a Fixed Positive-Sign Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0051","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0926f49df5a79a08b7997fcbc5ba8371cec3c667828940dc3a531ddb5e35ff9d","family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194051,"semantic_template":"lagrange-single-linear-constraint|{\"center\":[[0,1],[0,1]],\"constraint_gradient\":[[1,1],[1,1]],\"constraint_rhs\":[3,1]}"},"fields":{"title":"Squared Distance with One Regular Linear Constraint Has an Exact Solution","stem":"Minimize squared Euclidean distance to center=[0, 0] subject to normal a=[1, 1], a dot x=3; require a nonzero normal. Give (center, constraint_gradient, constraint_rhs, normal_displacement_coefficient, solution_point, multiplier, objective_gradient, feasible, stationary, specified_problem_scope).","answer":{"kind":"expression","value":"([0, 0], [1, 1], 3, 3/2, [3/2, 3/2], 3, [3, 3], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation":"Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([0, 0], [1, 1], 3, 3/2, [3/2, 3/2], 3, [3, 3], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)."},"skills":["Squared Distance with One Regular Linear Constraint Has an Exact Solution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0052","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa7254aaea46f6b87d5ce581eade7c0938656910921966e56879bd8023cd0a13","family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194052,"semantic_template":"lagrange-single-linear-constraint|{\"center\":[[1,1],[1,1]],\"constraint_gradient\":[[1,1],[2,1]],\"constraint_rhs\":[4,1]}"},"fields":{"title":"Squared Distance with One Regular Linear Constraint Has an Exact Solution","stem":"Minimize squared Euclidean distance to center=[1, 1] subject to normal a=[1, 2], a dot x=4; require a nonzero normal. Give (center, constraint_gradient, constraint_rhs, normal_displacement_coefficient, solution_point, multiplier, objective_gradient, feasible, stationary, specified_problem_scope).","answer":{"kind":"expression","value":"([1, 1], [1, 2], 4, 1/5, [6/5, 7/5], 2/5, [2/5, 4/5], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation":"Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([1, 1], [1, 2], 4, 1/5, [6/5, 7/5], 2/5, [2/5, 4/5], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)."},"skills":["Squared Distance with One Regular Linear Constraint Has an Exact Solution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0053","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1005c72bfe3a535e4cdd26eefda1aa9168349ad5d4e5c4b06004c88e2bc8be4","family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194053,"semantic_template":"lagrange-single-linear-constraint|{\"center\":[[2,1],[2,1]],\"constraint_gradient\":[[1,1],[3,1]],\"constraint_rhs\":[5,1]}"},"fields":{"title":"Squared Distance with One Regular Linear Constraint Has an Exact Solution","stem":"Minimize squared Euclidean distance to center=[2, 2] subject to normal a=[1, 3], a dot x=5; require a nonzero normal. Give (center, constraint_gradient, constraint_rhs, normal_displacement_coefficient, solution_point, multiplier, objective_gradient, feasible, stationary, specified_problem_scope).","answer":{"kind":"expression","value":"([2, 2], [1, 3], 5, -3/10, [17/10, 11/10], -3/5, [-3/5, -9/5], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation":"Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([2, 2], [1, 3], 5, -3/10, [17/10, 11/10], -3/5, [-3/5, -9/5], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)."},"skills":["Squared Distance with One Regular Linear Constraint Has an Exact Solution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0054","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d79054bce27737c746319af7eb8aa2bd9c1f8603bb69ffd32d6d12f37e94c840","family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194054,"semantic_template":"lagrange-single-linear-constraint|{\"center\":[[3,1],[3,1]],\"constraint_gradient\":[[1,1],[4,1]],\"constraint_rhs\":[6,1]}"},"fields":{"title":"Squared Distance with One Regular Linear Constraint Has an Exact Solution","stem":"Minimize squared Euclidean distance to center=[3, 3] subject to normal a=[1, 4], a dot x=6; require a nonzero normal. Give (center, constraint_gradient, constraint_rhs, normal_displacement_coefficient, solution_point, multiplier, objective_gradient, feasible, stationary, specified_problem_scope).","answer":{"kind":"expression","value":"([3, 3], [1, 4], 6, -9/17, [42/17, 15/17], -18/17, [-18/17, -72/17], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation":"Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([3, 3], [1, 4], 6, -9/17, [42/17, 15/17], -18/17, [-18/17, -72/17], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)."},"skills":["Squared Distance with One Regular Linear Constraint Has an Exact Solution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0055","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"905bb18587ac6a5cd9199eed24405c2abc802b3e6c3444bec5e1b8b37987d9ff","family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194055,"semantic_template":"lagrange-single-linear-constraint|{\"center\":[[4,1],[4,1]],\"constraint_gradient\":[[1,1],[5,1]],\"constraint_rhs\":[7,1]}"},"fields":{"title":"Squared Distance with One Regular Linear Constraint Has an Exact Solution","stem":"Minimize squared Euclidean distance to center=[4, 4] subject to normal a=[1, 5], a dot x=7; require a nonzero normal. Give (center, constraint_gradient, constraint_rhs, normal_displacement_coefficient, solution_point, multiplier, objective_gradient, feasible, stationary, specified_problem_scope).","answer":{"kind":"expression","value":"([4, 4], [1, 5], 7, -17/26, [87/26, 19/26], -17/13, [-17/13, -85/13], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation":"Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([4, 4], [1, 5], 7, -17/26, [87/26, 19/26], -17/13, [-17/13, -85/13], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)."},"skills":["Squared Distance with One Regular Linear Constraint Has an Exact Solution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0056","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cab4c75636a6e994db0fa5e27d3afe1519e1d1aedc72f5afe484014e3ad687f1","family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194056,"semantic_template":"lagrange-single-linear-constraint|{\"center\":[[0,1],[5,1]],\"constraint_gradient\":[[1,1],[6,1]],\"constraint_rhs\":[8,1]}"},"fields":{"title":"Squared Distance with One Regular Linear Constraint Has an Exact Solution","stem":"Minimize squared Euclidean distance to center=[0, 5] subject to normal a=[1, 6], a dot x=8; require a nonzero normal. Give (center, constraint_gradient, constraint_rhs, normal_displacement_coefficient, solution_point, multiplier, objective_gradient, feasible, stationary, specified_problem_scope).","answer":{"kind":"expression","value":"([0, 5], [1, 6], 8, -22/37, [-22/37, 53/37], -44/37, [-44/37, -264/37], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation":"Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([0, 5], [1, 6], 8, -22/37, [-22/37, 53/37], -44/37, [-44/37, -264/37], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)."},"skills":["Squared Distance with One Regular Linear Constraint Has an Exact Solution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0057","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea542253e39067f6206db6d43183ea5354381ac11e1feef31b65ea338d9701b3","family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194057,"semantic_template":"lagrange-single-linear-constraint|{\"center\":[[1,1],[6,1]],\"constraint_gradient\":[[1,1],[7,1]],\"constraint_rhs\":[9,1]}"},"fields":{"title":"Squared Distance with One Regular Linear Constraint Has an Exact Solution","stem":"Minimize squared Euclidean distance to center=[1, 6] subject to normal a=[1, 7], a dot x=9; require a nonzero normal. Give (center, constraint_gradient, constraint_rhs, normal_displacement_coefficient, solution_point, multiplier, objective_gradient, feasible, stationary, specified_problem_scope).","answer":{"kind":"expression","value":"([1, 6], [1, 7], 9, -17/25, [8/25, 31/25], -34/25, [-34/25, -238/25], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation":"Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([1, 6], [1, 7], 9, -17/25, [8/25, 31/25], -34/25, [-34/25, -238/25], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)."},"skills":["Squared Distance with One Regular Linear Constraint Has an Exact Solution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0058","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6991f6f229627d7dff42b2e0027efd19f04ab7c1145e691a8425051edcb0309","family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194058,"semantic_template":"lagrange-single-linear-constraint|{\"center\":[[2,1],[0,1]],\"constraint_gradient\":[[1,1],[8,1]],\"constraint_rhs\":[10,1]}"},"fields":{"title":"Squared Distance with One Regular Linear Constraint Has an Exact Solution","stem":"Minimize squared Euclidean distance to center=[2, 0] subject to normal a=[1, 8], a dot x=10; require a nonzero normal. Give (center, constraint_gradient, constraint_rhs, normal_displacement_coefficient, solution_point, multiplier, objective_gradient, feasible, stationary, specified_problem_scope).","answer":{"kind":"expression","value":"([2, 0], [1, 8], 10, 8/65, [138/65, 64/65], 16/65, [16/65, 128/65], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation":"Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([2, 0], [1, 8], 10, 8/65, [138/65, 64/65], 16/65, [16/65, 128/65], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)."},"skills":["Squared Distance with One Regular Linear Constraint Has an Exact Solution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0059","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f04af6845e43f5d05d770a27bde2843b3d6033798979ad417dd79caf83ddcc2","family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194059,"semantic_template":"lagrange-single-linear-constraint|{\"center\":[[3,1],[1,1]],\"constraint_gradient\":[[1,1],[9,1]],\"constraint_rhs\":[11,1]}"},"fields":{"title":"Squared Distance with One Regular Linear Constraint Has an Exact Solution","stem":"Minimize squared Euclidean distance to center=[3, 1] subject to normal a=[1, 9], a dot x=11; require a nonzero normal. Give (center, constraint_gradient, constraint_rhs, normal_displacement_coefficient, solution_point, multiplier, objective_gradient, feasible, stationary, specified_problem_scope).","answer":{"kind":"expression","value":"([3, 1], [1, 9], 11, -1/82, [245/82, 73/82], -1/41, [-1/41, -9/41], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation":"Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([3, 1], [1, 9], 11, -1/82, [245/82, 73/82], -1/41, [-1/41, -9/41], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)."},"skills":["Squared Distance with One Regular Linear Constraint Has an Exact Solution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0060","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85e19cde7446f2f5a9a5583cdb473767789ccce655ff627f52605d551209cf1a","family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194060,"semantic_template":"lagrange-single-linear-constraint|{\"center\":[[4,1],[2,1]],\"constraint_gradient\":[[1,1],[10,1]],\"constraint_rhs\":[12,1]}"},"fields":{"title":"Squared Distance with One Regular Linear Constraint Has an Exact Solution","stem":"Minimize squared Euclidean distance to center=[4, 2] subject to normal a=[1, 10], a dot x=12; require a nonzero normal. Give (center, constraint_gradient, constraint_rhs, normal_displacement_coefficient, solution_point, multiplier, objective_gradient, feasible, stationary, specified_problem_scope).","answer":{"kind":"expression","value":"([4, 2], [1, 10], 12, -12/101, [392/101, 82/101], -24/101, [-24/101, -240/101], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation":"Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([4, 2], [1, 10], 12, -12/101, [392/101, 82/101], -24/101, [-24/101, -240/101], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)."},"skills":["Squared Distance with One Regular Linear Constraint Has an Exact Solution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0061","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2033f3652de706727dbce5ebd92ff3ec1d83fd3003265b8205ef2e918ec52a33","family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194061,"semantic_template":"lagrange-single-linear-constraint|{\"center\":[[0,1],[3,1]],\"constraint_gradient\":[[1,1],[11,1]],\"constraint_rhs\":[13,1]}"},"fields":{"title":"Squared Distance with One Regular Linear Constraint Has an Exact Solution","stem":"Minimize squared Euclidean distance to center=[0, 3] subject to normal a=[1, 11], a dot x=13; require a nonzero normal. Give (center, constraint_gradient, constraint_rhs, normal_displacement_coefficient, solution_point, multiplier, objective_gradient, feasible, stationary, specified_problem_scope).","answer":{"kind":"expression","value":"([0, 3], [1, 11], 13, -10/61, [-10/61, 73/61], -20/61, [-20/61, -220/61], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation":"Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([0, 3], [1, 11], 13, -10/61, [-10/61, 73/61], -20/61, [-20/61, -220/61], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)."},"skills":["Squared Distance with One Regular Linear Constraint Has an Exact Solution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0062","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"700ae6e03d2f93cee749bd106d95e45c7577fe76160be06ce4f95fb2db37c74a","family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194062,"semantic_template":"lagrange-single-linear-constraint|{\"center\":[[1,1],[4,1]],\"constraint_gradient\":[[1,1],[12,1]],\"constraint_rhs\":[14,1]}"},"fields":{"title":"Squared Distance with One Regular Linear Constraint Has an Exact Solution","stem":"Minimize squared Euclidean distance to center=[1, 4] subject to normal a=[1, 12], a dot x=14; require a nonzero normal. Give (center, constraint_gradient, constraint_rhs, normal_displacement_coefficient, solution_point, multiplier, objective_gradient, feasible, stationary, specified_problem_scope).","answer":{"kind":"expression","value":"([1, 4], [1, 12], 14, -7/29, [22/29, 32/29], -14/29, [-14/29, -168/29], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation":"Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([1, 4], [1, 12], 14, -7/29, [22/29, 32/29], -14/29, [-14/29, -168/29], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)."},"skills":["Squared Distance with One Regular Linear Constraint Has an Exact Solution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0063","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28242779733973dac11d55c476b353a29a8b4b2d9c4e25784d72f2501b5f5459","family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194063,"semantic_template":"lagrange-single-linear-constraint|{\"center\":[[2,1],[5,1]],\"constraint_gradient\":[[1,1],[13,1]],\"constraint_rhs\":[15,1]}"},"fields":{"title":"Squared Distance with One Regular Linear Constraint Has an Exact Solution","stem":"Minimize squared Euclidean distance to center=[2, 5] subject to normal a=[1, 13], a dot x=15; require a nonzero normal. Give (center, constraint_gradient, constraint_rhs, normal_displacement_coefficient, solution_point, multiplier, objective_gradient, feasible, stationary, specified_problem_scope).","answer":{"kind":"expression","value":"([2, 5], [1, 13], 15, -26/85, [144/85, 87/85], -52/85, [-52/85, -676/85], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation":"Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([2, 5], [1, 13], 15, -26/85, [144/85, 87/85], -52/85, [-52/85, -676/85], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)."},"skills":["Squared Distance with One Regular Linear Constraint Has an Exact Solution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0064","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37a5124f7fc29d4c84c12d7070dd860f09bc08f94490b1e2bfbf2827d92c6f60","family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194064,"semantic_template":"lagrange-single-linear-constraint|{\"center\":[[3,1],[6,1]],\"constraint_gradient\":[[1,1],[14,1]],\"constraint_rhs\":[16,1]}"},"fields":{"title":"Squared Distance with One Regular Linear Constraint Has an Exact Solution","stem":"Minimize squared Euclidean distance to center=[3, 6] subject to normal a=[1, 14], a dot x=16; require a nonzero normal. Give (center, constraint_gradient, constraint_rhs, normal_displacement_coefficient, solution_point, multiplier, objective_gradient, feasible, stationary, specified_problem_scope).","answer":{"kind":"expression","value":"([3, 6], [1, 14], 16, -71/197, [520/197, 188/197], -142/197, [-142/197, -1988/197], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation":"Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([3, 6], [1, 14], 16, -71/197, [520/197, 188/197], -142/197, [-142/197, -1988/197], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)."},"skills":["Squared Distance with One Regular Linear Constraint Has an Exact Solution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0065","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31dbeafd2cbb491f42489f9a0ef86b6257b93f60fdc96e73e439e89b7c51bc87","family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194065,"semantic_template":"lagrange-single-linear-constraint|{\"center\":[[4,1],[0,1]],\"constraint_gradient\":[[1,1],[15,1]],\"constraint_rhs\":[17,1]}"},"fields":{"title":"Squared Distance with One Regular Linear Constraint Has an Exact Solution","stem":"Minimize squared Euclidean distance to center=[4, 0] subject to normal a=[1, 15], a dot x=17; require a nonzero normal. Give (center, constraint_gradient, constraint_rhs, normal_displacement_coefficient, solution_point, multiplier, objective_gradient, feasible, stationary, specified_problem_scope).","answer":{"kind":"expression","value":"([4, 0], [1, 15], 17, 13/226, [917/226, 195/226], 13/113, [13/113, 195/113], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation":"Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([4, 0], [1, 15], 17, 13/226, [917/226, 195/226], 13/113, [13/113, 195/113], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)."},"skills":["Squared Distance with One Regular Linear Constraint Has an Exact Solution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0066","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2db6253c11bd7f6187aa46b77c4bbb47dd861af6a6e5f78e38c43149cc2a5fc1","family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194066,"semantic_template":"lagrange-single-linear-constraint|{\"center\":[[0,1],[1,1]],\"constraint_gradient\":[[1,1],[16,1]],\"constraint_rhs\":[18,1]}"},"fields":{"title":"Squared Distance with One Regular Linear Constraint Has an Exact Solution","stem":"Minimize squared Euclidean distance to center=[0, 1] subject to normal a=[1, 16], a dot x=18; require a nonzero normal. Give (center, constraint_gradient, constraint_rhs, normal_displacement_coefficient, solution_point, multiplier, objective_gradient, feasible, stationary, specified_problem_scope).","answer":{"kind":"expression","value":"([0, 1], [1, 16], 18, 2/257, [2/257, 289/257], 4/257, [4/257, 64/257], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation":"Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([0, 1], [1, 16], 18, 2/257, [2/257, 289/257], 4/257, [4/257, 64/257], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)."},"skills":["Squared Distance with One Regular Linear Constraint Has an Exact Solution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0067","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee833138e9ab0d1928ac639cd36324e3f7ac5b6a02364f8a3574700adb69251c","family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194067,"semantic_template":"lagrange-single-linear-constraint|{\"center\":[[1,1],[2,1]],\"constraint_gradient\":[[1,1],[17,1]],\"constraint_rhs\":[19,1]}"},"fields":{"title":"Squared Distance with One Regular Linear Constraint Has an Exact Solution","stem":"Minimize squared Euclidean distance to center=[1, 2] subject to normal a=[1, 17], a dot x=19; require a nonzero normal. Give (center, constraint_gradient, constraint_rhs, normal_displacement_coefficient, solution_point, multiplier, objective_gradient, feasible, stationary, specified_problem_scope).","answer":{"kind":"expression","value":"([1, 2], [1, 17], 19, -8/145, [137/145, 154/145], -16/145, [-16/145, -272/145], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation":"Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([1, 2], [1, 17], 19, -8/145, [137/145, 154/145], -16/145, [-16/145, -272/145], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)."},"skills":["Squared Distance with One Regular Linear Constraint Has an Exact Solution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0068","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e557327dfaa5d1db6458ebebe59289e2ae0bfae59780668cde9663b405d0d17","family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194068,"semantic_template":"lagrange-single-linear-constraint|{\"center\":[[2,1],[3,1]],\"constraint_gradient\":[[1,1],[18,1]],\"constraint_rhs\":[20,1]}"},"fields":{"title":"Squared Distance with One Regular Linear Constraint Has an Exact Solution","stem":"Minimize squared Euclidean distance to center=[2, 3] subject to normal a=[1, 18], a dot x=20; require a nonzero normal. Give (center, constraint_gradient, constraint_rhs, normal_displacement_coefficient, solution_point, multiplier, objective_gradient, feasible, stationary, specified_problem_scope).","answer":{"kind":"expression","value":"([2, 3], [1, 18], 20, -36/325, [614/325, 327/325], -72/325, [-72/325, -1296/325], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation":"Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([2, 3], [1, 18], 20, -36/325, [614/325, 327/325], -72/325, [-72/325, -1296/325], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)."},"skills":["Squared Distance with One Regular Linear Constraint Has an Exact Solution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0069","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76b957a38afdbfbc72b09d26d47c188e9816ca18c876f7bda3e3306760074d94","family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194069,"semantic_template":"lagrange-single-linear-constraint|{\"center\":[[3,1],[4,1]],\"constraint_gradient\":[[1,1],[19,1]],\"constraint_rhs\":[21,1]}"},"fields":{"title":"Squared Distance with One Regular Linear Constraint Has an Exact Solution","stem":"Minimize squared Euclidean distance to center=[3, 4] subject to normal a=[1, 19], a dot x=21; require a nonzero normal. Give (center, constraint_gradient, constraint_rhs, normal_displacement_coefficient, solution_point, multiplier, objective_gradient, feasible, stationary, specified_problem_scope).","answer":{"kind":"expression","value":"([3, 4], [1, 19], 21, -29/181, [514/181, 173/181], -58/181, [-58/181, -1102/181], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation":"Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([3, 4], [1, 19], 21, -29/181, [514/181, 173/181], -58/181, [-58/181, -1102/181], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)."},"skills":["Squared Distance with One Regular Linear Constraint Has an Exact Solution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0070","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f86a61f44eabea7be4bcca7f7a301aa82fbf026207a27e639df627ad774c019","family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194070,"semantic_template":"lagrange-single-linear-constraint|{\"center\":[[4,1],[5,1]],\"constraint_gradient\":[[1,1],[20,1]],\"constraint_rhs\":[22,1]}"},"fields":{"title":"Squared Distance with One Regular Linear Constraint Has an Exact Solution","stem":"Minimize squared Euclidean distance to center=[4, 5] subject to normal a=[1, 20], a dot x=22; require a nonzero normal. Give (center, constraint_gradient, constraint_rhs, normal_displacement_coefficient, solution_point, multiplier, objective_gradient, feasible, stationary, specified_problem_scope).","answer":{"kind":"expression","value":"([4, 5], [1, 20], 22, -82/401, [1522/401, 365/401], -164/401, [-164/401, -3280/401], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation":"Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([4, 5], [1, 20], 22, -82/401, [1522/401, 365/401], -164/401, [-164/401, -3280/401], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)."},"skills":["Squared Distance with One Regular Linear Constraint Has an Exact Solution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0071","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f5cbf0fc77fc86f0b081515314b00b7ba7a798c7ce76a83704bd375f886133d","family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194071,"semantic_template":"lagrange-single-linear-constraint|{\"center\":[[0,1],[6,1]],\"constraint_gradient\":[[1,1],[21,1]],\"constraint_rhs\":[23,1]}"},"fields":{"title":"Squared Distance with One Regular Linear Constraint Has an Exact Solution","stem":"Minimize squared Euclidean distance to center=[0, 6] subject to normal a=[1, 21], a dot x=23; require a nonzero normal. Give (center, constraint_gradient, constraint_rhs, normal_displacement_coefficient, solution_point, multiplier, objective_gradient, feasible, stationary, specified_problem_scope).","answer":{"kind":"expression","value":"([0, 6], [1, 21], 23, -103/442, [-103/442, 489/442], -103/221, [-103/221, -2163/221], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation":"Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([0, 6], [1, 21], 23, -103/442, [-103/442, 489/442], -103/221, [-103/221, -2163/221], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)."},"skills":["Squared Distance with One Regular Linear Constraint Has an Exact Solution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0072","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f790634aca8aa93d21508344c9728ce3b7ea019f7049fb684fa6fc64c3bfe245","family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194072,"semantic_template":"lagrange-single-linear-constraint|{\"center\":[[1,1],[0,1]],\"constraint_gradient\":[[1,1],[22,1]],\"constraint_rhs\":[24,1]}"},"fields":{"title":"Squared Distance with One Regular Linear Constraint Has an Exact Solution","stem":"Minimize squared Euclidean distance to center=[1, 0] subject to normal a=[1, 22], a dot x=24; require a nonzero normal. Give (center, constraint_gradient, constraint_rhs, normal_displacement_coefficient, solution_point, multiplier, objective_gradient, feasible, stationary, specified_problem_scope).","answer":{"kind":"expression","value":"([1, 0], [1, 22], 24, 23/485, [508/485, 506/485], 46/485, [46/485, 1012/485], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation":"Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([1, 0], [1, 22], 24, 23/485, [508/485, 506/485], 46/485, [46/485, 1012/485], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)."},"skills":["Squared Distance with One Regular Linear Constraint Has an Exact Solution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0073","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f68ab027152b689dd189677adb53067f7c9f90cdc51cc9f75e2389ee37c8d0ae","family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194073,"semantic_template":"lagrange-single-linear-constraint|{\"center\":[[2,1],[1,1]],\"constraint_gradient\":[[1,1],[23,1]],\"constraint_rhs\":[25,1]}"},"fields":{"title":"Squared Distance with One Regular Linear Constraint Has an Exact Solution","stem":"Minimize squared Euclidean distance to center=[2, 1] subject to normal a=[1, 23], a dot x=25; require a nonzero normal. Give (center, constraint_gradient, constraint_rhs, normal_displacement_coefficient, solution_point, multiplier, objective_gradient, feasible, stationary, specified_problem_scope).","answer":{"kind":"expression","value":"([2, 1], [1, 23], 25, 0, [2, 1], 0, [0, 0], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation":"Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([2, 1], [1, 23], 25, 0, [2, 1], 0, [0, 0], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)."},"skills":["Squared Distance with One Regular Linear Constraint Has an Exact Solution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0074","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00d08999ade46712f13910bc85a5ac818ba8af4baac20f26fb2cb100e8331a00","family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194074,"semantic_template":"lagrange-single-linear-constraint|{\"center\":[[3,1],[2,1]],\"constraint_gradient\":[[1,1],[24,1]],\"constraint_rhs\":[26,1]}"},"fields":{"title":"Squared Distance with One Regular Linear Constraint Has an Exact Solution","stem":"Minimize squared Euclidean distance to center=[3, 2] subject to normal a=[1, 24], a dot x=26; require a nonzero normal. Give (center, constraint_gradient, constraint_rhs, normal_displacement_coefficient, solution_point, multiplier, objective_gradient, feasible, stationary, specified_problem_scope).","answer":{"kind":"expression","value":"([3, 2], [1, 24], 26, -25/577, [1706/577, 554/577], -50/577, [-50/577, -1200/577], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation":"Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([3, 2], [1, 24], 26, -25/577, [1706/577, 554/577], -50/577, [-50/577, -1200/577], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)."},"skills":["Squared Distance with One Regular Linear Constraint Has an Exact Solution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0075","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a0f30589adaf7b383e5b7d282732aadcddd504ffc145b2c2f5475a16f26da7a5","family_id":"qf-mod50-lagrange-single-linear-constraint","seed":202611194075,"semantic_template":"lagrange-single-linear-constraint|{\"center\":[[4,1],[3,1]],\"constraint_gradient\":[[1,1],[25,1]],\"constraint_rhs\":[27,1]}"},"fields":{"title":"Squared Distance with One Regular Linear Constraint Has an Exact Solution","stem":"Minimize squared Euclidean distance to center=[4, 3] subject to normal a=[1, 25], a dot x=27; require a nonzero normal. Give (center, constraint_gradient, constraint_rhs, normal_displacement_coefficient, solution_point, multiplier, objective_gradient, feasible, stationary, specified_problem_scope).","answer":{"kind":"expression","value":"([4, 3], [1, 25], 27, -26/313, [1226/313, 289/313], -52/313, [-52/313, -1300/313], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)","explanation":"Use displacement coefficient (rhs−a dot center)/(a dot a) along the normal. Under grad_f=J_g^T*lambda the multiplier is twice that coefficient. Verify feasibility and the complete stationarity equation. This specifically solved squared-distance/linear-constraint problem does not turn arbitrary multiplier equations into sufficient minimum, globality or uniqueness certificates. The exact result, retaining the requested order, is ([4, 3], [1, 25], 27, -26/313, [1226/313, 289/313], -52/313, [-52/313, -1300/313], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly)."},"skills":["Squared Distance with One Regular Linear Constraint Has an Exact Solution"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0076","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed0c81a0fc95b45a04aa54f6b570d83c55000e24c7d716da038502ef8fe3d8eb","family_id":"qf-mod50-lagrange-boundary","seed":202611194076,"semantic_template":"lagrange-boundary|{\"mode\":\"rank_deficient\",\"model\":{\"constraint_gradients\":[[[1,1],[1,1]],[[2,1],[2,1]]]}}"},"fields":{"title":"Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT","stem":"Use mode=rank_deficient, complete model={\"constraint_gradients\":[[1,1],[2,2]]}. Give (mode, constraint_gradient_rows, rank, constraint_count, regular_theorem_applicable, scope).","answer":{"kind":"expression","value":"(rank_deficient, [[1, 1], [2, 2]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)","explanation":"Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (rank_deficient, [[1, 1], [2, 2]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)."},"skills":["Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0077","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ca893ee1b8c23c5b7756868902b692347dc1acc8fd8f8d53aec45df4e8539a8","family_id":"qf-mod50-lagrange-boundary","seed":202611194077,"semantic_template":"lagrange-boundary|{\"mode\":\"stationary_not_sufficient\",\"model\":{\"scale\":[2,1]}}"},"fields":{"title":"Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT","stem":"Use mode=stationary_not_sufficient, complete model={\"scale\":2}. Give (mode, objective_description, constraint_description, point, stationarity_statement, stationary, actual_point_status, minimum_claim, boundary).","answer":{"kind":"expression","value":"(stationary_not_sufficient, f(x,y)=-2*x^2, g(x,y)=y=0, [0,0], grad_f=[0,0]=0*grad_g, Yes, strict_local_maximum_on_constraint, No, first_order_multiplier_equation_is_necessary_not_sufficient_for_a_minimum)","explanation":"Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (stationary_not_sufficient, f(x,y)=-2*x^2, g(x,y)=y=0, [0,0], grad_f=[0,0]=0*grad_g, Yes, strict_local_maximum_on_constraint, No, first_order_multiplier_equation_is_necessary_not_sufficient_for_a_minimum)."},"skills":["Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0078","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a677e4d2ed94a37741cb6993bca3eef876ad46c5d5c92dee9b386f3eb20b3cff","family_id":"qf-mod50-lagrange-boundary","seed":202611194078,"semantic_template":"lagrange-boundary|{\"mode\":\"infeasible_point\",\"model\":{\"constraint_gradient\":[[1,1],[3,1]],\"constraint_rhs\":[2,1],\"point\":[[3,1],[0,1]]}}"},"fields":{"title":"Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT","stem":"Use mode=infeasible_point, complete model={\"constraint_gradient\":[1,3],\"constraint_rhs\":2,\"point\":[3,0]}. Give (mode, constraint_gradient, constraint_rhs, point, constraint_value, feasible, boundary).","answer":{"kind":"expression","value":"(infeasible_point, [1, 3], 2, [3, 0], 3, No, stationarity_never_repairs_missing_feasibility)","explanation":"Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (infeasible_point, [1, 3], 2, [3, 0], 3, No, stationarity_never_repairs_missing_feasibility)."},"skills":["Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0079","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62b85abca5ea74a19029a4326cfa065fa312223fe68540efcc28a23180a80c95","family_id":"qf-mod50-lagrange-boundary","seed":202611194079,"semantic_template":"lagrange-boundary|{\"mode\":\"kkt_confusion\",\"model\":{\"constraint_kind\":\"inequality\"}}"},"fields":{"title":"Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT","stem":"Use mode=kkt_confusion, complete model={\"constraint_kind\":\"inequality\"}. Give (mode, constraint_kind, equality_theorem_supplies_KKT_claim, boundary).","answer":{"kind":"expression","value":"(kkt_confusion, inequality, No, equality_only_Lagrange_condition_does_not_supply_KKT_sign_or_complementarity_conditions)","explanation":"Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (kkt_confusion, inequality, No, equality_only_Lagrange_condition_does_not_supply_KKT_sign_or_complementarity_conditions)."},"skills":["Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0080","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14cb542ed763b0d9b8d48982bcffca48a9c85d0eae83d41b0b028f932206e95d","family_id":"qf-mod50-lagrange-boundary","seed":202611194080,"semantic_template":"lagrange-boundary|{\"mode\":\"rank_deficient\",\"model\":{\"constraint_gradients\":[[[1,1],[5,1]],[[2,1],[10,1]]]}}"},"fields":{"title":"Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT","stem":"Use mode=rank_deficient, complete model={\"constraint_gradients\":[[1,5],[2,10]]}. Give (mode, constraint_gradient_rows, rank, constraint_count, regular_theorem_applicable, scope).","answer":{"kind":"expression","value":"(rank_deficient, [[1, 5], [2, 10]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)","explanation":"Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (rank_deficient, [[1, 5], [2, 10]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)."},"skills":["Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0081","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"356403d7d0aa1f6a90ca9f8f278b7ab1bffc5f2ccc8bd27a84aac4228224d38a","family_id":"qf-mod50-lagrange-boundary","seed":202611194081,"semantic_template":"lagrange-boundary|{\"mode\":\"stationary_not_sufficient\",\"model\":{\"scale\":[6,1]}}"},"fields":{"title":"Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT","stem":"Use mode=stationary_not_sufficient, complete model={\"scale\":6}. Give (mode, objective_description, constraint_description, point, stationarity_statement, stationary, actual_point_status, minimum_claim, boundary).","answer":{"kind":"expression","value":"(stationary_not_sufficient, f(x,y)=-6*x^2, g(x,y)=y=0, [0,0], grad_f=[0,0]=0*grad_g, Yes, strict_local_maximum_on_constraint, No, first_order_multiplier_equation_is_necessary_not_sufficient_for_a_minimum)","explanation":"Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (stationary_not_sufficient, f(x,y)=-6*x^2, g(x,y)=y=0, [0,0], grad_f=[0,0]=0*grad_g, Yes, strict_local_maximum_on_constraint, No, first_order_multiplier_equation_is_necessary_not_sufficient_for_a_minimum)."},"skills":["Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0082","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a3e9e87d96d4ca2f9141dfed6d140e32b250e7630993fa0b4025b2661fffc78","family_id":"qf-mod50-lagrange-boundary","seed":202611194082,"semantic_template":"lagrange-boundary|{\"mode\":\"infeasible_point\",\"model\":{\"constraint_gradient\":[[1,1],[7,1]],\"constraint_rhs\":[6,1],\"point\":[[7,1],[0,1]]}}"},"fields":{"title":"Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT","stem":"Use mode=infeasible_point, complete model={\"constraint_gradient\":[1,7],\"constraint_rhs\":6,\"point\":[7,0]}. Give (mode, constraint_gradient, constraint_rhs, point, constraint_value, feasible, boundary).","answer":{"kind":"expression","value":"(infeasible_point, [1, 7], 6, [7, 0], 7, No, stationarity_never_repairs_missing_feasibility)","explanation":"Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (infeasible_point, [1, 7], 6, [7, 0], 7, No, stationarity_never_repairs_missing_feasibility)."},"skills":["Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0083","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43136b04a3e5fff1f2f70f590b6f9dcafb51ded93e595d91ee1143d162fa42ad","family_id":"qf-mod50-lagrange-boundary","seed":202611194083,"semantic_template":"lagrange-boundary|{\"mode\":\"rank_deficient\",\"model\":{\"constraint_gradients\":[[[1,1],[8,1]],[[2,1],[16,1]]]}}"},"fields":{"title":"Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT","stem":"Use mode=rank_deficient, complete model={\"constraint_gradients\":[[1,8],[2,16]]}. Give (mode, constraint_gradient_rows, rank, constraint_count, regular_theorem_applicable, scope).","answer":{"kind":"expression","value":"(rank_deficient, [[1, 8], [2, 16]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)","explanation":"Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (rank_deficient, [[1, 8], [2, 16]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)."},"skills":["Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0084","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85a0642acec4ec8177237a0d3f2d8b9077c9a1491efea386d255dd97bf51765c","family_id":"qf-mod50-lagrange-boundary","seed":202611194084,"semantic_template":"lagrange-boundary|{\"mode\":\"rank_deficient\",\"model\":{\"constraint_gradients\":[[[1,1],[9,1]],[[2,1],[18,1]]]}}"},"fields":{"title":"Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT","stem":"Use mode=rank_deficient, complete model={\"constraint_gradients\":[[1,9],[2,18]]}. Give (mode, constraint_gradient_rows, rank, constraint_count, regular_theorem_applicable, scope).","answer":{"kind":"expression","value":"(rank_deficient, [[1, 9], [2, 18]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)","explanation":"Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (rank_deficient, [[1, 9], [2, 18]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)."},"skills":["Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0085","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa7ae98c54a7c8faa95a23e7685845ff6e0c2cbb60e58ad30af7d2662d41233c","family_id":"qf-mod50-lagrange-boundary","seed":202611194085,"semantic_template":"lagrange-boundary|{\"mode\":\"stationary_not_sufficient\",\"model\":{\"scale\":[10,1]}}"},"fields":{"title":"Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT","stem":"Use mode=stationary_not_sufficient, complete model={\"scale\":10}. Give (mode, objective_description, constraint_description, point, stationarity_statement, stationary, actual_point_status, minimum_claim, boundary).","answer":{"kind":"expression","value":"(stationary_not_sufficient, f(x,y)=-10*x^2, g(x,y)=y=0, [0,0], grad_f=[0,0]=0*grad_g, Yes, strict_local_maximum_on_constraint, No, first_order_multiplier_equation_is_necessary_not_sufficient_for_a_minimum)","explanation":"Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (stationary_not_sufficient, f(x,y)=-10*x^2, g(x,y)=y=0, [0,0], grad_f=[0,0]=0*grad_g, Yes, strict_local_maximum_on_constraint, No, first_order_multiplier_equation_is_necessary_not_sufficient_for_a_minimum)."},"skills":["Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0086","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1485a2d7ee1ecd6851f327a4cc8e5a3c1ba8facd7cb8663496783805c739ce4","family_id":"qf-mod50-lagrange-boundary","seed":202611194086,"semantic_template":"lagrange-boundary|{\"mode\":\"infeasible_point\",\"model\":{\"constraint_gradient\":[[1,1],[11,1]],\"constraint_rhs\":[10,1],\"point\":[[11,1],[0,1]]}}"},"fields":{"title":"Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT","stem":"Use mode=infeasible_point, complete model={\"constraint_gradient\":[1,11],\"constraint_rhs\":10,\"point\":[11,0]}. Give (mode, constraint_gradient, constraint_rhs, point, constraint_value, feasible, boundary).","answer":{"kind":"expression","value":"(infeasible_point, [1, 11], 10, [11, 0], 11, No, stationarity_never_repairs_missing_feasibility)","explanation":"Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (infeasible_point, [1, 11], 10, [11, 0], 11, No, stationarity_never_repairs_missing_feasibility)."},"skills":["Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0087","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69c284fe695fe4f5f28d147522c6b311714e218198d75ece92525222a4e8b772","family_id":"qf-mod50-lagrange-boundary","seed":202611194087,"semantic_template":"lagrange-boundary|{\"mode\":\"rank_deficient\",\"model\":{\"constraint_gradients\":[[[1,1],[12,1]],[[2,1],[24,1]]]}}"},"fields":{"title":"Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT","stem":"Use mode=rank_deficient, complete model={\"constraint_gradients\":[[1,12],[2,24]]}. Give (mode, constraint_gradient_rows, rank, constraint_count, regular_theorem_applicable, scope).","answer":{"kind":"expression","value":"(rank_deficient, [[1, 12], [2, 24]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)","explanation":"Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (rank_deficient, [[1, 12], [2, 24]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)."},"skills":["Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0088","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8cf183e63fa9be35c308fd4384426a19739bff439527de670f6aac718cf18a72","family_id":"qf-mod50-lagrange-boundary","seed":202611194088,"semantic_template":"lagrange-boundary|{\"mode\":\"rank_deficient\",\"model\":{\"constraint_gradients\":[[[1,1],[13,1]],[[2,1],[26,1]]]}}"},"fields":{"title":"Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT","stem":"Use mode=rank_deficient, complete model={\"constraint_gradients\":[[1,13],[2,26]]}. Give (mode, constraint_gradient_rows, rank, constraint_count, regular_theorem_applicable, scope).","answer":{"kind":"expression","value":"(rank_deficient, [[1, 13], [2, 26]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)","explanation":"Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (rank_deficient, [[1, 13], [2, 26]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)."},"skills":["Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0089","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f85b030117b30692c939688c480a6f113e03ecb9dae3272997940081094fe4b2","family_id":"qf-mod50-lagrange-boundary","seed":202611194089,"semantic_template":"lagrange-boundary|{\"mode\":\"stationary_not_sufficient\",\"model\":{\"scale\":[14,1]}}"},"fields":{"title":"Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT","stem":"Use mode=stationary_not_sufficient, complete model={\"scale\":14}. Give (mode, objective_description, constraint_description, point, stationarity_statement, stationary, actual_point_status, minimum_claim, boundary).","answer":{"kind":"expression","value":"(stationary_not_sufficient, f(x,y)=-14*x^2, g(x,y)=y=0, [0,0], grad_f=[0,0]=0*grad_g, Yes, strict_local_maximum_on_constraint, No, first_order_multiplier_equation_is_necessary_not_sufficient_for_a_minimum)","explanation":"Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (stationary_not_sufficient, f(x,y)=-14*x^2, g(x,y)=y=0, [0,0], grad_f=[0,0]=0*grad_g, Yes, strict_local_maximum_on_constraint, No, first_order_multiplier_equation_is_necessary_not_sufficient_for_a_minimum)."},"skills":["Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0090","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"399c6354298c6e48f94649337c2c40cf8f577092a424c1be425c072d107bee4d","family_id":"qf-mod50-lagrange-boundary","seed":202611194090,"semantic_template":"lagrange-boundary|{\"mode\":\"infeasible_point\",\"model\":{\"constraint_gradient\":[[1,1],[15,1]],\"constraint_rhs\":[14,1],\"point\":[[15,1],[0,1]]}}"},"fields":{"title":"Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT","stem":"Use mode=infeasible_point, complete model={\"constraint_gradient\":[1,15],\"constraint_rhs\":14,\"point\":[15,0]}. Give (mode, constraint_gradient, constraint_rhs, point, constraint_value, feasible, boundary).","answer":{"kind":"expression","value":"(infeasible_point, [1, 15], 14, [15, 0], 15, No, stationarity_never_repairs_missing_feasibility)","explanation":"Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (infeasible_point, [1, 15], 14, [15, 0], 15, No, stationarity_never_repairs_missing_feasibility)."},"skills":["Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0091","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18b23c34468e2128a97e92cb3b0f7430d975f063e0af2ae12435066e018f9ae7","family_id":"qf-mod50-lagrange-boundary","seed":202611194091,"semantic_template":"lagrange-boundary|{\"mode\":\"rank_deficient\",\"model\":{\"constraint_gradients\":[[[1,1],[16,1]],[[2,1],[32,1]]]}}"},"fields":{"title":"Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT","stem":"Use mode=rank_deficient, complete model={\"constraint_gradients\":[[1,16],[2,32]]}. Give (mode, constraint_gradient_rows, rank, constraint_count, regular_theorem_applicable, scope).","answer":{"kind":"expression","value":"(rank_deficient, [[1, 16], [2, 32]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)","explanation":"Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (rank_deficient, [[1, 16], [2, 32]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)."},"skills":["Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0092","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79592785e9c208168b2dec9017b75cf0dee1365f09422f0c9d25c69cd6bd2a5b","family_id":"qf-mod50-lagrange-boundary","seed":202611194092,"semantic_template":"lagrange-boundary|{\"mode\":\"rank_deficient\",\"model\":{\"constraint_gradients\":[[[1,1],[17,1]],[[2,1],[34,1]]]}}"},"fields":{"title":"Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT","stem":"Use mode=rank_deficient, complete model={\"constraint_gradients\":[[1,17],[2,34]]}. Give (mode, constraint_gradient_rows, rank, constraint_count, regular_theorem_applicable, scope).","answer":{"kind":"expression","value":"(rank_deficient, [[1, 17], [2, 34]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)","explanation":"Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (rank_deficient, [[1, 17], [2, 34]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)."},"skills":["Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0093","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed5f0381c5688dc53cec7535cb662686f73a098f09f84a9f1e09a4376b01c713","family_id":"qf-mod50-lagrange-boundary","seed":202611194093,"semantic_template":"lagrange-boundary|{\"mode\":\"stationary_not_sufficient\",\"model\":{\"scale\":[18,1]}}"},"fields":{"title":"Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT","stem":"Use mode=stationary_not_sufficient, complete model={\"scale\":18}. Give (mode, objective_description, constraint_description, point, stationarity_statement, stationary, actual_point_status, minimum_claim, boundary).","answer":{"kind":"expression","value":"(stationary_not_sufficient, f(x,y)=-18*x^2, g(x,y)=y=0, [0,0], grad_f=[0,0]=0*grad_g, Yes, strict_local_maximum_on_constraint, No, first_order_multiplier_equation_is_necessary_not_sufficient_for_a_minimum)","explanation":"Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (stationary_not_sufficient, f(x,y)=-18*x^2, g(x,y)=y=0, [0,0], grad_f=[0,0]=0*grad_g, Yes, strict_local_maximum_on_constraint, No, first_order_multiplier_equation_is_necessary_not_sufficient_for_a_minimum)."},"skills":["Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0094","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54c07d4dd3b759ebe3f40a73f8683e97a33a3bb515b5102861822b67e042ac43","family_id":"qf-mod50-lagrange-boundary","seed":202611194094,"semantic_template":"lagrange-boundary|{\"mode\":\"infeasible_point\",\"model\":{\"constraint_gradient\":[[1,1],[19,1]],\"constraint_rhs\":[18,1],\"point\":[[19,1],[0,1]]}}"},"fields":{"title":"Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT","stem":"Use mode=infeasible_point, complete model={\"constraint_gradient\":[1,19],\"constraint_rhs\":18,\"point\":[19,0]}. Give (mode, constraint_gradient, constraint_rhs, point, constraint_value, feasible, boundary).","answer":{"kind":"expression","value":"(infeasible_point, [1, 19], 18, [19, 0], 19, No, stationarity_never_repairs_missing_feasibility)","explanation":"Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (infeasible_point, [1, 19], 18, [19, 0], 19, No, stationarity_never_repairs_missing_feasibility)."},"skills":["Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0095","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5248e01902422f22182767114554d41bf897141dae78e48851f7e5ac68372e24","family_id":"qf-mod50-lagrange-boundary","seed":202611194095,"semantic_template":"lagrange-boundary|{\"mode\":\"rank_deficient\",\"model\":{\"constraint_gradients\":[[[1,1],[20,1]],[[2,1],[40,1]]]}}"},"fields":{"title":"Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT","stem":"Use mode=rank_deficient, complete model={\"constraint_gradients\":[[1,20],[2,40]]}. Give (mode, constraint_gradient_rows, rank, constraint_count, regular_theorem_applicable, scope).","answer":{"kind":"expression","value":"(rank_deficient, [[1, 20], [2, 40]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)","explanation":"Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (rank_deficient, [[1, 20], [2, 40]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)."},"skills":["Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0096","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59ab1121861219ca43e75921f522a3b368d42a8a6143d562a62f411cb6176a38","family_id":"qf-mod50-lagrange-boundary","seed":202611194096,"semantic_template":"lagrange-boundary|{\"mode\":\"rank_deficient\",\"model\":{\"constraint_gradients\":[[[1,1],[21,1]],[[2,1],[42,1]]]}}"},"fields":{"title":"Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT","stem":"Use mode=rank_deficient, complete model={\"constraint_gradients\":[[1,21],[2,42]]}. Give (mode, constraint_gradient_rows, rank, constraint_count, regular_theorem_applicable, scope).","answer":{"kind":"expression","value":"(rank_deficient, [[1, 21], [2, 42]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)","explanation":"Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (rank_deficient, [[1, 21], [2, 42]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)."},"skills":["Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0097","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"09c7965ba4076a2dccaabbcc371ebce9703f550b0612397b93f30986edf6062b","family_id":"qf-mod50-lagrange-boundary","seed":202611194097,"semantic_template":"lagrange-boundary|{\"mode\":\"stationary_not_sufficient\",\"model\":{\"scale\":[22,1]}}"},"fields":{"title":"Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT","stem":"Use mode=stationary_not_sufficient, complete model={\"scale\":22}. Give (mode, objective_description, constraint_description, point, stationarity_statement, stationary, actual_point_status, minimum_claim, boundary).","answer":{"kind":"expression","value":"(stationary_not_sufficient, f(x,y)=-22*x^2, g(x,y)=y=0, [0,0], grad_f=[0,0]=0*grad_g, Yes, strict_local_maximum_on_constraint, No, first_order_multiplier_equation_is_necessary_not_sufficient_for_a_minimum)","explanation":"Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (stationary_not_sufficient, f(x,y)=-22*x^2, g(x,y)=y=0, [0,0], grad_f=[0,0]=0*grad_g, Yes, strict_local_maximum_on_constraint, No, first_order_multiplier_equation_is_necessary_not_sufficient_for_a_minimum)."},"skills":["Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0098","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b754624c855aadfc089d69ed8dffe6f76f5ad0edd36173718ee80b156456e60","family_id":"qf-mod50-lagrange-boundary","seed":202611194098,"semantic_template":"lagrange-boundary|{\"mode\":\"infeasible_point\",\"model\":{\"constraint_gradient\":[[1,1],[23,1]],\"constraint_rhs\":[22,1],\"point\":[[23,1],[0,1]]}}"},"fields":{"title":"Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT","stem":"Use mode=infeasible_point, complete model={\"constraint_gradient\":[1,23],\"constraint_rhs\":22,\"point\":[23,0]}. Give (mode, constraint_gradient, constraint_rhs, point, constraint_value, feasible, boundary).","answer":{"kind":"expression","value":"(infeasible_point, [1, 23], 22, [23, 0], 23, No, stationarity_never_repairs_missing_feasibility)","explanation":"Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (infeasible_point, [1, 23], 22, [23, 0], 23, No, stationarity_never_repairs_missing_feasibility)."},"skills":["Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0099","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05eea3d790c9622b7690d5e816cffb93de01c47121de4ab246769d6c63a4e311","family_id":"qf-mod50-lagrange-boundary","seed":202611194099,"semantic_template":"lagrange-boundary|{\"mode\":\"rank_deficient\",\"model\":{\"constraint_gradients\":[[[1,1],[24,1]],[[2,1],[48,1]]]}}"},"fields":{"title":"Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT","stem":"Use mode=rank_deficient, complete model={\"constraint_gradients\":[[1,24],[2,48]]}. Give (mode, constraint_gradient_rows, rank, constraint_count, regular_theorem_applicable, scope).","answer":{"kind":"expression","value":"(rank_deficient, [[1, 24], [2, 48]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)","explanation":"Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (rank_deficient, [[1, 24], [2, 48]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)."},"skills":["Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r04-0100","batch_id":"batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3274f0709138afb7d10151fd40c4af8a3e49619ed7e7e16e2ac100e26364623","family_id":"qf-mod50-lagrange-boundary","seed":202611194100,"semantic_template":"lagrange-boundary|{\"mode\":\"rank_deficient\",\"model\":{\"constraint_gradients\":[[[1,1],[25,1]],[[2,1],[50,1]]]}}"},"fields":{"title":"Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT","stem":"Use mode=rank_deficient, complete model={\"constraint_gradients\":[[1,25],[2,50]]}. Give (mode, constraint_gradient_rows, rank, constraint_count, regular_theorem_applicable, scope).","answer":{"kind":"expression","value":"(rank_deficient, [[1, 25], [2, 50]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)","explanation":"Dependent constraint gradients withhold this regular theorem but do not imply absence of extrema. A feasible stationary point can be a strict constrained maximum. Stationarity never repairs infeasibility. Equality-only multipliers do not supply inequality KKT signs or complementarity, nor general global or unique primal optimality. The exact result, retaining the requested order, is (rank_deficient, [[1, 25], [2, 50]], 1, 2, No, regular_Lagrange_theorem_not_applicable_does_not_mean_no_extremum)."},"skills":["Regular Equality Conditions Are Necessary, Not Sufficient, and Not Inequality KKT"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}