{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0001","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a1a663e0bc0bdb63996a377a50a86a180ca0b34da21a1ee1b9260afdcad57ad","family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154001,"semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[2,1],[3,1]],\"c\":[[1,1],[1,1]]}"},"fields":{"title":"Construct the Dual from One Fixed Primal Convention and the Exact Transpose","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[2, 3], c=[1, 1]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [2, 3], [1, 1], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation":"Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [2, 3], [1, 1], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)."},"skills":["Construct the Dual from One Fixed Primal Convention and the Exact Transpose"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0002","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3c9a8ddf9f6c42949adebdcbb44ddcf405845be389aeea155a19f7fa4b751c0","family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154002,"semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[3,1],[4,1]],\"c\":[[2,1],[2,1]]}"},"fields":{"title":"Construct the Dual from One Fixed Primal Convention and the Exact Transpose","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[3, 4], c=[2, 2]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [3, 4], [2, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation":"Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [3, 4], [2, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)."},"skills":["Construct the Dual from One Fixed Primal Convention and the Exact Transpose"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0003","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36a301533cb3061f9dceebabe13754097cda8a9b25b3e215bfc7c23f05d64166","family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154003,"semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[4,1],[5,1]],\"c\":[[3,1],[3,1]]}"},"fields":{"title":"Construct the Dual from One Fixed Primal Convention and the Exact Transpose","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[4, 5], c=[3, 3]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [4, 5], [3, 3], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation":"Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [4, 5], [3, 3], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)."},"skills":["Construct the Dual from One Fixed Primal Convention and the Exact Transpose"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0004","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a92e3206e9f1ae873eacb22cb6b00bdf6925ddae06cb3f736dbe0dd2a9289867","family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154004,"semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[5,1],[6,1]],\"c\":[[4,1],[4,1]]}"},"fields":{"title":"Construct the Dual from One Fixed Primal Convention and the Exact Transpose","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[5, 6], c=[4, 4]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [5, 6], [4, 4], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation":"Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [5, 6], [4, 4], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)."},"skills":["Construct the Dual from One Fixed Primal Convention and the Exact Transpose"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0005","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cbb42dbfaae73e34e9c2bad7f2499417cd6c9f9971c0d362cb08b1861cc3a43a","family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154005,"semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[6,1],[7,1]],\"c\":[[1,1],[5,1]]}"},"fields":{"title":"Construct the Dual from One Fixed Primal Convention and the Exact Transpose","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[6, 7], c=[1, 5]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [6, 7], [1, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation":"Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [6, 7], [1, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)."},"skills":["Construct the Dual from One Fixed Primal Convention and the Exact Transpose"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0006","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8181ed24a836be0ac12c605524ab1009e280c49c225bc52074712b5436f0384b","family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154006,"semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[7,1],[8,1]],\"c\":[[2,1],[1,1]]}"},"fields":{"title":"Construct the Dual from One Fixed Primal Convention and the Exact Transpose","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[7, 8], c=[2, 1]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [7, 8], [2, 1], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation":"Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [7, 8], [2, 1], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)."},"skills":["Construct the Dual from One Fixed Primal Convention and the Exact Transpose"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0007","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ad213747518f4699ccbb35b136da5b21d94f49551a82ae842f4385033e5a9e7","family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154007,"semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[8,1],[9,1]],\"c\":[[3,1],[2,1]]}"},"fields":{"title":"Construct the Dual from One Fixed Primal Convention and the Exact Transpose","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[8, 9], c=[3, 2]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [8, 9], [3, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation":"Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [8, 9], [3, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)."},"skills":["Construct the Dual from One Fixed Primal Convention and the Exact Transpose"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0008","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1dfe068dffe84d67913c84b997b8a89f2f35ae6d3925941296375291ff50bb4e","family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154008,"semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[9,1],[10,1]],\"c\":[[4,1],[3,1]]}"},"fields":{"title":"Construct the Dual from One Fixed Primal Convention and the Exact Transpose","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[9, 10], c=[4, 3]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [9, 10], [4, 3], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation":"Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [9, 10], [4, 3], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)."},"skills":["Construct the Dual from One Fixed Primal Convention and the Exact Transpose"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0009","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b07163f755483f3ddb619e056540247c832212c17e8e2bd72731bc214407528","family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154009,"semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[10,1],[11,1]],\"c\":[[1,1],[4,1]]}"},"fields":{"title":"Construct the Dual from One Fixed Primal Convention and the Exact Transpose","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[10, 11], c=[1, 4]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [10, 11], [1, 4], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation":"Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [10, 11], [1, 4], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)."},"skills":["Construct the Dual from One Fixed Primal Convention and the Exact Transpose"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0010","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e810e92fd8f9be80b99b3978a633d99bf7d17273e4f368bd6e05062f20f681b","family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154010,"semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[11,1],[12,1]],\"c\":[[2,1],[5,1]]}"},"fields":{"title":"Construct the Dual from One Fixed Primal Convention and the Exact Transpose","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[11, 12], c=[2, 5]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [11, 12], [2, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation":"Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [11, 12], [2, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)."},"skills":["Construct the Dual from One Fixed Primal Convention and the Exact Transpose"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0011","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5053773fbf1ca78c4fbf7efeb27f3f36e3a2f3e1866071988de4a5ddf7fa7ec5","family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154011,"semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[12,1],[13,1]],\"c\":[[3,1],[1,1]]}"},"fields":{"title":"Construct the Dual from One Fixed Primal Convention and the Exact Transpose","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[12, 13], c=[3, 1]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [12, 13], [3, 1], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation":"Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [12, 13], [3, 1], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)."},"skills":["Construct the Dual from One Fixed Primal Convention and the Exact Transpose"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0012","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad63f5ae680f5f8a170105ab796615115165f332b0f2336454ace25b174690e0","family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154012,"semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[13,1],[14,1]],\"c\":[[4,1],[2,1]]}"},"fields":{"title":"Construct the Dual from One Fixed Primal Convention and the Exact Transpose","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[13, 14], c=[4, 2]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [13, 14], [4, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation":"Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [13, 14], [4, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)."},"skills":["Construct the Dual from One Fixed Primal Convention and the Exact Transpose"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0013","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31fbf68182450ad9036737a9c465c849ceabeff6da3935d881f7d1f80de3b41f","family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154013,"semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[14,1],[15,1]],\"c\":[[1,1],[3,1]]}"},"fields":{"title":"Construct the Dual from One Fixed Primal Convention and the Exact Transpose","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[14, 15], c=[1, 3]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [14, 15], [1, 3], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation":"Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [14, 15], [1, 3], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)."},"skills":["Construct the Dual from One Fixed Primal Convention and the Exact Transpose"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0014","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fcb0e2854a6606b237ac79713c51708ab714591f85ac8ac1badc8841beaad7ea","family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154014,"semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[15,1],[16,1]],\"c\":[[2,1],[4,1]]}"},"fields":{"title":"Construct the Dual from One Fixed Primal Convention and the Exact Transpose","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[15, 16], c=[2, 4]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [15, 16], [2, 4], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation":"Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [15, 16], [2, 4], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)."},"skills":["Construct the Dual from One Fixed Primal Convention and the Exact Transpose"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0015","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aaeffdedeb2be3e3cb12e9d016435ad671653cb1d5f099ee5a0b61363c383906","family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154015,"semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[16,1],[17,1]],\"c\":[[3,1],[5,1]]}"},"fields":{"title":"Construct the Dual from One Fixed Primal Convention and the Exact Transpose","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[16, 17], c=[3, 5]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [16, 17], [3, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation":"Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [16, 17], [3, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)."},"skills":["Construct the Dual from One Fixed Primal Convention and the Exact Transpose"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0016","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b50bffe6b270c7723ba8cf2f5d398cf689bf83dd35f2f83820061959a680a6c","family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154016,"semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[17,1],[18,1]],\"c\":[[4,1],[1,1]]}"},"fields":{"title":"Construct the Dual from One Fixed Primal Convention and the Exact Transpose","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[17, 18], c=[4, 1]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [17, 18], [4, 1], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation":"Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [17, 18], [4, 1], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)."},"skills":["Construct the Dual from One Fixed Primal Convention and the Exact Transpose"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0017","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"98a2f4ebdcca7df787ae2819e53e9a06e1f78c8db022fc7eff4fc01fcbd1da24","family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154017,"semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[18,1],[19,1]],\"c\":[[1,1],[2,1]]}"},"fields":{"title":"Construct the Dual from One Fixed Primal Convention and the Exact Transpose","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[18, 19], c=[1, 2]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [18, 19], [1, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation":"Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [18, 19], [1, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)."},"skills":["Construct the Dual from One Fixed Primal Convention and the Exact Transpose"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0018","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63a3f59d07b89729c17b9ce2a47dd2ba7ef283b87624d452c126729427fbad3a","family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154018,"semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[19,1],[20,1]],\"c\":[[2,1],[3,1]]}"},"fields":{"title":"Construct the Dual from One Fixed Primal Convention and the Exact Transpose","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[19, 20], c=[2, 3]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [19, 20], [2, 3], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation":"Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [19, 20], [2, 3], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)."},"skills":["Construct the Dual from One Fixed Primal Convention and the Exact Transpose"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0019","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7a084ae8f8d4f29c322f38a2fcae98ce346cf734541388335869667f25e6abd","family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154019,"semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[20,1],[21,1]],\"c\":[[3,1],[4,1]]}"},"fields":{"title":"Construct the Dual from One Fixed Primal Convention and the Exact Transpose","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[20, 21], c=[3, 4]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [20, 21], [3, 4], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation":"Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [20, 21], [3, 4], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)."},"skills":["Construct the Dual from One Fixed Primal Convention and the Exact Transpose"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0020","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"057734565a3290b88404cd7c6555b6b00c9236decdcdc0209a3776dfb81163c7","family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154020,"semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[21,1],[22,1]],\"c\":[[4,1],[5,1]]}"},"fields":{"title":"Construct the Dual from One Fixed Primal Convention and the Exact Transpose","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[21, 22], c=[4, 5]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [21, 22], [4, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation":"Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [21, 22], [4, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)."},"skills":["Construct the Dual from One Fixed Primal Convention and the Exact Transpose"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0021","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a997ca4e5ebbdeb73bc7df935af4fe311886ce222ee7155fa8ad73a4883d7c0d","family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154021,"semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[22,1],[23,1]],\"c\":[[1,1],[1,1]]}"},"fields":{"title":"Construct the Dual from One Fixed Primal Convention and the Exact Transpose","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[22, 23], c=[1, 1]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [22, 23], [1, 1], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation":"Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [22, 23], [1, 1], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)."},"skills":["Construct the Dual from One Fixed Primal Convention and the Exact Transpose"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0022","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18ddf804a0263ae035374a54dfd6382eff854f2c0a9dbca060462cdc686679c2","family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154022,"semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[23,1],[24,1]],\"c\":[[2,1],[2,1]]}"},"fields":{"title":"Construct the Dual from One Fixed Primal Convention and the Exact Transpose","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[23, 24], c=[2, 2]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [23, 24], [2, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation":"Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [23, 24], [2, 2], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)."},"skills":["Construct the Dual from One Fixed Primal Convention and the Exact Transpose"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0023","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59588a83ff648ab4c65a89b1a8273b07196b95c44ef01a98dc374a31f66bc80b","family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154023,"semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[24,1],[25,1]],\"c\":[[3,1],[3,1]]}"},"fields":{"title":"Construct the Dual from One Fixed Primal Convention and the Exact Transpose","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[24, 25], c=[3, 3]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [24, 25], [3, 3], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation":"Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [24, 25], [3, 3], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)."},"skills":["Construct the Dual from One Fixed Primal Convention and the Exact Transpose"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0024","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c97b40552095fc6e36dedcbf291906f008d13cd41249d2282459f0c73ee4238","family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154024,"semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[25,1],[26,1]],\"c\":[[4,1],[4,1]]}"},"fields":{"title":"Construct the Dual from One Fixed Primal Convention and the Exact Transpose","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[25, 26], c=[4, 4]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [25, 26], [4, 4], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation":"Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [25, 26], [4, 4], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)."},"skills":["Construct the Dual from One Fixed Primal Convention and the Exact Transpose"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0025","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"300015383b61f771058e45c950e91d11e47fd36c5a7882a6cfe97a13095bc93d","family_id":"qf-mod46-lpdual-construction-qualification","seed":202611154025,"semantic_template":"qf-mod46-lpdual-construction-qualification/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[26,1],[27,1]],\"c\":[[1,1],[5,1]]}"},"fields":{"title":"Construct the Dual from One Fixed Primal Convention and the Exact Transpose","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[26, 27], c=[1, 5]. Give (constraint_count, variable_count, A, b, c, transpose_A, primal_convention, dual_convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[1, 0], [0, 1]], [26, 27], [1, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)","explanation":"Check rectangular dimensions m,n and matching vectors. The dual has m variables and n lower-bound constraints with transposed A. Coefficients may have either sign; dimensions and signs do not certify feasibility or optimality. The exact result, retaining the requested order, is (2, 2, [[1, 0], [0, 1]], [26, 27], [1, 5], [[1, 0], [0, 1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes, dimensions_signs_and_transpose_are_fixed_by_one_convention)."},"skills":["Construct the Dual from One Fixed Primal Convention and the Exact Transpose"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0026","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93eae591d254dad25539d2e72f3a725f8acef5f5b89248c24b50b314ceaa434f","family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154026,"semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[2,1],[3,1]],\"c\":[[1,1],[1,1]],\"x\":[[1,1],[3,2]],\"y\":[[2,1],[2,1]]}"},"fields":{"title":"Both Feasibility Checks Precede the Weak-Duality Inequality Chain","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[2, 3], c=[1, 1]. Candidate x=[2/2, 3/2], y=[2, 2]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).","answer":{"kind":"expression","value":"([1, 3/2], [2, 2], [1, 3/2], [2, 2], Yes, Yes, 5/2, 5, 10, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation":"Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([1, 3/2], [2, 2], [1, 3/2], [2, 2], Yes, Yes, 5/2, 5, 10, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)."},"skills":["Both Feasibility Checks Precede the Weak-Duality Inequality Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0027","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9c1de334480d1854eba3a533175e07bb1d56862518437197aa22fae6588ae22","family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154027,"semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[3,1],[4,1]],\"c\":[[2,1],[2,1]],\"x\":[[3,2],[2,1]],\"y\":[[3,1],[3,1]]}"},"fields":{"title":"Both Feasibility Checks Precede the Weak-Duality Inequality Chain","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[3, 4], c=[2, 2]. Candidate x=[3/2, 4/2], y=[3, 3]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).","answer":{"kind":"expression","value":"([3/2, 2], [3, 3], [3/2, 2], [3, 3], Yes, Yes, 7, 21/2, 21, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation":"Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([3/2, 2], [3, 3], [3/2, 2], [3, 3], Yes, Yes, 7, 21/2, 21, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)."},"skills":["Both Feasibility Checks Precede the Weak-Duality Inequality Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0028","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"331c18fa0c462b2e5cea1e41d0da1456523c09154c199f8dcf345d6897725b47","family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154028,"semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[4,1],[5,1]],\"c\":[[3,1],[3,1]],\"x\":[[2,1],[5,2]],\"y\":[[4,1],[4,1]]}"},"fields":{"title":"Both Feasibility Checks Precede the Weak-Duality Inequality Chain","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[4, 5], c=[3, 3]. Candidate x=[4/2, 5/2], y=[4, 4]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).","answer":{"kind":"expression","value":"([2, 5/2], [4, 4], [2, 5/2], [4, 4], Yes, Yes, 27/2, 18, 36, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation":"Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([2, 5/2], [4, 4], [2, 5/2], [4, 4], Yes, Yes, 27/2, 18, 36, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)."},"skills":["Both Feasibility Checks Precede the Weak-Duality Inequality Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0029","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1bcd9929d14ece8c579dc4a72ff7be0600b81375cd0e9f9a723684cfb7f960fa","family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154029,"semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[5,1],[6,1]],\"c\":[[4,1],[4,1]],\"x\":[[5,2],[3,1]],\"y\":[[5,1],[5,1]]}"},"fields":{"title":"Both Feasibility Checks Precede the Weak-Duality Inequality Chain","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[5, 6], c=[4, 4]. Candidate x=[5/2, 6/2], y=[5, 5]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).","answer":{"kind":"expression","value":"([5/2, 3], [5, 5], [5/2, 3], [5, 5], Yes, Yes, 22, 55/2, 55, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation":"Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([5/2, 3], [5, 5], [5/2, 3], [5, 5], Yes, Yes, 22, 55/2, 55, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)."},"skills":["Both Feasibility Checks Precede the Weak-Duality Inequality Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0030","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"859b4b1aedb96c616c4e26d5691485d71fb9ea503d8666ace0e3a723fe97f0eb","family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154030,"semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[6,1],[7,1]],\"c\":[[1,1],[5,1]],\"x\":[[3,1],[7,2]],\"y\":[[2,1],[6,1]]}"},"fields":{"title":"Both Feasibility Checks Precede the Weak-Duality Inequality Chain","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[6, 7], c=[1, 5]. Candidate x=[6/2, 7/2], y=[2, 6]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).","answer":{"kind":"expression","value":"([3, 7/2], [2, 6], [3, 7/2], [2, 6], Yes, Yes, 41/2, 27, 54, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation":"Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([3, 7/2], [2, 6], [3, 7/2], [2, 6], Yes, Yes, 41/2, 27, 54, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)."},"skills":["Both Feasibility Checks Precede the Weak-Duality Inequality Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0031","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab66cd560ce18bc82a5d4f4e69f527476e7be32e2dcb0cb5c53baf911d464e04","family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154031,"semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[7,1],[8,1]],\"c\":[[2,1],[1,1]],\"x\":[[7,2],[4,1]],\"y\":[[3,1],[2,1]]}"},"fields":{"title":"Both Feasibility Checks Precede the Weak-Duality Inequality Chain","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[7, 8], c=[2, 1]. Candidate x=[7/2, 8/2], y=[3, 2]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).","answer":{"kind":"expression","value":"([7/2, 4], [3, 2], [7/2, 4], [3, 2], Yes, Yes, 11, 37/2, 37, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation":"Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([7/2, 4], [3, 2], [7/2, 4], [3, 2], Yes, Yes, 11, 37/2, 37, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)."},"skills":["Both Feasibility Checks Precede the Weak-Duality Inequality Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0032","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc0fedd642b6850aacfb76ba2d27a4d9405c863b9c83886b2fa7fd738d4a2886","family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154032,"semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[8,1],[9,1]],\"c\":[[3,1],[2,1]],\"x\":[[4,1],[9,2]],\"y\":[[4,1],[3,1]]}"},"fields":{"title":"Both Feasibility Checks Precede the Weak-Duality Inequality Chain","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[8, 9], c=[3, 2]. Candidate x=[8/2, 9/2], y=[4, 3]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).","answer":{"kind":"expression","value":"([4, 9/2], [4, 3], [4, 9/2], [4, 3], Yes, Yes, 21, 59/2, 59, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation":"Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([4, 9/2], [4, 3], [4, 9/2], [4, 3], Yes, Yes, 21, 59/2, 59, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)."},"skills":["Both Feasibility Checks Precede the Weak-Duality Inequality Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0033","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a932d3f5c7d0e63caf3b6980c68a1044103476ddf110f7c20b1b13bb01f107a7","family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154033,"semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[9,1],[10,1]],\"c\":[[4,1],[3,1]],\"x\":[[9,2],[5,1]],\"y\":[[5,1],[4,1]]}"},"fields":{"title":"Both Feasibility Checks Precede the Weak-Duality Inequality Chain","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[9, 10], c=[4, 3]. Candidate x=[9/2, 10/2], y=[5, 4]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).","answer":{"kind":"expression","value":"([9/2, 5], [5, 4], [9/2, 5], [5, 4], Yes, Yes, 33, 85/2, 85, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation":"Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([9/2, 5], [5, 4], [9/2, 5], [5, 4], Yes, Yes, 33, 85/2, 85, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)."},"skills":["Both Feasibility Checks Precede the Weak-Duality Inequality Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0034","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54eb03b654561edc64ffa26dff134f201022cdf75372c2b43ee565825e834190","family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154034,"semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[10,1],[11,1]],\"c\":[[1,1],[4,1]],\"x\":[[5,1],[11,2]],\"y\":[[2,1],[5,1]]}"},"fields":{"title":"Both Feasibility Checks Precede the Weak-Duality Inequality Chain","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[10, 11], c=[1, 4]. Candidate x=[10/2, 11/2], y=[2, 5]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).","answer":{"kind":"expression","value":"([5, 11/2], [2, 5], [5, 11/2], [2, 5], Yes, Yes, 27, 75/2, 75, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation":"Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([5, 11/2], [2, 5], [5, 11/2], [2, 5], Yes, Yes, 27, 75/2, 75, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)."},"skills":["Both Feasibility Checks Precede the Weak-Duality Inequality Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0035","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c744e1228c7aa7e9519dea784a319dd37ed73c5ae552470938d149e34792c708","family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154035,"semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[11,1],[12,1]],\"c\":[[2,1],[5,1]],\"x\":[[11,2],[6,1]],\"y\":[[3,1],[6,1]]}"},"fields":{"title":"Both Feasibility Checks Precede the Weak-Duality Inequality Chain","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[11, 12], c=[2, 5]. Candidate x=[11/2, 12/2], y=[3, 6]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).","answer":{"kind":"expression","value":"([11/2, 6], [3, 6], [11/2, 6], [3, 6], Yes, Yes, 41, 105/2, 105, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation":"Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([11/2, 6], [3, 6], [11/2, 6], [3, 6], Yes, Yes, 41, 105/2, 105, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)."},"skills":["Both Feasibility Checks Precede the Weak-Duality Inequality Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0036","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"44c5d09022636f120936805b0951330a8d15467ba37727e8f7de9b096d13a68e","family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154036,"semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[12,1],[13,1]],\"c\":[[3,1],[1,1]],\"x\":[[6,1],[13,2]],\"y\":[[4,1],[2,1]]}"},"fields":{"title":"Both Feasibility Checks Precede the Weak-Duality Inequality Chain","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[12, 13], c=[3, 1]. Candidate x=[12/2, 13/2], y=[4, 2]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).","answer":{"kind":"expression","value":"([6, 13/2], [4, 2], [6, 13/2], [4, 2], Yes, Yes, 49/2, 37, 74, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation":"Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([6, 13/2], [4, 2], [6, 13/2], [4, 2], Yes, Yes, 49/2, 37, 74, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)."},"skills":["Both Feasibility Checks Precede the Weak-Duality Inequality Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0037","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f9076fc735d2a8d7a7fc408984cb5172c14999ada7e2209bcf62cc3f1fbddce5","family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154037,"semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[13,1],[14,1]],\"c\":[[4,1],[2,1]],\"x\":[[13,2],[7,1]],\"y\":[[5,1],[3,1]]}"},"fields":{"title":"Both Feasibility Checks Precede the Weak-Duality Inequality Chain","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[13, 14], c=[4, 2]. Candidate x=[13/2, 14/2], y=[5, 3]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).","answer":{"kind":"expression","value":"([13/2, 7], [5, 3], [13/2, 7], [5, 3], Yes, Yes, 40, 107/2, 107, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation":"Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([13/2, 7], [5, 3], [13/2, 7], [5, 3], Yes, Yes, 40, 107/2, 107, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)."},"skills":["Both Feasibility Checks Precede the Weak-Duality Inequality Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0038","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08f285dad3b074a2f9af38ef92eb4198c7778327411aad861ab99210c5b0e795","family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154038,"semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[14,1],[15,1]],\"c\":[[1,1],[3,1]],\"x\":[[7,1],[15,2]],\"y\":[[2,1],[4,1]]}"},"fields":{"title":"Both Feasibility Checks Precede the Weak-Duality Inequality Chain","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[14, 15], c=[1, 3]. Candidate x=[14/2, 15/2], y=[2, 4]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).","answer":{"kind":"expression","value":"([7, 15/2], [2, 4], [7, 15/2], [2, 4], Yes, Yes, 59/2, 44, 88, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation":"Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([7, 15/2], [2, 4], [7, 15/2], [2, 4], Yes, Yes, 59/2, 44, 88, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)."},"skills":["Both Feasibility Checks Precede the Weak-Duality Inequality Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0039","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee71ba88193a661b3f3cd6b5e705e01e32e26c719474ce4ed50fd58a81f9fb47","family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154039,"semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[15,1],[16,1]],\"c\":[[2,1],[4,1]],\"x\":[[15,2],[8,1]],\"y\":[[3,1],[5,1]]}"},"fields":{"title":"Both Feasibility Checks Precede the Weak-Duality Inequality Chain","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[15, 16], c=[2, 4]. Candidate x=[15/2, 16/2], y=[3, 5]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).","answer":{"kind":"expression","value":"([15/2, 8], [3, 5], [15/2, 8], [3, 5], Yes, Yes, 47, 125/2, 125, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation":"Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([15/2, 8], [3, 5], [15/2, 8], [3, 5], Yes, Yes, 47, 125/2, 125, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)."},"skills":["Both Feasibility Checks Precede the Weak-Duality Inequality Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0040","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d963a32c83c4e5770369cf5a0b8e527f30f655f3ed5940416786c045e3f22c2b","family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154040,"semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[16,1],[17,1]],\"c\":[[3,1],[5,1]],\"x\":[[8,1],[17,2]],\"y\":[[4,1],[6,1]]}"},"fields":{"title":"Both Feasibility Checks Precede the Weak-Duality Inequality Chain","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[16, 17], c=[3, 5]. Candidate x=[16/2, 17/2], y=[4, 6]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).","answer":{"kind":"expression","value":"([8, 17/2], [4, 6], [8, 17/2], [4, 6], Yes, Yes, 133/2, 83, 166, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation":"Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([8, 17/2], [4, 6], [8, 17/2], [4, 6], Yes, Yes, 133/2, 83, 166, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)."},"skills":["Both Feasibility Checks Precede the Weak-Duality Inequality Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0041","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"475dc69b192b2a9e0d227bf99c5095e582fbc98ded3164a10fe53db462069c2a","family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154041,"semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[17,1],[18,1]],\"c\":[[4,1],[1,1]],\"x\":[[17,2],[9,1]],\"y\":[[5,1],[2,1]]}"},"fields":{"title":"Both Feasibility Checks Precede the Weak-Duality Inequality Chain","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[17, 18], c=[4, 1]. Candidate x=[17/2, 18/2], y=[5, 2]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).","answer":{"kind":"expression","value":"([17/2, 9], [5, 2], [17/2, 9], [5, 2], Yes, Yes, 43, 121/2, 121, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation":"Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([17/2, 9], [5, 2], [17/2, 9], [5, 2], Yes, Yes, 43, 121/2, 121, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)."},"skills":["Both Feasibility Checks Precede the Weak-Duality Inequality Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0042","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71f65e5964b5b8ebc21e1bfc5f9f7309f6af588f446826ec1d6ba0ba92522cc5","family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154042,"semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[18,1],[19,1]],\"c\":[[1,1],[2,1]],\"x\":[[9,1],[19,2]],\"y\":[[2,1],[3,1]]}"},"fields":{"title":"Both Feasibility Checks Precede the Weak-Duality Inequality Chain","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[18, 19], c=[1, 2]. Candidate x=[18/2, 19/2], y=[2, 3]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).","answer":{"kind":"expression","value":"([9, 19/2], [2, 3], [9, 19/2], [2, 3], Yes, Yes, 28, 93/2, 93, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation":"Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([9, 19/2], [2, 3], [9, 19/2], [2, 3], Yes, Yes, 28, 93/2, 93, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)."},"skills":["Both Feasibility Checks Precede the Weak-Duality Inequality Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0043","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84b1e77bfc9b47a5d939064bccb47083acbe6e851e70c96f8d333ea216e080f2","family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154043,"semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[19,1],[20,1]],\"c\":[[2,1],[3,1]],\"x\":[[19,2],[10,1]],\"y\":[[3,1],[4,1]]}"},"fields":{"title":"Both Feasibility Checks Precede the Weak-Duality Inequality Chain","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[19, 20], c=[2, 3]. Candidate x=[19/2, 20/2], y=[3, 4]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).","answer":{"kind":"expression","value":"([19/2, 10], [3, 4], [19/2, 10], [3, 4], Yes, Yes, 49, 137/2, 137, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation":"Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([19/2, 10], [3, 4], [19/2, 10], [3, 4], Yes, Yes, 49, 137/2, 137, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)."},"skills":["Both Feasibility Checks Precede the Weak-Duality Inequality Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0044","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"98bde0673ac2cc87ba2b18e7cef93fcd391b5883ce8b5252ce82e0503acc952f","family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154044,"semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[20,1],[21,1]],\"c\":[[3,1],[4,1]],\"x\":[[10,1],[21,2]],\"y\":[[4,1],[5,1]]}"},"fields":{"title":"Both Feasibility Checks Precede the Weak-Duality Inequality Chain","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[20, 21], c=[3, 4]. Candidate x=[20/2, 21/2], y=[4, 5]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).","answer":{"kind":"expression","value":"([10, 21/2], [4, 5], [10, 21/2], [4, 5], Yes, Yes, 72, 185/2, 185, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation":"Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([10, 21/2], [4, 5], [10, 21/2], [4, 5], Yes, Yes, 72, 185/2, 185, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)."},"skills":["Both Feasibility Checks Precede the Weak-Duality Inequality Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0045","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73de7ea28acb73a0e3df11c24563ce3851715872f7f996cfac9604875c55a246","family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154045,"semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[21,1],[22,1]],\"c\":[[4,1],[5,1]],\"x\":[[21,2],[11,1]],\"y\":[[5,1],[6,1]]}"},"fields":{"title":"Both Feasibility Checks Precede the Weak-Duality Inequality Chain","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[21, 22], c=[4, 5]. Candidate x=[21/2, 22/2], y=[5, 6]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).","answer":{"kind":"expression","value":"([21/2, 11], [5, 6], [21/2, 11], [5, 6], Yes, Yes, 97, 237/2, 237, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation":"Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([21/2, 11], [5, 6], [21/2, 11], [5, 6], Yes, Yes, 97, 237/2, 237, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)."},"skills":["Both Feasibility Checks Precede the Weak-Duality Inequality Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0046","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dfd5b0049f2e8d8922ed71623caefc105d3ef60837678c8c998bfa28b7601edc","family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154046,"semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[22,1],[23,1]],\"c\":[[1,1],[1,1]],\"x\":[[11,1],[23,2]],\"y\":[[2,1],[2,1]]}"},"fields":{"title":"Both Feasibility Checks Precede the Weak-Duality Inequality Chain","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[22, 23], c=[1, 1]. Candidate x=[22/2, 23/2], y=[2, 2]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).","answer":{"kind":"expression","value":"([11, 23/2], [2, 2], [11, 23/2], [2, 2], Yes, Yes, 45/2, 45, 90, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation":"Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([11, 23/2], [2, 2], [11, 23/2], [2, 2], Yes, Yes, 45/2, 45, 90, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)."},"skills":["Both Feasibility Checks Precede the Weak-Duality Inequality Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0047","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d343c3bf6602bb3ae57d2c5a553d715039e369a0d245a31fe0c000daac7eb377","family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154047,"semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[23,1],[24,1]],\"c\":[[2,1],[2,1]],\"x\":[[23,2],[12,1]],\"y\":[[3,1],[3,1]]}"},"fields":{"title":"Both Feasibility Checks Precede the Weak-Duality Inequality Chain","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[23, 24], c=[2, 2]. Candidate x=[23/2, 24/2], y=[3, 3]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).","answer":{"kind":"expression","value":"([23/2, 12], [3, 3], [23/2, 12], [3, 3], Yes, Yes, 47, 141/2, 141, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation":"Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([23/2, 12], [3, 3], [23/2, 12], [3, 3], Yes, Yes, 47, 141/2, 141, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)."},"skills":["Both Feasibility Checks Precede the Weak-Duality Inequality Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0048","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ae1678daacda930de4384f88707d86be5dcd7320ad7adf10f43c2daf127335c","family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154048,"semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[24,1],[25,1]],\"c\":[[3,1],[3,1]],\"x\":[[12,1],[25,2]],\"y\":[[4,1],[4,1]]}"},"fields":{"title":"Both Feasibility Checks Precede the Weak-Duality Inequality Chain","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[24, 25], c=[3, 3]. Candidate x=[24/2, 25/2], y=[4, 4]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).","answer":{"kind":"expression","value":"([12, 25/2], [4, 4], [12, 25/2], [4, 4], Yes, Yes, 147/2, 98, 196, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation":"Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([12, 25/2], [4, 4], [12, 25/2], [4, 4], Yes, Yes, 147/2, 98, 196, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)."},"skills":["Both Feasibility Checks Precede the Weak-Duality Inequality Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0049","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4aaaf2a327f3c9214e3f3b7125e47c62494bac92d3e266970ed498d7e5a8c40","family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154049,"semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[25,1],[26,1]],\"c\":[[4,1],[4,1]],\"x\":[[25,2],[13,1]],\"y\":[[5,1],[5,1]]}"},"fields":{"title":"Both Feasibility Checks Precede the Weak-Duality Inequality Chain","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[25, 26], c=[4, 4]. Candidate x=[25/2, 26/2], y=[5, 5]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).","answer":{"kind":"expression","value":"([25/2, 13], [5, 5], [25/2, 13], [5, 5], Yes, Yes, 102, 255/2, 255, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation":"Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([25/2, 13], [5, 5], [25/2, 13], [5, 5], Yes, Yes, 102, 255/2, 255, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)."},"skills":["Both Feasibility Checks Precede the Weak-Duality Inequality Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0050","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a4f02308ac6dd67fc903e0ac71c6b9bb6c27861e3606161e2e1ceb8b729e1ed","family_id":"qf-mod46-lpdual-weak-duality-chain","seed":202611154050,"semantic_template":"qf-mod46-lpdual-weak-duality-chain/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[26,1],[27,1]],\"c\":[[1,1],[5,1]],\"x\":[[13,1],[27,2]],\"y\":[[2,1],[6,1]]}"},"fields":{"title":"Both Feasibility Checks Precede the Weak-Duality Inequality Chain","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[26, 27], c=[1, 5]. Candidate x=[26/2, 27/2], y=[2, 6]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, weak_chain_holds, certificate).","answer":{"kind":"expression","value":"([13, 27/2], [2, 6], [13, 27/2], [2, 6], Yes, Yes, 161/2, 107, 214, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)","explanation":"Check x≥0,A*x≤b and y≥0,A^T*y≥c, then c^T*x≤y^T*A*x≤b^T*y. The middle value is not an extra objective. A feasible pair and the chain do not alone establish zero gap or attainment. The exact result, retaining the requested order, is ([13, 27/2], [2, 6], [13, 27/2], [2, 6], Yes, Yes, 161/2, 107, 214, Yes, Yes, Yes, cTx_le_yTAx_le_bTy_for_every_feasible_pair)."},"skills":["Both Feasibility Checks Precede the Weak-Duality Inequality Chain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0051","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"658df43d526744db4dc2c50fb2f931e48c6f8337a6e694e8a87c68863cff2387","family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154051,"semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[2,1],[3,1]],\"c\":[[1,1],[1,1]],\"x\":[[2,1],[3,1]],\"y\":[[1,1],[1,1]]}"},"fields":{"title":"A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[2, 3], c=[1, 1]. Candidate x=[2, 3], y=[1, 1]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).","answer":{"kind":"expression","value":"([2, 3], [1, 1], [2, 3], [1, 1], Yes, Yes, 5, 5, 5, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation":"Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([2, 3], [1, 1], [2, 3], [1, 1], Yes, Yes, 5, 5, 5, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)."},"skills":["A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0052","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"219f4eb8231b0d3c8357754ba78583a17b0c00192a335e81cb377bf1a9f9f395","family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154052,"semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[3,1],[4,1]],\"c\":[[2,1],[2,1]],\"x\":[[3,1],[4,1]],\"y\":[[2,1],[2,1]]}"},"fields":{"title":"A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[3, 4], c=[2, 2]. Candidate x=[3, 4], y=[2, 2]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).","answer":{"kind":"expression","value":"([3, 4], [2, 2], [3, 4], [2, 2], Yes, Yes, 14, 14, 14, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation":"Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([3, 4], [2, 2], [3, 4], [2, 2], Yes, Yes, 14, 14, 14, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)."},"skills":["A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0053","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb12cb04999d8c157015c4bb21d9b48d448858456d883a81d6400b8c744bf0f9","family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154053,"semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[4,1],[5,1]],\"c\":[[3,1],[3,1]],\"x\":[[4,1],[5,1]],\"y\":[[3,1],[3,1]]}"},"fields":{"title":"A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[4, 5], c=[3, 3]. Candidate x=[4, 5], y=[3, 3]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).","answer":{"kind":"expression","value":"([4, 5], [3, 3], [4, 5], [3, 3], Yes, Yes, 27, 27, 27, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation":"Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([4, 5], [3, 3], [4, 5], [3, 3], Yes, Yes, 27, 27, 27, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)."},"skills":["A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0054","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4310d0018ba3266011f9753329899aa5a946b04ceef145595b3f227467374185","family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154054,"semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[5,1],[6,1]],\"c\":[[4,1],[4,1]],\"x\":[[5,1],[6,1]],\"y\":[[4,1],[4,1]]}"},"fields":{"title":"A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[5, 6], c=[4, 4]. Candidate x=[5, 6], y=[4, 4]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).","answer":{"kind":"expression","value":"([5, 6], [4, 4], [5, 6], [4, 4], Yes, Yes, 44, 44, 44, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation":"Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([5, 6], [4, 4], [5, 6], [4, 4], Yes, Yes, 44, 44, 44, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)."},"skills":["A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0055","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89ac9270496920fd1cc7bb4c860daabf56ddd089adb3f5b49c9d84ea9a30ab21","family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154055,"semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[6,1],[7,1]],\"c\":[[1,1],[5,1]],\"x\":[[6,1],[7,1]],\"y\":[[1,1],[5,1]]}"},"fields":{"title":"A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[6, 7], c=[1, 5]. Candidate x=[6, 7], y=[1, 5]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).","answer":{"kind":"expression","value":"([6, 7], [1, 5], [6, 7], [1, 5], Yes, Yes, 41, 41, 41, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation":"Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([6, 7], [1, 5], [6, 7], [1, 5], Yes, Yes, 41, 41, 41, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)."},"skills":["A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0056","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ded564efcef7f8a48dcd3b3c6eec0025efe75ac95bea019b13875d53aa3c1bb","family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154056,"semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[7,1],[8,1]],\"c\":[[2,1],[1,1]],\"x\":[[7,1],[8,1]],\"y\":[[2,1],[1,1]]}"},"fields":{"title":"A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[7, 8], c=[2, 1]. Candidate x=[7, 8], y=[2, 1]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).","answer":{"kind":"expression","value":"([7, 8], [2, 1], [7, 8], [2, 1], Yes, Yes, 22, 22, 22, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation":"Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([7, 8], [2, 1], [7, 8], [2, 1], Yes, Yes, 22, 22, 22, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)."},"skills":["A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0057","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f60f229c6685b1151e3c9382c9f9f03c4a4597a4ebb2c0afe082c0ba284ccf9","family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154057,"semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[8,1],[9,1]],\"c\":[[3,1],[2,1]],\"x\":[[8,1],[9,1]],\"y\":[[3,1],[2,1]]}"},"fields":{"title":"A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[8, 9], c=[3, 2]. Candidate x=[8, 9], y=[3, 2]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).","answer":{"kind":"expression","value":"([8, 9], [3, 2], [8, 9], [3, 2], Yes, Yes, 42, 42, 42, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation":"Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([8, 9], [3, 2], [8, 9], [3, 2], Yes, Yes, 42, 42, 42, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)."},"skills":["A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0058","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c292cc56b078a391a4225b38a65a5704b5f1b711763753f3fb05cdc2ae80cf8","family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154058,"semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[9,1],[10,1]],\"c\":[[4,1],[3,1]],\"x\":[[9,1],[10,1]],\"y\":[[4,1],[3,1]]}"},"fields":{"title":"A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[9, 10], c=[4, 3]. Candidate x=[9, 10], y=[4, 3]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).","answer":{"kind":"expression","value":"([9, 10], [4, 3], [9, 10], [4, 3], Yes, Yes, 66, 66, 66, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation":"Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([9, 10], [4, 3], [9, 10], [4, 3], Yes, Yes, 66, 66, 66, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)."},"skills":["A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0059","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6eaacb8bce44f909056d95e171064489a52f7faa41a11b1cc310f3dd526f0045","family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154059,"semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[10,1],[11,1]],\"c\":[[1,1],[4,1]],\"x\":[[10,1],[11,1]],\"y\":[[1,1],[4,1]]}"},"fields":{"title":"A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[10, 11], c=[1, 4]. Candidate x=[10, 11], y=[1, 4]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).","answer":{"kind":"expression","value":"([10, 11], [1, 4], [10, 11], [1, 4], Yes, Yes, 54, 54, 54, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation":"Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([10, 11], [1, 4], [10, 11], [1, 4], Yes, Yes, 54, 54, 54, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)."},"skills":["A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0060","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"605bd2650c719b094b48327f0286f9e5c8b7b90450901a9ab6ac82388f53aa78","family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154060,"semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[11,1],[12,1]],\"c\":[[2,1],[5,1]],\"x\":[[11,1],[12,1]],\"y\":[[2,1],[5,1]]}"},"fields":{"title":"A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[11, 12], c=[2, 5]. Candidate x=[11, 12], y=[2, 5]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).","answer":{"kind":"expression","value":"([11, 12], [2, 5], [11, 12], [2, 5], Yes, Yes, 82, 82, 82, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation":"Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([11, 12], [2, 5], [11, 12], [2, 5], Yes, Yes, 82, 82, 82, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)."},"skills":["A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0061","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f669a78a17d4b4b48e045ab81d5fa606322c97633079e859de3279e2ab4253b","family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154061,"semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[12,1],[13,1]],\"c\":[[3,1],[1,1]],\"x\":[[12,1],[13,1]],\"y\":[[3,1],[1,1]]}"},"fields":{"title":"A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[12, 13], c=[3, 1]. Candidate x=[12, 13], y=[3, 1]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).","answer":{"kind":"expression","value":"([12, 13], [3, 1], [12, 13], [3, 1], Yes, Yes, 49, 49, 49, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation":"Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([12, 13], [3, 1], [12, 13], [3, 1], Yes, Yes, 49, 49, 49, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)."},"skills":["A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0062","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"551ed2520cb54ad1d8b801c59e069d39c067caf0f9d96308032e5fd326ce7515","family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154062,"semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[13,1],[14,1]],\"c\":[[4,1],[2,1]],\"x\":[[13,1],[14,1]],\"y\":[[4,1],[2,1]]}"},"fields":{"title":"A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[13, 14], c=[4, 2]. Candidate x=[13, 14], y=[4, 2]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).","answer":{"kind":"expression","value":"([13, 14], [4, 2], [13, 14], [4, 2], Yes, Yes, 80, 80, 80, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation":"Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([13, 14], [4, 2], [13, 14], [4, 2], Yes, Yes, 80, 80, 80, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)."},"skills":["A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0063","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2d36210ec2d51de04f12a07dc8967e26fe7d0e9d51573a115e4f5d49e0909d1","family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154063,"semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[14,1],[15,1]],\"c\":[[1,1],[3,1]],\"x\":[[14,1],[15,1]],\"y\":[[1,1],[3,1]]}"},"fields":{"title":"A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[14, 15], c=[1, 3]. Candidate x=[14, 15], y=[1, 3]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).","answer":{"kind":"expression","value":"([14, 15], [1, 3], [14, 15], [1, 3], Yes, Yes, 59, 59, 59, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation":"Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([14, 15], [1, 3], [14, 15], [1, 3], Yes, Yes, 59, 59, 59, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)."},"skills":["A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0064","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"faf423563e2c159d918d52875c63950ab4da9abf7f02e34f23d5fa5766248db4","family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154064,"semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[15,1],[16,1]],\"c\":[[2,1],[4,1]],\"x\":[[15,1],[16,1]],\"y\":[[2,1],[4,1]]}"},"fields":{"title":"A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[15, 16], c=[2, 4]. Candidate x=[15, 16], y=[2, 4]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).","answer":{"kind":"expression","value":"([15, 16], [2, 4], [15, 16], [2, 4], Yes, Yes, 94, 94, 94, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation":"Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([15, 16], [2, 4], [15, 16], [2, 4], Yes, Yes, 94, 94, 94, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)."},"skills":["A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0065","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5afb6702c1cb30fb452fd6fcd74d48aa592f7478178e91ab6eb37fe3ed8c2d2","family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154065,"semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[16,1],[17,1]],\"c\":[[3,1],[5,1]],\"x\":[[16,1],[17,1]],\"y\":[[3,1],[5,1]]}"},"fields":{"title":"A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[16, 17], c=[3, 5]. Candidate x=[16, 17], y=[3, 5]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).","answer":{"kind":"expression","value":"([16, 17], [3, 5], [16, 17], [3, 5], Yes, Yes, 133, 133, 133, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation":"Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([16, 17], [3, 5], [16, 17], [3, 5], Yes, Yes, 133, 133, 133, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)."},"skills":["A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0066","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ea171f27842e152844207250cb557a39e30b2d023414f47b6d0b367d5bacf0a","family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154066,"semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[17,1],[18,1]],\"c\":[[4,1],[1,1]],\"x\":[[17,1],[18,1]],\"y\":[[4,1],[1,1]]}"},"fields":{"title":"A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[17, 18], c=[4, 1]. Candidate x=[17, 18], y=[4, 1]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).","answer":{"kind":"expression","value":"([17, 18], [4, 1], [17, 18], [4, 1], Yes, Yes, 86, 86, 86, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation":"Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([17, 18], [4, 1], [17, 18], [4, 1], Yes, Yes, 86, 86, 86, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)."},"skills":["A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0067","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42df97ef0ceb356c120a92d8f29f84b1bb2c9081b36fee47b68d054ba3a560d1","family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154067,"semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[18,1],[19,1]],\"c\":[[1,1],[2,1]],\"x\":[[18,1],[19,1]],\"y\":[[1,1],[2,1]]}"},"fields":{"title":"A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[18, 19], c=[1, 2]. Candidate x=[18, 19], y=[1, 2]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).","answer":{"kind":"expression","value":"([18, 19], [1, 2], [18, 19], [1, 2], Yes, Yes, 56, 56, 56, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation":"Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([18, 19], [1, 2], [18, 19], [1, 2], Yes, Yes, 56, 56, 56, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)."},"skills":["A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0068","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e03cf8c21cdf84cfb81716b21ce279376d0386bbc8941d767f69447f71a620a0","family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154068,"semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[19,1],[20,1]],\"c\":[[2,1],[3,1]],\"x\":[[19,1],[20,1]],\"y\":[[2,1],[3,1]]}"},"fields":{"title":"A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[19, 20], c=[2, 3]. Candidate x=[19, 20], y=[2, 3]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).","answer":{"kind":"expression","value":"([19, 20], [2, 3], [19, 20], [2, 3], Yes, Yes, 98, 98, 98, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation":"Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([19, 20], [2, 3], [19, 20], [2, 3], Yes, Yes, 98, 98, 98, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)."},"skills":["A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0069","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85da59d2df15328ae3bbb0b99503047eccef1bc34a61610962a5221dfab89b90","family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154069,"semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[20,1],[21,1]],\"c\":[[3,1],[4,1]],\"x\":[[20,1],[21,1]],\"y\":[[3,1],[4,1]]}"},"fields":{"title":"A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[20, 21], c=[3, 4]. Candidate x=[20, 21], y=[3, 4]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).","answer":{"kind":"expression","value":"([20, 21], [3, 4], [20, 21], [3, 4], Yes, Yes, 144, 144, 144, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation":"Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([20, 21], [3, 4], [20, 21], [3, 4], Yes, Yes, 144, 144, 144, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)."},"skills":["A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0070","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"013b041463804cdfbf0c72f222dc2a6afa665111676308dec5d72b6cb793bad5","family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154070,"semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[21,1],[22,1]],\"c\":[[4,1],[5,1]],\"x\":[[21,1],[22,1]],\"y\":[[4,1],[5,1]]}"},"fields":{"title":"A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[21, 22], c=[4, 5]. Candidate x=[21, 22], y=[4, 5]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).","answer":{"kind":"expression","value":"([21, 22], [4, 5], [21, 22], [4, 5], Yes, Yes, 194, 194, 194, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation":"Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([21, 22], [4, 5], [21, 22], [4, 5], Yes, Yes, 194, 194, 194, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)."},"skills":["A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0071","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ace9a07b195c1038392aa11a5c22848d4f6565ccc87aadd396df69a77a9cd558","family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154071,"semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[22,1],[23,1]],\"c\":[[1,1],[1,1]],\"x\":[[22,1],[23,1]],\"y\":[[1,1],[1,1]]}"},"fields":{"title":"A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[22, 23], c=[1, 1]. Candidate x=[22, 23], y=[1, 1]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).","answer":{"kind":"expression","value":"([22, 23], [1, 1], [22, 23], [1, 1], Yes, Yes, 45, 45, 45, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation":"Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([22, 23], [1, 1], [22, 23], [1, 1], Yes, Yes, 45, 45, 45, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)."},"skills":["A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0072","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19f91c03cf76e008573911d1cbcea7983f212ab26a3e7a9e7cdae50a2f7f67e8","family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154072,"semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[23,1],[24,1]],\"c\":[[2,1],[2,1]],\"x\":[[23,1],[24,1]],\"y\":[[2,1],[2,1]]}"},"fields":{"title":"A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[23, 24], c=[2, 2]. Candidate x=[23, 24], y=[2, 2]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).","answer":{"kind":"expression","value":"([23, 24], [2, 2], [23, 24], [2, 2], Yes, Yes, 94, 94, 94, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation":"Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([23, 24], [2, 2], [23, 24], [2, 2], Yes, Yes, 94, 94, 94, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)."},"skills":["A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0073","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23710e7815a68dfcd59624c55d05413a22f112253299d99bf1ba48921741cd0f","family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154073,"semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[24,1],[25,1]],\"c\":[[3,1],[3,1]],\"x\":[[24,1],[25,1]],\"y\":[[3,1],[3,1]]}"},"fields":{"title":"A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[24, 25], c=[3, 3]. Candidate x=[24, 25], y=[3, 3]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).","answer":{"kind":"expression","value":"([24, 25], [3, 3], [24, 25], [3, 3], Yes, Yes, 147, 147, 147, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation":"Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([24, 25], [3, 3], [24, 25], [3, 3], Yes, Yes, 147, 147, 147, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)."},"skills":["A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0074","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b1f118dfbdb35c7097784bd41c1ea0ac72a04c919af2e3751805445d65201c4","family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154074,"semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[25,1],[26,1]],\"c\":[[4,1],[4,1]],\"x\":[[25,1],[26,1]],\"y\":[[4,1],[4,1]]}"},"fields":{"title":"A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[25, 26], c=[4, 4]. Candidate x=[25, 26], y=[4, 4]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).","answer":{"kind":"expression","value":"([25, 26], [4, 4], [25, 26], [4, 4], Yes, Yes, 204, 204, 204, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation":"Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([25, 26], [4, 4], [25, 26], [4, 4], Yes, Yes, 204, 204, 204, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)."},"skills":["A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0075","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f19c7e9bbe547032c392bbb74807e589681e31372117cc8784065f7b3dbfe2bd","family_id":"qf-mod46-lpdual-zero-gap-certificate","seed":202611154075,"semantic_template":"qf-mod46-lpdual-zero-gap-certificate/{\"A\":[[[1,1],[0,1]],[[0,1],[1,1]]],\"b\":[[26,1],[27,1]],\"c\":[[1,1],[5,1]],\"x\":[[26,1],[27,1]],\"y\":[[1,1],[5,1]]}"},"fields":{"title":"A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1, 0], [0, 1]], b=[26, 27], c=[1, 5]. Candidate x=[26, 27], y=[1, 5]. Give (x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, primal_optimal, dual_optimal, certificate).","answer":{"kind":"expression","value":"([26, 27], [1, 5], [26, 27], [1, 5], Yes, Yes, 161, 161, 161, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)","explanation":"Require both points feasible and their exact objective values equal. Weak duality then certifies each point optimal. Neither bare numerical equality nor a finite search grid supplies these feasibility obligations or a general strong-duality proof. The exact result, retaining the requested order, is ([26, 27], [1, 5], [26, 27], [1, 5], Yes, Yes, 161, 161, 161, Yes, Yes, Yes, Yes, Yes, feasible_equal_objectives_certify_both_points_optimal_by_weak_duality)."},"skills":["A Feasible Zero-Gap Pair Certifies Both Primal and Dual Optimality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0076","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff0faa2cb16c223c7d6f6cae147ee7f3bb9b4b627c7df730555e245d74d81437","family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154076,"semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[2,1]],\"c\":[[1,1]],\"mode\":\"primal_infeasible_equal_objective\",\"x\":[[3,1]],\"y\":[[3,2]]}"},"fields":{"title":"Equal Objective Values and Finite Grids Do Not Replace Theorem Premises","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[2], c=[1]. Candidate x=[3], y=[3/2]. Boundary mode=primal_infeasible_equal_objective. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).","answer":{"kind":"expression","value":"(primal_infeasible_equal_objective, [3], [3/2], [3], [3/2], No, Yes, 3, 9/2, 3, Yes, No, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation":"Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (primal_infeasible_equal_objective, [3], [3/2], [3], [3/2], No, Yes, 3, 9/2, 3, Yes, No, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)."},"skills":["Equal Objective Values and Finite Grids Do Not Replace Theorem Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0077","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e19b773745e9f9f38fe280d0b365b2afb19735199a81c1f5ccebb3fa62066b8","family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154077,"semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[1,1]],\"c\":[[3,1]],\"mode\":\"dual_infeasible_equal_objective\",\"x\":[[1,3]],\"y\":[[1,1]]}"},"fields":{"title":"Equal Objective Values and Finite Grids Do Not Replace Theorem Premises","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[1], c=[3]. Candidate x=[1/3], y=[1]. Boundary mode=dual_infeasible_equal_objective. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).","answer":{"kind":"expression","value":"(dual_infeasible_equal_objective, [1/3], [1], [1/3], [1], Yes, No, 1, 1/3, 1, No, Yes, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation":"Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (dual_infeasible_equal_objective, [1/3], [1], [1/3], [1], Yes, No, 1, 1/3, 1, No, Yes, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)."},"skills":["Equal Objective Values and Finite Grids Do Not Replace Theorem Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0078","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"191042e421a9796bda99b795ba06f567e802257353e59f89f1e6431174c57250","family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154078,"semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[4,1]],\"c\":[[1,1]],\"mode\":\"finite_grid_not_general\",\"x\":[[1,1]],\"y\":[[2,1]]}"},"fields":{"title":"Equal Objective Values and Finite Grids Do Not Replace Theorem Premises","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[4], c=[1]. Candidate x=[1], y=[2]. Boundary mode=finite_grid_not_general. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).","answer":{"kind":"expression","value":"(finite_grid_not_general, [1], [2], [1], [2], Yes, Yes, 1, 2, 8, Yes, Yes, No, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation":"Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (finite_grid_not_general, [1], [2], [1], [2], Yes, Yes, 1, 2, 8, Yes, Yes, No, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)."},"skills":["Equal Objective Values and Finite Grids Do Not Replace Theorem Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0079","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"956783ead5380fa5244a9f9078c8ea87b6aa7f47869020f8adc3eb4f80f0d62a","family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154079,"semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[5,1]],\"c\":[[1,1]],\"mode\":\"primal_infeasible_equal_objective\",\"x\":[[6,1]],\"y\":[[6,5]]}"},"fields":{"title":"Equal Objective Values and Finite Grids Do Not Replace Theorem Premises","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[5], c=[1]. Candidate x=[6], y=[6/5]. Boundary mode=primal_infeasible_equal_objective. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).","answer":{"kind":"expression","value":"(primal_infeasible_equal_objective, [6], [6/5], [6], [6/5], No, Yes, 6, 36/5, 6, Yes, No, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation":"Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (primal_infeasible_equal_objective, [6], [6/5], [6], [6/5], No, Yes, 6, 36/5, 6, Yes, No, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)."},"skills":["Equal Objective Values and Finite Grids Do Not Replace Theorem Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0080","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5713f582388c6db14a76574bd185df8286885e23a86e628bb78067917a24a013","family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154080,"semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[1,1]],\"c\":[[6,1]],\"mode\":\"dual_infeasible_equal_objective\",\"x\":[[1,6]],\"y\":[[1,1]]}"},"fields":{"title":"Equal Objective Values and Finite Grids Do Not Replace Theorem Premises","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[1], c=[6]. Candidate x=[1/6], y=[1]. Boundary mode=dual_infeasible_equal_objective. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).","answer":{"kind":"expression","value":"(dual_infeasible_equal_objective, [1/6], [1], [1/6], [1], Yes, No, 1, 1/6, 1, No, Yes, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation":"Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (dual_infeasible_equal_objective, [1/6], [1], [1/6], [1], Yes, No, 1, 1/6, 1, No, Yes, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)."},"skills":["Equal Objective Values and Finite Grids Do Not Replace Theorem Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0081","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae046abab175b9381afa1f063d1973daa50946b9c394e824b198c301363ca164","family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154081,"semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[7,1]],\"c\":[[1,1]],\"mode\":\"finite_grid_not_general\",\"x\":[[1,1]],\"y\":[[2,1]]}"},"fields":{"title":"Equal Objective Values and Finite Grids Do Not Replace Theorem Premises","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[7], c=[1]. Candidate x=[1], y=[2]. Boundary mode=finite_grid_not_general. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).","answer":{"kind":"expression","value":"(finite_grid_not_general, [1], [2], [1], [2], Yes, Yes, 1, 2, 14, Yes, Yes, No, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation":"Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (finite_grid_not_general, [1], [2], [1], [2], Yes, Yes, 1, 2, 14, Yes, Yes, No, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)."},"skills":["Equal Objective Values and Finite Grids Do Not Replace Theorem Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0082","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f849b933301d9d749f3038e1bd7011c8b1396e9d55594537298f0925dc6d355","family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154082,"semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[8,1]],\"c\":[[1,1]],\"mode\":\"primal_infeasible_equal_objective\",\"x\":[[9,1]],\"y\":[[9,8]]}"},"fields":{"title":"Equal Objective Values and Finite Grids Do Not Replace Theorem Premises","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[8], c=[1]. Candidate x=[9], y=[9/8]. Boundary mode=primal_infeasible_equal_objective. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).","answer":{"kind":"expression","value":"(primal_infeasible_equal_objective, [9], [9/8], [9], [9/8], No, Yes, 9, 81/8, 9, Yes, No, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation":"Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (primal_infeasible_equal_objective, [9], [9/8], [9], [9/8], No, Yes, 9, 81/8, 9, Yes, No, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)."},"skills":["Equal Objective Values and Finite Grids Do Not Replace Theorem Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0083","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4d1f9ab3c5ed3c7987701a670e365858cf1dd32f22379fa051975fbd1c961bb","family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154083,"semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[1,1]],\"c\":[[9,1]],\"mode\":\"dual_infeasible_equal_objective\",\"x\":[[1,9]],\"y\":[[1,1]]}"},"fields":{"title":"Equal Objective Values and Finite Grids Do Not Replace Theorem Premises","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[1], c=[9]. Candidate x=[1/9], y=[1]. Boundary mode=dual_infeasible_equal_objective. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).","answer":{"kind":"expression","value":"(dual_infeasible_equal_objective, [1/9], [1], [1/9], [1], Yes, No, 1, 1/9, 1, No, Yes, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation":"Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (dual_infeasible_equal_objective, [1/9], [1], [1/9], [1], Yes, No, 1, 1/9, 1, No, Yes, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)."},"skills":["Equal Objective Values and Finite Grids Do Not Replace Theorem Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0084","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f3ffe3948f6f35ba7dc4b66703cf29b57bd7e2a357de4ac59a0d15841624b6e","family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154084,"semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[10,1]],\"c\":[[1,1]],\"mode\":\"finite_grid_not_general\",\"x\":[[1,1]],\"y\":[[2,1]]}"},"fields":{"title":"Equal Objective Values and Finite Grids Do Not Replace Theorem Premises","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[10], c=[1]. Candidate x=[1], y=[2]. Boundary mode=finite_grid_not_general. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).","answer":{"kind":"expression","value":"(finite_grid_not_general, [1], [2], [1], [2], Yes, Yes, 1, 2, 20, Yes, Yes, No, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation":"Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (finite_grid_not_general, [1], [2], [1], [2], Yes, Yes, 1, 2, 20, Yes, Yes, No, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)."},"skills":["Equal Objective Values and Finite Grids Do Not Replace Theorem Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0085","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b2c0ae7191a6bcef720198661695005ee75021e49625249d9f7fa109284b5d6","family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154085,"semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[11,1]],\"c\":[[1,1]],\"mode\":\"primal_infeasible_equal_objective\",\"x\":[[12,1]],\"y\":[[12,11]]}"},"fields":{"title":"Equal Objective Values and Finite Grids Do Not Replace Theorem Premises","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[11], c=[1]. Candidate x=[12], y=[12/11]. Boundary mode=primal_infeasible_equal_objective. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).","answer":{"kind":"expression","value":"(primal_infeasible_equal_objective, [12], [12/11], [12], [12/11], No, Yes, 12, 144/11, 12, Yes, No, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation":"Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (primal_infeasible_equal_objective, [12], [12/11], [12], [12/11], No, Yes, 12, 144/11, 12, Yes, No, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)."},"skills":["Equal Objective Values and Finite Grids Do Not Replace Theorem Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0086","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e999ec158111d81ab19f06ef0463348baaaef19acb746385708f9730f33a22d","family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154086,"semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[1,1]],\"c\":[[12,1]],\"mode\":\"dual_infeasible_equal_objective\",\"x\":[[1,12]],\"y\":[[1,1]]}"},"fields":{"title":"Equal Objective Values and Finite Grids Do Not Replace Theorem Premises","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[1], c=[12]. Candidate x=[1/12], y=[1]. Boundary mode=dual_infeasible_equal_objective. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).","answer":{"kind":"expression","value":"(dual_infeasible_equal_objective, [1/12], [1], [1/12], [1], Yes, No, 1, 1/12, 1, No, Yes, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation":"Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (dual_infeasible_equal_objective, [1/12], [1], [1/12], [1], Yes, No, 1, 1/12, 1, No, Yes, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)."},"skills":["Equal Objective Values and Finite Grids Do Not Replace Theorem Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0087","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3e4524f3267231becc670bb05b8ae67da6ec7e832e61417b5c7bc18b8db9547","family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154087,"semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[13,1]],\"c\":[[1,1]],\"mode\":\"finite_grid_not_general\",\"x\":[[1,1]],\"y\":[[2,1]]}"},"fields":{"title":"Equal Objective Values and Finite Grids Do Not Replace Theorem Premises","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[13], c=[1]. Candidate x=[1], y=[2]. Boundary mode=finite_grid_not_general. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).","answer":{"kind":"expression","value":"(finite_grid_not_general, [1], [2], [1], [2], Yes, Yes, 1, 2, 26, Yes, Yes, No, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation":"Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (finite_grid_not_general, [1], [2], [1], [2], Yes, Yes, 1, 2, 26, Yes, Yes, No, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)."},"skills":["Equal Objective Values and Finite Grids Do Not Replace Theorem Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0088","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d59b76e3c8e9cd9b6d620c46355b58394cd146a41a8410bbfd3e10b73e55708","family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154088,"semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[14,1]],\"c\":[[1,1]],\"mode\":\"primal_infeasible_equal_objective\",\"x\":[[15,1]],\"y\":[[15,14]]}"},"fields":{"title":"Equal Objective Values and Finite Grids Do Not Replace Theorem Premises","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[14], c=[1]. Candidate x=[15], y=[15/14]. Boundary mode=primal_infeasible_equal_objective. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).","answer":{"kind":"expression","value":"(primal_infeasible_equal_objective, [15], [15/14], [15], [15/14], No, Yes, 15, 225/14, 15, Yes, No, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation":"Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (primal_infeasible_equal_objective, [15], [15/14], [15], [15/14], No, Yes, 15, 225/14, 15, Yes, No, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)."},"skills":["Equal Objective Values and Finite Grids Do Not Replace Theorem Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0089","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb02ffa69858f0bf29829eaeea67cd744254bea3136d4de3e64dc3835a8563e4","family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154089,"semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[1,1]],\"c\":[[15,1]],\"mode\":\"dual_infeasible_equal_objective\",\"x\":[[1,15]],\"y\":[[1,1]]}"},"fields":{"title":"Equal Objective Values and Finite Grids Do Not Replace Theorem Premises","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[1], c=[15]. Candidate x=[1/15], y=[1]. Boundary mode=dual_infeasible_equal_objective. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).","answer":{"kind":"expression","value":"(dual_infeasible_equal_objective, [1/15], [1], [1/15], [1], Yes, No, 1, 1/15, 1, No, Yes, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation":"Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (dual_infeasible_equal_objective, [1/15], [1], [1/15], [1], Yes, No, 1, 1/15, 1, No, Yes, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)."},"skills":["Equal Objective Values and Finite Grids Do Not Replace Theorem Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0090","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9335c45c436d4e8b8f17b815b8631004384744ae9b928612d3451f69066b07e5","family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154090,"semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[16,1]],\"c\":[[1,1]],\"mode\":\"finite_grid_not_general\",\"x\":[[1,1]],\"y\":[[2,1]]}"},"fields":{"title":"Equal Objective Values and Finite Grids Do Not Replace Theorem Premises","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[16], c=[1]. Candidate x=[1], y=[2]. Boundary mode=finite_grid_not_general. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).","answer":{"kind":"expression","value":"(finite_grid_not_general, [1], [2], [1], [2], Yes, Yes, 1, 2, 32, Yes, Yes, No, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation":"Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (finite_grid_not_general, [1], [2], [1], [2], Yes, Yes, 1, 2, 32, Yes, Yes, No, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)."},"skills":["Equal Objective Values and Finite Grids Do Not Replace Theorem Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0091","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"404aff779104ae5c88d322192ef5d053f4349cbf6cea58522dbc658effbb567c","family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154091,"semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[17,1]],\"c\":[[1,1]],\"mode\":\"primal_infeasible_equal_objective\",\"x\":[[18,1]],\"y\":[[18,17]]}"},"fields":{"title":"Equal Objective Values and Finite Grids Do Not Replace Theorem Premises","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[17], c=[1]. Candidate x=[18], y=[18/17]. Boundary mode=primal_infeasible_equal_objective. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).","answer":{"kind":"expression","value":"(primal_infeasible_equal_objective, [18], [18/17], [18], [18/17], No, Yes, 18, 324/17, 18, Yes, No, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation":"Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (primal_infeasible_equal_objective, [18], [18/17], [18], [18/17], No, Yes, 18, 324/17, 18, Yes, No, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)."},"skills":["Equal Objective Values and Finite Grids Do Not Replace Theorem Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0092","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a262ab3fb993f80a8badf468eaecb1c128cfb8565bbdf3701f25432f2ba045a","family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154092,"semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[1,1]],\"c\":[[18,1]],\"mode\":\"dual_infeasible_equal_objective\",\"x\":[[1,18]],\"y\":[[1,1]]}"},"fields":{"title":"Equal Objective Values and Finite Grids Do Not Replace Theorem Premises","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[1], c=[18]. Candidate x=[1/18], y=[1]. Boundary mode=dual_infeasible_equal_objective. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).","answer":{"kind":"expression","value":"(dual_infeasible_equal_objective, [1/18], [1], [1/18], [1], Yes, No, 1, 1/18, 1, No, Yes, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation":"Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (dual_infeasible_equal_objective, [1/18], [1], [1/18], [1], Yes, No, 1, 1/18, 1, No, Yes, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)."},"skills":["Equal Objective Values and Finite Grids Do Not Replace Theorem Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0093","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3556e5ad7a0658a01c5787375576e4bc3c1a0e4c7a82cdec7748eb5831529630","family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154093,"semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[19,1]],\"c\":[[1,1]],\"mode\":\"finite_grid_not_general\",\"x\":[[1,1]],\"y\":[[2,1]]}"},"fields":{"title":"Equal Objective Values and Finite Grids Do Not Replace Theorem Premises","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[19], c=[1]. Candidate x=[1], y=[2]. Boundary mode=finite_grid_not_general. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).","answer":{"kind":"expression","value":"(finite_grid_not_general, [1], [2], [1], [2], Yes, Yes, 1, 2, 38, Yes, Yes, No, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation":"Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (finite_grid_not_general, [1], [2], [1], [2], Yes, Yes, 1, 2, 38, Yes, Yes, No, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)."},"skills":["Equal Objective Values and Finite Grids Do Not Replace Theorem Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0094","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14eda03132ab5f80377f37c788e6c1634e19687a4d543b466356838db997949f","family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154094,"semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[20,1]],\"c\":[[1,1]],\"mode\":\"primal_infeasible_equal_objective\",\"x\":[[21,1]],\"y\":[[21,20]]}"},"fields":{"title":"Equal Objective Values and Finite Grids Do Not Replace Theorem Premises","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[20], c=[1]. Candidate x=[21], y=[21/20]. Boundary mode=primal_infeasible_equal_objective. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).","answer":{"kind":"expression","value":"(primal_infeasible_equal_objective, [21], [21/20], [21], [21/20], No, Yes, 21, 441/20, 21, Yes, No, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation":"Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (primal_infeasible_equal_objective, [21], [21/20], [21], [21/20], No, Yes, 21, 441/20, 21, Yes, No, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)."},"skills":["Equal Objective Values and Finite Grids Do Not Replace Theorem Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0095","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8d9aed53a57d743cc28cea4b0adb0d0650442493a8d2a0d05a8bc89f1d59165","family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154095,"semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[1,1]],\"c\":[[21,1]],\"mode\":\"dual_infeasible_equal_objective\",\"x\":[[1,21]],\"y\":[[1,1]]}"},"fields":{"title":"Equal Objective Values and Finite Grids Do Not Replace Theorem Premises","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[1], c=[21]. Candidate x=[1/21], y=[1]. Boundary mode=dual_infeasible_equal_objective. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).","answer":{"kind":"expression","value":"(dual_infeasible_equal_objective, [1/21], [1], [1/21], [1], Yes, No, 1, 1/21, 1, No, Yes, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation":"Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (dual_infeasible_equal_objective, [1/21], [1], [1/21], [1], Yes, No, 1, 1/21, 1, No, Yes, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)."},"skills":["Equal Objective Values and Finite Grids Do Not Replace Theorem Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0096","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"841a1d3d026622b3e676c7c43e31c527b746a8961bf6600b4a525714ec3b1405","family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154096,"semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[22,1]],\"c\":[[1,1]],\"mode\":\"finite_grid_not_general\",\"x\":[[1,1]],\"y\":[[2,1]]}"},"fields":{"title":"Equal Objective Values and Finite Grids Do Not Replace Theorem Premises","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[22], c=[1]. Candidate x=[1], y=[2]. Boundary mode=finite_grid_not_general. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).","answer":{"kind":"expression","value":"(finite_grid_not_general, [1], [2], [1], [2], Yes, Yes, 1, 2, 44, Yes, Yes, No, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation":"Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (finite_grid_not_general, [1], [2], [1], [2], Yes, Yes, 1, 2, 44, Yes, Yes, No, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)."},"skills":["Equal Objective Values and Finite Grids Do Not Replace Theorem Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0097","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d56c314a758a7e5e740ebb8946317cfd77e81dc8acff8f649832e13429abfeea","family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154097,"semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[23,1]],\"c\":[[1,1]],\"mode\":\"primal_infeasible_equal_objective\",\"x\":[[24,1]],\"y\":[[24,23]]}"},"fields":{"title":"Equal Objective Values and Finite Grids Do Not Replace Theorem Premises","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[23], c=[1]. Candidate x=[24], y=[24/23]. Boundary mode=primal_infeasible_equal_objective. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).","answer":{"kind":"expression","value":"(primal_infeasible_equal_objective, [24], [24/23], [24], [24/23], No, Yes, 24, 576/23, 24, Yes, No, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation":"Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (primal_infeasible_equal_objective, [24], [24/23], [24], [24/23], No, Yes, 24, 576/23, 24, Yes, No, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)."},"skills":["Equal Objective Values and Finite Grids Do Not Replace Theorem Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0098","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"483755ad7cea447d62b482e173e2d178c492917063eb46699928fdf76084775b","family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154098,"semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[1,1]],\"c\":[[24,1]],\"mode\":\"dual_infeasible_equal_objective\",\"x\":[[1,24]],\"y\":[[1,1]]}"},"fields":{"title":"Equal Objective Values and Finite Grids Do Not Replace Theorem Premises","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[1], c=[24]. Candidate x=[1/24], y=[1]. Boundary mode=dual_infeasible_equal_objective. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).","answer":{"kind":"expression","value":"(dual_infeasible_equal_objective, [1/24], [1], [1/24], [1], Yes, No, 1, 1/24, 1, No, Yes, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation":"Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (dual_infeasible_equal_objective, [1/24], [1], [1/24], [1], Yes, No, 1, 1/24, 1, No, Yes, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)."},"skills":["Equal Objective Values and Finite Grids Do Not Replace Theorem Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0099","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48e0517b72f54bb0ae4186b5d65f62a26f172ca0269f5cac4ae67c96e153eea4","family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154099,"semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[25,1]],\"c\":[[1,1]],\"mode\":\"finite_grid_not_general\",\"x\":[[1,1]],\"y\":[[2,1]]}"},"fields":{"title":"Equal Objective Values and Finite Grids Do Not Replace Theorem Premises","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[25], c=[1]. Candidate x=[1], y=[2]. Boundary mode=finite_grid_not_general. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).","answer":{"kind":"expression","value":"(finite_grid_not_general, [1], [2], [1], [2], Yes, Yes, 1, 2, 50, Yes, Yes, No, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation":"Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (finite_grid_not_general, [1], [2], [1], [2], Yes, Yes, 1, 2, 50, Yes, Yes, No, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)."},"skills":["Equal Objective Values and Finite Grids Do Not Replace Theorem Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261115-r04-linear-program-duality-0100","batch_id":"batch-20261015-advance-20261115-r04-linear-program-duality-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f0a4b23824cf64b3b2eb6624a1934556ddde3f8dd5ebf51a0fb09a679314c76","family_id":"qf-mod46-lpdual-strong-boundary","seed":202611154100,"semantic_template":"qf-mod46-lpdual-strong-boundary/{\"A\":[[[1,1]]],\"b\":[[26,1]],\"c\":[[1,1]],\"mode\":\"primal_infeasible_equal_objective\",\"x\":[[27,1]],\"y\":[[27,26]]}"},"fields":{"title":"Equal Objective Values and Finite Grids Do Not Replace Theorem Premises","stem":"Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[26], c=[1]. Candidate x=[27], y=[27/26]. Boundary mode=primal_infeasible_equal_objective. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary).","answer":{"kind":"expression","value":"(primal_infeasible_equal_objective, [27], [27/26], [27], [27/26], No, Yes, 27, 729/26, 27, Yes, No, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)","explanation":"Retain each feasibility flag, both inequalities and objective equality separately. Infeasible-equal-objective modes reject an optimality inference; finite_grid_not_general rejects treating finite checking as a general strong-duality proof, even when a particular feasible zero-gap pair happens to be optimal. Infeasible/unbounded classifications require additional theorem-level work. The exact result, retaining the requested order, is (primal_infeasible_equal_objective, [27], [27/26], [27], [27/26], No, Yes, 27, 729/26, 27, Yes, No, Yes, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases)."},"skills":["Equal Objective Values and Finite Grids Do Not Replace Theorem Premises"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}