{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r06-linear-program-standard-form-0001","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5989d3833beb5d9a869256354f5e0c095823d0b26c90f973710f560812a0062","family_id":"qf-mod45-lp-standard-dimension-sign-qualification","seed":202611146001,"semantic_template":"qf-mod45-lp-standard-dimension-sign-qualification/{\"A\":[[[1,1],[1,1]],[[-1,1],[2,1]]],\"b\":[[4,1],[2,1]],\"c\":[[3,1],[-1,1]]}"},"fields":{"title":"Qualify Dimensions and the Fixed Convention without Claiming Feasibility","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[1, 1], [-1, 2]], b=[4, 2], c=[3, -1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[1, 1], [-1, 2]], [4, 2], [3, -1], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)","explanation":"Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[1, 1], [-1, 2]], [4, 2], [3, -1], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)."},"skills":["Qualify Dimensions and the Fixed Convention without Claiming Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0002","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0589bf41be32db6dfea2d2ed41795a116be4c883d087ae30cff7fa9a8715580e","family_id":"qf-mod45-lp-standard-dimension-sign-qualification","seed":202611146002,"semantic_template":"qf-mod45-lp-standard-dimension-sign-qualification/{\"A\":[[[2,1],[1,1]],[[-1,1],[3,1]]],\"b\":[[6,1],[3,1]],\"c\":[[4,1],[-2,1]]}"},"fields":{"title":"Qualify Dimensions and the Fixed Convention without Claiming Feasibility","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[2, 1], [-1, 3]], b=[6, 3], c=[4, -2]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[2, 1], [-1, 3]], [6, 3], [4, -2], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)","explanation":"Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[2, 1], [-1, 3]], [6, 3], [4, -2], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)."},"skills":["Qualify Dimensions and the Fixed Convention without Claiming Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0003","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c7e5a0f62b74ef6750bf7d43c8bd5b9bffea6a198d7d93e2668a9a922fde2ef","family_id":"qf-mod45-lp-standard-dimension-sign-qualification","seed":202611146003,"semantic_template":"qf-mod45-lp-standard-dimension-sign-qualification/{\"A\":[[[3,1],[1,1]],[[-1,1],[4,1]]],\"b\":[[8,1],[4,1]],\"c\":[[5,1],[-3,1]]}"},"fields":{"title":"Qualify Dimensions and the Fixed Convention without Claiming Feasibility","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[3, 1], [-1, 4]], b=[8, 4], c=[5, -3]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[3, 1], [-1, 4]], [8, 4], [5, -3], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)","explanation":"Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[3, 1], [-1, 4]], [8, 4], [5, -3], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)."},"skills":["Qualify Dimensions and the Fixed Convention without Claiming Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0004","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4cfa099d9e769b6413473cc8b1f0ee1cf18f6e2a28f5681091988e97d755070","family_id":"qf-mod45-lp-standard-dimension-sign-qualification","seed":202611146004,"semantic_template":"qf-mod45-lp-standard-dimension-sign-qualification/{\"A\":[[[4,1],[1,1]],[[-1,1],[5,1]]],\"b\":[[10,1],[5,1]],\"c\":[[6,1],[-4,1]]}"},"fields":{"title":"Qualify Dimensions and the Fixed Convention without Claiming Feasibility","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[4, 1], [-1, 5]], b=[10, 5], c=[6, -4]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[4, 1], [-1, 5]], [10, 5], [6, -4], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)","explanation":"Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[4, 1], [-1, 5]], [10, 5], [6, -4], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)."},"skills":["Qualify Dimensions and the Fixed Convention without Claiming Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0005","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78a84adb577c9f216fc8db4052597ec795804662d8cc2729f988d96f94fbf7f9","family_id":"qf-mod45-lp-standard-dimension-sign-qualification","seed":202611146005,"semantic_template":"qf-mod45-lp-standard-dimension-sign-qualification/{\"A\":[[[5,1],[1,1]],[[-1,1],[6,1]]],\"b\":[[12,1],[6,1]],\"c\":[[7,1],[-5,1]]}"},"fields":{"title":"Qualify Dimensions and the Fixed Convention without Claiming Feasibility","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[5, 1], [-1, 6]], b=[12, 6], c=[7, -5]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[5, 1], [-1, 6]], [12, 6], [7, -5], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)","explanation":"Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[5, 1], [-1, 6]], [12, 6], [7, -5], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)."},"skills":["Qualify Dimensions and the Fixed Convention without Claiming Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0006","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca8c6b7aef9ed5b5a59b8c20eb0b6b7279980d37fae21d41117f94ac5e1ee7bd","family_id":"qf-mod45-lp-standard-dimension-sign-qualification","seed":202611146006,"semantic_template":"qf-mod45-lp-standard-dimension-sign-qualification/{\"A\":[[[6,1],[1,1]],[[-1,1],[7,1]]],\"b\":[[14,1],[7,1]],\"c\":[[8,1],[-6,1]]}"},"fields":{"title":"Qualify Dimensions and the Fixed Convention without Claiming Feasibility","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[6, 1], [-1, 7]], b=[14, 7], c=[8, -6]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[6, 1], [-1, 7]], [14, 7], [8, -6], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)","explanation":"Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[6, 1], [-1, 7]], [14, 7], [8, -6], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)."},"skills":["Qualify Dimensions and the Fixed Convention without Claiming Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0007","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69d6a27b6db1f9792528b173d84c6a45d06b6ac73a553f6167ee3993822ed3ad","family_id":"qf-mod45-lp-standard-dimension-sign-qualification","seed":202611146007,"semantic_template":"qf-mod45-lp-standard-dimension-sign-qualification/{\"A\":[[[7,1],[1,1]],[[-1,1],[8,1]]],\"b\":[[16,1],[8,1]],\"c\":[[9,1],[-7,1]]}"},"fields":{"title":"Qualify Dimensions and the Fixed Convention without Claiming Feasibility","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[7, 1], [-1, 8]], b=[16, 8], c=[9, -7]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[7, 1], [-1, 8]], [16, 8], [9, -7], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)","explanation":"Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[7, 1], [-1, 8]], [16, 8], [9, -7], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)."},"skills":["Qualify Dimensions and the Fixed Convention without Claiming Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0008","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7edeaad4a39b7766a85ecc69db9734c1518caa60eb3dd610f765ba208e8cd0e","family_id":"qf-mod45-lp-standard-dimension-sign-qualification","seed":202611146008,"semantic_template":"qf-mod45-lp-standard-dimension-sign-qualification/{\"A\":[[[8,1],[1,1]],[[-1,1],[9,1]]],\"b\":[[18,1],[9,1]],\"c\":[[10,1],[-8,1]]}"},"fields":{"title":"Qualify Dimensions and the Fixed Convention without Claiming Feasibility","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[8, 1], [-1, 9]], b=[18, 9], c=[10, -8]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[8, 1], [-1, 9]], [18, 9], [10, -8], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)","explanation":"Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[8, 1], [-1, 9]], [18, 9], [10, -8], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)."},"skills":["Qualify Dimensions and the Fixed Convention without Claiming Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0009","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74334cf31d2de4690815c66ac9ed4a30e4864775c877cec64f710c8ffe3e8472","family_id":"qf-mod45-lp-standard-dimension-sign-qualification","seed":202611146009,"semantic_template":"qf-mod45-lp-standard-dimension-sign-qualification/{\"A\":[[[9,1],[1,1]],[[-1,1],[10,1]]],\"b\":[[20,1],[10,1]],\"c\":[[11,1],[-9,1]]}"},"fields":{"title":"Qualify Dimensions and the Fixed Convention without Claiming Feasibility","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[9, 1], [-1, 10]], b=[20, 10], c=[11, -9]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[9, 1], [-1, 10]], [20, 10], [11, -9], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)","explanation":"Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[9, 1], [-1, 10]], [20, 10], [11, -9], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)."},"skills":["Qualify Dimensions and the Fixed Convention without Claiming Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0010","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5419d9c0ef471cb32d0e15c08ae2988b87076020be4da41459ac7c23fa931fdb","family_id":"qf-mod45-lp-standard-dimension-sign-qualification","seed":202611146010,"semantic_template":"qf-mod45-lp-standard-dimension-sign-qualification/{\"A\":[[[10,1],[1,1]],[[-1,1],[11,1]]],\"b\":[[22,1],[11,1]],\"c\":[[12,1],[-10,1]]}"},"fields":{"title":"Qualify Dimensions and the Fixed Convention without Claiming Feasibility","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[10, 1], [-1, 11]], b=[22, 11], c=[12, -10]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[10, 1], [-1, 11]], [22, 11], [12, -10], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)","explanation":"Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[10, 1], [-1, 11]], [22, 11], [12, -10], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)."},"skills":["Qualify Dimensions and the Fixed Convention without Claiming Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0011","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aed1da2e9c0167999064d9d12caaab5134bc79c862fdbfdb07f2c868c159117e","family_id":"qf-mod45-lp-standard-dimension-sign-qualification","seed":202611146011,"semantic_template":"qf-mod45-lp-standard-dimension-sign-qualification/{\"A\":[[[11,1],[1,1]],[[-1,1],[12,1]]],\"b\":[[24,1],[12,1]],\"c\":[[13,1],[-11,1]]}"},"fields":{"title":"Qualify Dimensions and the Fixed Convention without Claiming Feasibility","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[11, 1], [-1, 12]], b=[24, 12], c=[13, -11]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[11, 1], [-1, 12]], [24, 12], [13, -11], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)","explanation":"Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[11, 1], [-1, 12]], [24, 12], [13, -11], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)."},"skills":["Qualify Dimensions and the Fixed Convention without Claiming Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0012","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1499966bf61dc5d1dd9d231d4049efba51b0fcff53400c6e563e6b58b67c3aa9","family_id":"qf-mod45-lp-standard-dimension-sign-qualification","seed":202611146012,"semantic_template":"qf-mod45-lp-standard-dimension-sign-qualification/{\"A\":[[[12,1],[1,1]],[[-1,1],[13,1]]],\"b\":[[26,1],[13,1]],\"c\":[[14,1],[-12,1]]}"},"fields":{"title":"Qualify Dimensions and the Fixed Convention without Claiming Feasibility","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[12, 1], [-1, 13]], b=[26, 13], c=[14, -12]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[12, 1], [-1, 13]], [26, 13], [14, -12], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)","explanation":"Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[12, 1], [-1, 13]], [26, 13], [14, -12], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)."},"skills":["Qualify Dimensions and the Fixed Convention without Claiming Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0013","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"906d9157b4a77392b15cadb279abf6d8fcf5ef5920802df02c6cad9f00969742","family_id":"qf-mod45-lp-standard-dimension-sign-qualification","seed":202611146013,"semantic_template":"qf-mod45-lp-standard-dimension-sign-qualification/{\"A\":[[[13,1],[1,1]],[[-1,1],[14,1]]],\"b\":[[28,1],[14,1]],\"c\":[[15,1],[-13,1]]}"},"fields":{"title":"Qualify Dimensions and the Fixed Convention without Claiming Feasibility","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[13, 1], [-1, 14]], b=[28, 14], c=[15, -13]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[13, 1], [-1, 14]], [28, 14], [15, -13], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)","explanation":"Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[13, 1], [-1, 14]], [28, 14], [15, -13], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)."},"skills":["Qualify Dimensions and the Fixed Convention without Claiming Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0014","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f7ffba0d491a8825260789929609dda81fe44e5ca380d73f10f54ae71fb4f77","family_id":"qf-mod45-lp-standard-dimension-sign-qualification","seed":202611146014,"semantic_template":"qf-mod45-lp-standard-dimension-sign-qualification/{\"A\":[[[14,1],[1,1]],[[-1,1],[15,1]]],\"b\":[[30,1],[15,1]],\"c\":[[16,1],[-14,1]]}"},"fields":{"title":"Qualify Dimensions and the Fixed Convention without Claiming Feasibility","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[14, 1], [-1, 15]], b=[30, 15], c=[16, -14]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[14, 1], [-1, 15]], [30, 15], [16, -14], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)","explanation":"Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[14, 1], [-1, 15]], [30, 15], [16, -14], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)."},"skills":["Qualify Dimensions and the Fixed Convention without Claiming Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0015","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9164dde3d9c7fdb0feeefa56184dd588e424069439a02ff139faa36454e15e85","family_id":"qf-mod45-lp-standard-dimension-sign-qualification","seed":202611146015,"semantic_template":"qf-mod45-lp-standard-dimension-sign-qualification/{\"A\":[[[15,1],[1,1]],[[-1,1],[16,1]]],\"b\":[[32,1],[16,1]],\"c\":[[17,1],[-15,1]]}"},"fields":{"title":"Qualify Dimensions and the Fixed Convention without Claiming Feasibility","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[15, 1], [-1, 16]], b=[32, 16], c=[17, -15]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[15, 1], [-1, 16]], [32, 16], [17, -15], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)","explanation":"Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[15, 1], [-1, 16]], [32, 16], [17, -15], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)."},"skills":["Qualify Dimensions and the Fixed Convention without Claiming Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0016","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"541abda97bda0aacb7061edee1184d43bd0ac9fd16dd557ffe29bb54dadae05a","family_id":"qf-mod45-lp-standard-dimension-sign-qualification","seed":202611146016,"semantic_template":"qf-mod45-lp-standard-dimension-sign-qualification/{\"A\":[[[16,1],[1,1]],[[-1,1],[17,1]]],\"b\":[[34,1],[17,1]],\"c\":[[18,1],[-16,1]]}"},"fields":{"title":"Qualify Dimensions and the Fixed Convention without Claiming Feasibility","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[16, 1], [-1, 17]], b=[34, 17], c=[18, -16]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[16, 1], [-1, 17]], [34, 17], [18, -16], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)","explanation":"Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[16, 1], [-1, 17]], [34, 17], [18, -16], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)."},"skills":["Qualify Dimensions and the Fixed Convention without Claiming Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0017","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b828386d5b90dda60f79c05945a917f3faaaf5d7a3f98c72f5f8a7e575ec0d78","family_id":"qf-mod45-lp-standard-dimension-sign-qualification","seed":202611146017,"semantic_template":"qf-mod45-lp-standard-dimension-sign-qualification/{\"A\":[[[17,1],[1,1]],[[-1,1],[18,1]]],\"b\":[[36,1],[18,1]],\"c\":[[19,1],[-17,1]]}"},"fields":{"title":"Qualify Dimensions and the Fixed Convention without Claiming Feasibility","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[17, 1], [-1, 18]], b=[36, 18], c=[19, -17]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[17, 1], [-1, 18]], [36, 18], [19, -17], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)","explanation":"Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[17, 1], [-1, 18]], [36, 18], [19, -17], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)."},"skills":["Qualify Dimensions and the Fixed Convention without Claiming Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0018","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6afcae4c7591ad35dce4b9d5136160d8fe3426433a1ca9f7972c1951c595be09","family_id":"qf-mod45-lp-standard-dimension-sign-qualification","seed":202611146018,"semantic_template":"qf-mod45-lp-standard-dimension-sign-qualification/{\"A\":[[[18,1],[1,1]],[[-1,1],[19,1]]],\"b\":[[38,1],[19,1]],\"c\":[[20,1],[-18,1]]}"},"fields":{"title":"Qualify Dimensions and the Fixed Convention without Claiming Feasibility","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[18, 1], [-1, 19]], b=[38, 19], c=[20, -18]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[18, 1], [-1, 19]], [38, 19], [20, -18], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)","explanation":"Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[18, 1], [-1, 19]], [38, 19], [20, -18], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)."},"skills":["Qualify Dimensions and the Fixed Convention without Claiming Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0019","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb8fe9e993488a35e53cf61852b77dad9b3d87d4f1ea844922ba04322e8ac306","family_id":"qf-mod45-lp-standard-dimension-sign-qualification","seed":202611146019,"semantic_template":"qf-mod45-lp-standard-dimension-sign-qualification/{\"A\":[[[19,1],[1,1]],[[-1,1],[20,1]]],\"b\":[[40,1],[20,1]],\"c\":[[21,1],[-19,1]]}"},"fields":{"title":"Qualify Dimensions and the Fixed Convention without Claiming Feasibility","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[19, 1], [-1, 20]], b=[40, 20], c=[21, -19]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[19, 1], [-1, 20]], [40, 20], [21, -19], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)","explanation":"Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[19, 1], [-1, 20]], [40, 20], [21, -19], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)."},"skills":["Qualify Dimensions and the Fixed Convention without Claiming Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0020","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00fc617d0008b4574c4655a27196e7dadb7619da7cd9211d27d63f25d8f1de00","family_id":"qf-mod45-lp-standard-dimension-sign-qualification","seed":202611146020,"semantic_template":"qf-mod45-lp-standard-dimension-sign-qualification/{\"A\":[[[20,1],[1,1]],[[-1,1],[21,1]]],\"b\":[[42,1],[21,1]],\"c\":[[22,1],[-20,1]]}"},"fields":{"title":"Qualify Dimensions and the Fixed Convention without Claiming Feasibility","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[20, 1], [-1, 21]], b=[42, 21], c=[22, -20]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[20, 1], [-1, 21]], [42, 21], [22, -20], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)","explanation":"Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[20, 1], [-1, 21]], [42, 21], [22, -20], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)."},"skills":["Qualify Dimensions and the Fixed Convention without Claiming Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0021","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0dc0dca2a0972cc0486f46647850d2176f1e6a5b97439b64ab70330131857f8","family_id":"qf-mod45-lp-standard-dimension-sign-qualification","seed":202611146021,"semantic_template":"qf-mod45-lp-standard-dimension-sign-qualification/{\"A\":[[[21,1],[1,1]],[[-1,1],[22,1]]],\"b\":[[44,1],[22,1]],\"c\":[[23,1],[-21,1]]}"},"fields":{"title":"Qualify Dimensions and the Fixed Convention without Claiming Feasibility","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[21, 1], [-1, 22]], b=[44, 22], c=[23, -21]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[21, 1], [-1, 22]], [44, 22], [23, -21], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)","explanation":"Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[21, 1], [-1, 22]], [44, 22], [23, -21], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)."},"skills":["Qualify Dimensions and the Fixed Convention without Claiming Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0022","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf58e136235b40c7445e67decef72afd6f0e74065ed09962f83f9cd54297cb8d","family_id":"qf-mod45-lp-standard-dimension-sign-qualification","seed":202611146022,"semantic_template":"qf-mod45-lp-standard-dimension-sign-qualification/{\"A\":[[[22,1],[1,1]],[[-1,1],[23,1]]],\"b\":[[46,1],[23,1]],\"c\":[[24,1],[-22,1]]}"},"fields":{"title":"Qualify Dimensions and the Fixed Convention without Claiming Feasibility","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[22, 1], [-1, 23]], b=[46, 23], c=[24, -22]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[22, 1], [-1, 23]], [46, 23], [24, -22], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)","explanation":"Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[22, 1], [-1, 23]], [46, 23], [24, -22], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)."},"skills":["Qualify Dimensions and the Fixed Convention without Claiming Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0023","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91d32caaf41eaa51ab6b437321bd029d1d74a0184a347145006bf6465a835633","family_id":"qf-mod45-lp-standard-dimension-sign-qualification","seed":202611146023,"semantic_template":"qf-mod45-lp-standard-dimension-sign-qualification/{\"A\":[[[23,1],[1,1]],[[-1,1],[24,1]]],\"b\":[[48,1],[24,1]],\"c\":[[25,1],[-23,1]]}"},"fields":{"title":"Qualify Dimensions and the Fixed Convention without Claiming Feasibility","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[23, 1], [-1, 24]], b=[48, 24], c=[25, -23]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[23, 1], [-1, 24]], [48, 24], [25, -23], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)","explanation":"Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[23, 1], [-1, 24]], [48, 24], [25, -23], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)."},"skills":["Qualify Dimensions and the Fixed Convention without Claiming Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0024","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7009d8a84337eb0326964b43082ede7299f78573e1512a7368f2ebb580c260db","family_id":"qf-mod45-lp-standard-dimension-sign-qualification","seed":202611146024,"semantic_template":"qf-mod45-lp-standard-dimension-sign-qualification/{\"A\":[[[24,1],[1,1]],[[-1,1],[25,1]]],\"b\":[[50,1],[25,1]],\"c\":[[26,1],[-24,1]]}"},"fields":{"title":"Qualify Dimensions and the Fixed Convention without Claiming Feasibility","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[24, 1], [-1, 25]], b=[50, 25], c=[26, -24]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[24, 1], [-1, 25]], [50, 25], [26, -24], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)","explanation":"Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[24, 1], [-1, 25]], [50, 25], [26, -24], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)."},"skills":["Qualify Dimensions and the Fixed Convention without Claiming Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0025","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f65be1042c9c7f8ddf3d225db108fe2cbf99daccce1dc8dd37123c239a8e5288","family_id":"qf-mod45-lp-standard-dimension-sign-qualification","seed":202611146025,"semantic_template":"qf-mod45-lp-standard-dimension-sign-qualification/{\"A\":[[[25,1],[1,1]],[[-1,1],[26,1]]],\"b\":[[52,1],[26,1]],\"c\":[[27,1],[-25,1]]}"},"fields":{"title":"Qualify Dimensions and the Fixed Convention without Claiming Feasibility","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[25, 1], [-1, 26]], b=[52, 26], c=[27, -25]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Give (constraint_count, variable_count, A, b, c, convention, dimensions_qualified, certificate).","answer":{"kind":"expression","value":"(2, 2, [[25, 1], [-1, 26]], [52, 26], [27, -25], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)","explanation":"Report m,n and all exact data. The convention is one chosen meaning of standard form, not an assertion that all textbooks use it. Rectangular dimensions do not guarantee a feasible, bounded or optimizable problem; b need not be nonnegative. The exact result, retaining the requested order, is (2, 2, [[25, 1], [-1, 26]], [52, 26], [27, -25], maximize_cTx_subject_to_Ax_less_equal_b_and_x_nonnegative, Yes, one_fixed_standard_form_convention)."},"skills":["Qualify Dimensions and the Fixed Convention without Claiming Feasibility"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0026","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28f860720a860f0cc0305fac08c27915b4c241f248aeefdcd9b6d23fc47c12c8","family_id":"qf-mod45-lp-standard-slack-conversion","seed":202611146026,"semantic_template":"qf-mod45-lp-standard-slack-conversion/{\"A\":[[[1,1],[1,1]],[[1,1],[1,1]]],\"b\":[[4,1],[4,1]],\"c\":[[2,1],[1,1]],\"x\":[[1,1],[1,1]]}"},"fields":{"title":"The Slack Equality Representation Preserves Feasibility in Both Directions","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[1, 1], [1, 1]], b=[4, 4], c=[2, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).","answer":{"kind":"expression","value":"([1, 1], [2, 2], [4, 4], [2, 2], Yes, Yes, Yes, Yes, Yes, 3, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)","explanation":"Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [2, 2], [4, 4], [2, 2], Yes, Yes, Yes, Yes, Yes, 3, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)."},"skills":["The Slack Equality Representation Preserves Feasibility in Both Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0027","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81840a23b2cf1809d93fe5723e361a463abcdea95556651567be760bab392c45","family_id":"qf-mod45-lp-standard-slack-conversion","seed":202611146027,"semantic_template":"qf-mod45-lp-standard-slack-conversion/{\"A\":[[[2,1],[1,1]],[[1,1],[2,1]]],\"b\":[[5,1],[5,1]],\"c\":[[3,1],[1,1]],\"x\":[[1,1],[1,1]]}"},"fields":{"title":"The Slack Equality Representation Preserves Feasibility in Both Directions","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[2, 1], [1, 2]], b=[5, 5], c=[3, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).","answer":{"kind":"expression","value":"([1, 1], [3, 3], [5, 5], [2, 2], Yes, Yes, Yes, Yes, Yes, 4, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)","explanation":"Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [3, 3], [5, 5], [2, 2], Yes, Yes, Yes, Yes, Yes, 4, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)."},"skills":["The Slack Equality Representation Preserves Feasibility in Both Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0028","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eedb39596a1b4c0a2b6a3490fb1422f209fe6e91a37d2a1234d4b9272c72256d","family_id":"qf-mod45-lp-standard-slack-conversion","seed":202611146028,"semantic_template":"qf-mod45-lp-standard-slack-conversion/{\"A\":[[[3,1],[1,1]],[[1,1],[3,1]]],\"b\":[[6,1],[6,1]],\"c\":[[4,1],[1,1]],\"x\":[[1,1],[1,1]]}"},"fields":{"title":"The Slack Equality Representation Preserves Feasibility in Both Directions","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[3, 1], [1, 3]], b=[6, 6], c=[4, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).","answer":{"kind":"expression","value":"([1, 1], [4, 4], [6, 6], [2, 2], Yes, Yes, Yes, Yes, Yes, 5, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)","explanation":"Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [4, 4], [6, 6], [2, 2], Yes, Yes, Yes, Yes, Yes, 5, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)."},"skills":["The Slack Equality Representation Preserves Feasibility in Both Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0029","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42c981f1c998a64eaa1aa7ec53d12b7b6a80e70e0fbea31fefb3b70f37527368","family_id":"qf-mod45-lp-standard-slack-conversion","seed":202611146029,"semantic_template":"qf-mod45-lp-standard-slack-conversion/{\"A\":[[[4,1],[1,1]],[[1,1],[4,1]]],\"b\":[[7,1],[7,1]],\"c\":[[5,1],[1,1]],\"x\":[[1,1],[1,1]]}"},"fields":{"title":"The Slack Equality Representation Preserves Feasibility in Both Directions","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[4, 1], [1, 4]], b=[7, 7], c=[5, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).","answer":{"kind":"expression","value":"([1, 1], [5, 5], [7, 7], [2, 2], Yes, Yes, Yes, Yes, Yes, 6, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)","explanation":"Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [5, 5], [7, 7], [2, 2], Yes, Yes, Yes, Yes, Yes, 6, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)."},"skills":["The Slack Equality Representation Preserves Feasibility in Both Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0030","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d41b7a9be30cdaa68158342f84f81fd3d6a24df24fb0cc113bdd7b88d34d2c16","family_id":"qf-mod45-lp-standard-slack-conversion","seed":202611146030,"semantic_template":"qf-mod45-lp-standard-slack-conversion/{\"A\":[[[5,1],[1,1]],[[1,1],[5,1]]],\"b\":[[8,1],[8,1]],\"c\":[[6,1],[1,1]],\"x\":[[1,1],[1,1]]}"},"fields":{"title":"The Slack Equality Representation Preserves Feasibility in Both Directions","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[5, 1], [1, 5]], b=[8, 8], c=[6, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).","answer":{"kind":"expression","value":"([1, 1], [6, 6], [8, 8], [2, 2], Yes, Yes, Yes, Yes, Yes, 7, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)","explanation":"Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [6, 6], [8, 8], [2, 2], Yes, Yes, Yes, Yes, Yes, 7, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)."},"skills":["The Slack Equality Representation Preserves Feasibility in Both Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0031","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b6f3b6da7eb82dc3dd48a7bcb28703b51339d4bfd970c54a33b7b74ae84b9826","family_id":"qf-mod45-lp-standard-slack-conversion","seed":202611146031,"semantic_template":"qf-mod45-lp-standard-slack-conversion/{\"A\":[[[6,1],[1,1]],[[1,1],[6,1]]],\"b\":[[9,1],[9,1]],\"c\":[[7,1],[1,1]],\"x\":[[1,1],[1,1]]}"},"fields":{"title":"The Slack Equality Representation Preserves Feasibility in Both Directions","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[6, 1], [1, 6]], b=[9, 9], c=[7, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).","answer":{"kind":"expression","value":"([1, 1], [7, 7], [9, 9], [2, 2], Yes, Yes, Yes, Yes, Yes, 8, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)","explanation":"Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [7, 7], [9, 9], [2, 2], Yes, Yes, Yes, Yes, Yes, 8, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)."},"skills":["The Slack Equality Representation Preserves Feasibility in Both Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0032","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0791f2a5af8d4cc8a2110300361c88b8bb3f9a128d6f66bbb6e17caaf2e23189","family_id":"qf-mod45-lp-standard-slack-conversion","seed":202611146032,"semantic_template":"qf-mod45-lp-standard-slack-conversion/{\"A\":[[[7,1],[1,1]],[[1,1],[7,1]]],\"b\":[[10,1],[10,1]],\"c\":[[8,1],[1,1]],\"x\":[[1,1],[1,1]]}"},"fields":{"title":"The Slack Equality Representation Preserves Feasibility in Both Directions","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[7, 1], [1, 7]], b=[10, 10], c=[8, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).","answer":{"kind":"expression","value":"([1, 1], [8, 8], [10, 10], [2, 2], Yes, Yes, Yes, Yes, Yes, 9, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)","explanation":"Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [8, 8], [10, 10], [2, 2], Yes, Yes, Yes, Yes, Yes, 9, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)."},"skills":["The Slack Equality Representation Preserves Feasibility in Both Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0033","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"782558a1b2e40fefe99217a23f9754cd8d5160b40f8a256a0f2978e8427823ec","family_id":"qf-mod45-lp-standard-slack-conversion","seed":202611146033,"semantic_template":"qf-mod45-lp-standard-slack-conversion/{\"A\":[[[8,1],[1,1]],[[1,1],[8,1]]],\"b\":[[11,1],[11,1]],\"c\":[[9,1],[1,1]],\"x\":[[1,1],[1,1]]}"},"fields":{"title":"The Slack Equality Representation Preserves Feasibility in Both Directions","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[8, 1], [1, 8]], b=[11, 11], c=[9, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).","answer":{"kind":"expression","value":"([1, 1], [9, 9], [11, 11], [2, 2], Yes, Yes, Yes, Yes, Yes, 10, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)","explanation":"Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [9, 9], [11, 11], [2, 2], Yes, Yes, Yes, Yes, Yes, 10, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)."},"skills":["The Slack Equality Representation Preserves Feasibility in Both Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0034","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b35c6cdb826d4754e103fb2963233f252eb0914bd38f515712b77a5405dd6db5","family_id":"qf-mod45-lp-standard-slack-conversion","seed":202611146034,"semantic_template":"qf-mod45-lp-standard-slack-conversion/{\"A\":[[[9,1],[1,1]],[[1,1],[9,1]]],\"b\":[[12,1],[12,1]],\"c\":[[10,1],[1,1]],\"x\":[[1,1],[1,1]]}"},"fields":{"title":"The Slack Equality Representation Preserves Feasibility in Both Directions","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[9, 1], [1, 9]], b=[12, 12], c=[10, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).","answer":{"kind":"expression","value":"([1, 1], [10, 10], [12, 12], [2, 2], Yes, Yes, Yes, Yes, Yes, 11, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)","explanation":"Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [10, 10], [12, 12], [2, 2], Yes, Yes, Yes, Yes, Yes, 11, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)."},"skills":["The Slack Equality Representation Preserves Feasibility in Both Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0035","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e03add1c2ad3f79d88725fe3f466dce33f498e1e03c3635e14969ded9e418364","family_id":"qf-mod45-lp-standard-slack-conversion","seed":202611146035,"semantic_template":"qf-mod45-lp-standard-slack-conversion/{\"A\":[[[10,1],[1,1]],[[1,1],[10,1]]],\"b\":[[13,1],[13,1]],\"c\":[[11,1],[1,1]],\"x\":[[1,1],[1,1]]}"},"fields":{"title":"The Slack Equality Representation Preserves Feasibility in Both Directions","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[10, 1], [1, 10]], b=[13, 13], c=[11, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).","answer":{"kind":"expression","value":"([1, 1], [11, 11], [13, 13], [2, 2], Yes, Yes, Yes, Yes, Yes, 12, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)","explanation":"Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [11, 11], [13, 13], [2, 2], Yes, Yes, Yes, Yes, Yes, 12, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)."},"skills":["The Slack Equality Representation Preserves Feasibility in Both Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0036","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7a2e7f4e1d3ddbadbd1251d6a2e730939bbf59c26ebb0391e9cef680791f232","family_id":"qf-mod45-lp-standard-slack-conversion","seed":202611146036,"semantic_template":"qf-mod45-lp-standard-slack-conversion/{\"A\":[[[11,1],[1,1]],[[1,1],[11,1]]],\"b\":[[14,1],[14,1]],\"c\":[[12,1],[1,1]],\"x\":[[1,1],[1,1]]}"},"fields":{"title":"The Slack Equality Representation Preserves Feasibility in Both Directions","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[11, 1], [1, 11]], b=[14, 14], c=[12, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).","answer":{"kind":"expression","value":"([1, 1], [12, 12], [14, 14], [2, 2], Yes, Yes, Yes, Yes, Yes, 13, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)","explanation":"Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [12, 12], [14, 14], [2, 2], Yes, Yes, Yes, Yes, Yes, 13, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)."},"skills":["The Slack Equality Representation Preserves Feasibility in Both Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0037","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0955a7219acf3db2762cc63680ed2d089b6f527c2abfd4206c484aad05f15251","family_id":"qf-mod45-lp-standard-slack-conversion","seed":202611146037,"semantic_template":"qf-mod45-lp-standard-slack-conversion/{\"A\":[[[12,1],[1,1]],[[1,1],[12,1]]],\"b\":[[15,1],[15,1]],\"c\":[[13,1],[1,1]],\"x\":[[1,1],[1,1]]}"},"fields":{"title":"The Slack Equality Representation Preserves Feasibility in Both Directions","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[12, 1], [1, 12]], b=[15, 15], c=[13, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).","answer":{"kind":"expression","value":"([1, 1], [13, 13], [15, 15], [2, 2], Yes, Yes, Yes, Yes, Yes, 14, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)","explanation":"Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [13, 13], [15, 15], [2, 2], Yes, Yes, Yes, Yes, Yes, 14, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)."},"skills":["The Slack Equality Representation Preserves Feasibility in Both Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0038","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d865602f53bf872a51715baf21a0ea6947d7aa79cd5b7006cf3eed25815f0c4","family_id":"qf-mod45-lp-standard-slack-conversion","seed":202611146038,"semantic_template":"qf-mod45-lp-standard-slack-conversion/{\"A\":[[[13,1],[1,1]],[[1,1],[13,1]]],\"b\":[[16,1],[16,1]],\"c\":[[14,1],[1,1]],\"x\":[[1,1],[1,1]]}"},"fields":{"title":"The Slack Equality Representation Preserves Feasibility in Both Directions","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[13, 1], [1, 13]], b=[16, 16], c=[14, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).","answer":{"kind":"expression","value":"([1, 1], [14, 14], [16, 16], [2, 2], Yes, Yes, Yes, Yes, Yes, 15, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)","explanation":"Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [14, 14], [16, 16], [2, 2], Yes, Yes, Yes, Yes, Yes, 15, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)."},"skills":["The Slack Equality Representation Preserves Feasibility in Both Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0039","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf45a62afc902e4543f15cff049bd23bb99c48700af29c50680d64ba06ae814f","family_id":"qf-mod45-lp-standard-slack-conversion","seed":202611146039,"semantic_template":"qf-mod45-lp-standard-slack-conversion/{\"A\":[[[14,1],[1,1]],[[1,1],[14,1]]],\"b\":[[17,1],[17,1]],\"c\":[[15,1],[1,1]],\"x\":[[1,1],[1,1]]}"},"fields":{"title":"The Slack Equality Representation Preserves Feasibility in Both Directions","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[14, 1], [1, 14]], b=[17, 17], c=[15, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).","answer":{"kind":"expression","value":"([1, 1], [15, 15], [17, 17], [2, 2], Yes, Yes, Yes, Yes, Yes, 16, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)","explanation":"Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [15, 15], [17, 17], [2, 2], Yes, Yes, Yes, Yes, Yes, 16, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)."},"skills":["The Slack Equality Representation Preserves Feasibility in Both Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0040","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3cf995db95020e23e8ec5b1706be77a351fb427493c2ed54ef17586c6a441192","family_id":"qf-mod45-lp-standard-slack-conversion","seed":202611146040,"semantic_template":"qf-mod45-lp-standard-slack-conversion/{\"A\":[[[15,1],[1,1]],[[1,1],[15,1]]],\"b\":[[18,1],[18,1]],\"c\":[[16,1],[1,1]],\"x\":[[1,1],[1,1]]}"},"fields":{"title":"The Slack Equality Representation Preserves Feasibility in Both Directions","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[15, 1], [1, 15]], b=[18, 18], c=[16, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).","answer":{"kind":"expression","value":"([1, 1], [16, 16], [18, 18], [2, 2], Yes, Yes, Yes, Yes, Yes, 17, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)","explanation":"Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [16, 16], [18, 18], [2, 2], Yes, Yes, Yes, Yes, Yes, 17, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)."},"skills":["The Slack Equality Representation Preserves Feasibility in Both Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0041","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce4bd0ab964882838900ade771fa742494160ef7350ec7ea49c2845297c3bb68","family_id":"qf-mod45-lp-standard-slack-conversion","seed":202611146041,"semantic_template":"qf-mod45-lp-standard-slack-conversion/{\"A\":[[[16,1],[1,1]],[[1,1],[16,1]]],\"b\":[[19,1],[19,1]],\"c\":[[17,1],[1,1]],\"x\":[[1,1],[1,1]]}"},"fields":{"title":"The Slack Equality Representation Preserves Feasibility in Both Directions","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[16, 1], [1, 16]], b=[19, 19], c=[17, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).","answer":{"kind":"expression","value":"([1, 1], [17, 17], [19, 19], [2, 2], Yes, Yes, Yes, Yes, Yes, 18, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)","explanation":"Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [17, 17], [19, 19], [2, 2], Yes, Yes, Yes, Yes, Yes, 18, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)."},"skills":["The Slack Equality Representation Preserves Feasibility in Both Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0042","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9eb58e4ccf20cc16cbab886b5dff107e5e22a8f0a5e7d39ed717c8c3a5967172","family_id":"qf-mod45-lp-standard-slack-conversion","seed":202611146042,"semantic_template":"qf-mod45-lp-standard-slack-conversion/{\"A\":[[[17,1],[1,1]],[[1,1],[17,1]]],\"b\":[[20,1],[20,1]],\"c\":[[18,1],[1,1]],\"x\":[[1,1],[1,1]]}"},"fields":{"title":"The Slack Equality Representation Preserves Feasibility in Both Directions","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[17, 1], [1, 17]], b=[20, 20], c=[18, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).","answer":{"kind":"expression","value":"([1, 1], [18, 18], [20, 20], [2, 2], Yes, Yes, Yes, Yes, Yes, 19, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)","explanation":"Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [18, 18], [20, 20], [2, 2], Yes, Yes, Yes, Yes, Yes, 19, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)."},"skills":["The Slack Equality Representation Preserves Feasibility in Both Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0043","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90d1ef1407e8cab57f67b2284bad3d9933dc1c91805d43bf0c17ecf3f8ae42ca","family_id":"qf-mod45-lp-standard-slack-conversion","seed":202611146043,"semantic_template":"qf-mod45-lp-standard-slack-conversion/{\"A\":[[[18,1],[1,1]],[[1,1],[18,1]]],\"b\":[[21,1],[21,1]],\"c\":[[19,1],[1,1]],\"x\":[[1,1],[1,1]]}"},"fields":{"title":"The Slack Equality Representation Preserves Feasibility in Both Directions","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[18, 1], [1, 18]], b=[21, 21], c=[19, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).","answer":{"kind":"expression","value":"([1, 1], [19, 19], [21, 21], [2, 2], Yes, Yes, Yes, Yes, Yes, 20, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)","explanation":"Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [19, 19], [21, 21], [2, 2], Yes, Yes, Yes, Yes, Yes, 20, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)."},"skills":["The Slack Equality Representation Preserves Feasibility in Both Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0044","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c00bdb633887d349530c4df6f0fd45c4ac12ebeb5663822542384d71bfaf4332","family_id":"qf-mod45-lp-standard-slack-conversion","seed":202611146044,"semantic_template":"qf-mod45-lp-standard-slack-conversion/{\"A\":[[[19,1],[1,1]],[[1,1],[19,1]]],\"b\":[[22,1],[22,1]],\"c\":[[20,1],[1,1]],\"x\":[[1,1],[1,1]]}"},"fields":{"title":"The Slack Equality Representation Preserves Feasibility in Both Directions","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[19, 1], [1, 19]], b=[22, 22], c=[20, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).","answer":{"kind":"expression","value":"([1, 1], [20, 20], [22, 22], [2, 2], Yes, Yes, Yes, Yes, Yes, 21, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)","explanation":"Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [20, 20], [22, 22], [2, 2], Yes, Yes, Yes, Yes, Yes, 21, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)."},"skills":["The Slack Equality Representation Preserves Feasibility in Both Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0045","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9664e1e5d935e73cff0847610e46c51f9a45dcf4d22715af716bd55e580b0949","family_id":"qf-mod45-lp-standard-slack-conversion","seed":202611146045,"semantic_template":"qf-mod45-lp-standard-slack-conversion/{\"A\":[[[20,1],[1,1]],[[1,1],[20,1]]],\"b\":[[23,1],[23,1]],\"c\":[[21,1],[1,1]],\"x\":[[1,1],[1,1]]}"},"fields":{"title":"The Slack Equality Representation Preserves Feasibility in Both Directions","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[20, 1], [1, 20]], b=[23, 23], c=[21, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).","answer":{"kind":"expression","value":"([1, 1], [21, 21], [23, 23], [2, 2], Yes, Yes, Yes, Yes, Yes, 22, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)","explanation":"Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [21, 21], [23, 23], [2, 2], Yes, Yes, Yes, Yes, Yes, 22, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)."},"skills":["The Slack Equality Representation Preserves Feasibility in Both Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0046","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a0717038e583f35526135af1c4521bf4282b94e1948a985241f6f292d95202e7","family_id":"qf-mod45-lp-standard-slack-conversion","seed":202611146046,"semantic_template":"qf-mod45-lp-standard-slack-conversion/{\"A\":[[[21,1],[1,1]],[[1,1],[21,1]]],\"b\":[[24,1],[24,1]],\"c\":[[22,1],[1,1]],\"x\":[[1,1],[1,1]]}"},"fields":{"title":"The Slack Equality Representation Preserves Feasibility in Both Directions","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[21, 1], [1, 21]], b=[24, 24], c=[22, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).","answer":{"kind":"expression","value":"([1, 1], [22, 22], [24, 24], [2, 2], Yes, Yes, Yes, Yes, Yes, 23, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)","explanation":"Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [22, 22], [24, 24], [2, 2], Yes, Yes, Yes, Yes, Yes, 23, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)."},"skills":["The Slack Equality Representation Preserves Feasibility in Both Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0047","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"408cd9522c8983fd39ca7c8ec4f7a151ff497b7764966b2e5b9c275450d17e3d","family_id":"qf-mod45-lp-standard-slack-conversion","seed":202611146047,"semantic_template":"qf-mod45-lp-standard-slack-conversion/{\"A\":[[[22,1],[1,1]],[[1,1],[22,1]]],\"b\":[[25,1],[25,1]],\"c\":[[23,1],[1,1]],\"x\":[[1,1],[1,1]]}"},"fields":{"title":"The Slack Equality Representation Preserves Feasibility in Both Directions","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[22, 1], [1, 22]], b=[25, 25], c=[23, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).","answer":{"kind":"expression","value":"([1, 1], [23, 23], [25, 25], [2, 2], Yes, Yes, Yes, Yes, Yes, 24, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)","explanation":"Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [23, 23], [25, 25], [2, 2], Yes, Yes, Yes, Yes, Yes, 24, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)."},"skills":["The Slack Equality Representation Preserves Feasibility in Both Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0048","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38cb4944d6cf981966df480c8faeaf27fa8b7caac10ba11c246b31e2b43c3659","family_id":"qf-mod45-lp-standard-slack-conversion","seed":202611146048,"semantic_template":"qf-mod45-lp-standard-slack-conversion/{\"A\":[[[23,1],[1,1]],[[1,1],[23,1]]],\"b\":[[26,1],[26,1]],\"c\":[[24,1],[1,1]],\"x\":[[1,1],[1,1]]}"},"fields":{"title":"The Slack Equality Representation Preserves Feasibility in Both Directions","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[23, 1], [1, 23]], b=[26, 26], c=[24, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).","answer":{"kind":"expression","value":"([1, 1], [24, 24], [26, 26], [2, 2], Yes, Yes, Yes, Yes, Yes, 25, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)","explanation":"Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [24, 24], [26, 26], [2, 2], Yes, Yes, Yes, Yes, Yes, 25, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)."},"skills":["The Slack Equality Representation Preserves Feasibility in Both Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0049","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45674f1b11ffa9aef0043ba80fe76a299861da9df526276807656b8dd5b72308","family_id":"qf-mod45-lp-standard-slack-conversion","seed":202611146049,"semantic_template":"qf-mod45-lp-standard-slack-conversion/{\"A\":[[[24,1],[1,1]],[[1,1],[24,1]]],\"b\":[[27,1],[27,1]],\"c\":[[25,1],[1,1]],\"x\":[[1,1],[1,1]]}"},"fields":{"title":"The Slack Equality Representation Preserves Feasibility in Both Directions","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[24, 1], [1, 24]], b=[27, 27], c=[25, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).","answer":{"kind":"expression","value":"([1, 1], [25, 25], [27, 27], [2, 2], Yes, Yes, Yes, Yes, Yes, 26, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)","explanation":"Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [25, 25], [27, 27], [2, 2], Yes, Yes, Yes, Yes, Yes, 26, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)."},"skills":["The Slack Equality Representation Preserves Feasibility in Both Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0050","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"279820d9745077a8784cc0f31711caac9749be7f882e8fbcd3e66d6aca169129","family_id":"qf-mod45-lp-standard-slack-conversion","seed":202611146050,"semantic_template":"qf-mod45-lp-standard-slack-conversion/{\"A\":[[[25,1],[1,1]],[[1,1],[25,1]]],\"b\":[[28,1],[28,1]],\"c\":[[26,1],[1,1]],\"x\":[[1,1],[1,1]]}"},"fields":{"title":"The Slack Equality Representation Preserves Feasibility in Both Directions","stem":"Use the one fixed convention maximize c^T*x, subject to A*x≤b and x≥0, with A=[[25, 1], [1, 25]], b=[28, 28], c=[26, 1]. Require a nonempty rectangular exact rational matrix and matching vectors. Coefficients and b may have either sign; variable nonnegativity is a constraint, not an assumption that every supplied point is feasible. Supplied candidate x=[1, 1]. Give (x, Ax, b, slack, x_nonnegative, inequalities_hold, slack_nonnegative, feasible, feasibility_equivalence_holds, objective_value, certificate).","answer":{"kind":"expression","value":"([1, 1], [26, 26], [28, 28], [2, 2], Yes, Yes, Yes, Yes, Yes, 27, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)","explanation":"Compute A*x and the unique slack s=b−A*x. Compare x≥0, A*x≤b and s≥0 separately, then joint feasibility and its equivalence with nonnegative x/slack. The equivalence flag can be Yes for an infeasible point. Projection and x→(x,b−Ax) give the slack-point bijection, preserving c^T*x; this is not an optimality proof. The exact result, retaining the requested order, is ([1, 1], [26, 26], [28, 28], [2, 2], Yes, Yes, Yes, Yes, Yes, 27, x_maps_to_(x,b-Ax)_and_every_nonnegative_slack_equality_point_maps_back_bijectively)."},"skills":["The Slack Equality Representation Preserves Feasibility in Both Directions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0051","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32fa6c089025c3e7d5de46dbdab36dab1ffb7279df4ac41c4f9347c7b45c4415","family_id":"qf-mod45-lp-standard-unrestricted-split","seed":202611146051,"semantic_template":"qf-mod45-lp-standard-unrestricted-split/{\"column\":[[1,1],[-2,1],[3,1]],\"objective_coefficient\":[4,1],\"value\":[1,1]}"},"fields":{"title":"A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split","stem":"Use column=[1, -2, 3], objective coefficient=4, unrestricted value=1. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).","answer":{"kind":"expression","value":"([1, -2, 3], 4, 1, 1, 0, 1, [1, -2, 3], [1, -2, 3], Yes, 4, 4, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)","explanation":"Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([1, -2, 3], 4, 1, 1, 0, 1, [1, -2, 3], [1, -2, 3], Yes, 4, 4, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)."},"skills":["A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0052","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7aea7bd69c917624fda928a1126c4afa06bca6d4e61082dde7af40c1fafc40ad","family_id":"qf-mod45-lp-standard-unrestricted-split","seed":202611146052,"semantic_template":"qf-mod45-lp-standard-unrestricted-split/{\"column\":[[2,1],[-3,1],[4,1]],\"objective_coefficient\":[5,1],\"value\":[-2,1]}"},"fields":{"title":"A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split","stem":"Use column=[2, -3, 4], objective coefficient=5, unrestricted value=-2. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).","answer":{"kind":"expression","value":"([2, -3, 4], 5, -2, 0, 2, -2, [-4, 6, -8], [-4, 6, -8], Yes, -10, -10, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)","explanation":"Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([2, -3, 4], 5, -2, 0, 2, -2, [-4, 6, -8], [-4, 6, -8], Yes, -10, -10, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)."},"skills":["A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0053","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"87855fab24a4d541834a0787990c7743beb10e742beaad5fda418ecb51ac17e6","family_id":"qf-mod45-lp-standard-unrestricted-split","seed":202611146053,"semantic_template":"qf-mod45-lp-standard-unrestricted-split/{\"column\":[[3,1],[-4,1],[5,1]],\"objective_coefficient\":[6,1],\"value\":[3,1]}"},"fields":{"title":"A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split","stem":"Use column=[3, -4, 5], objective coefficient=6, unrestricted value=3. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).","answer":{"kind":"expression","value":"([3, -4, 5], 6, 3, 3, 0, 3, [9, -12, 15], [9, -12, 15], Yes, 18, 18, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)","explanation":"Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([3, -4, 5], 6, 3, 3, 0, 3, [9, -12, 15], [9, -12, 15], Yes, 18, 18, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)."},"skills":["A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0054","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3097480180e87e18c1df51e7a3f644ce9efb1041ec87e2fa9db680955522f53","family_id":"qf-mod45-lp-standard-unrestricted-split","seed":202611146054,"semantic_template":"qf-mod45-lp-standard-unrestricted-split/{\"column\":[[4,1],[-5,1],[6,1]],\"objective_coefficient\":[7,1],\"value\":[-4,1]}"},"fields":{"title":"A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split","stem":"Use column=[4, -5, 6], objective coefficient=7, unrestricted value=-4. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).","answer":{"kind":"expression","value":"([4, -5, 6], 7, -4, 0, 4, -4, [-16, 20, -24], [-16, 20, -24], Yes, -28, -28, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)","explanation":"Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([4, -5, 6], 7, -4, 0, 4, -4, [-16, 20, -24], [-16, 20, -24], Yes, -28, -28, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)."},"skills":["A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0055","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"218cc23db39e2b569c8169f92fdd9138e6fff4d3d44210589ec64539688b8fe4","family_id":"qf-mod45-lp-standard-unrestricted-split","seed":202611146055,"semantic_template":"qf-mod45-lp-standard-unrestricted-split/{\"column\":[[5,1],[-6,1],[7,1]],\"objective_coefficient\":[8,1],\"value\":[5,1]}"},"fields":{"title":"A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split","stem":"Use column=[5, -6, 7], objective coefficient=8, unrestricted value=5. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).","answer":{"kind":"expression","value":"([5, -6, 7], 8, 5, 5, 0, 5, [25, -30, 35], [25, -30, 35], Yes, 40, 40, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)","explanation":"Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([5, -6, 7], 8, 5, 5, 0, 5, [25, -30, 35], [25, -30, 35], Yes, 40, 40, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)."},"skills":["A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0056","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce04cbc34ec31638348d588f727f04de992fc767f51753723099eb943ea9904f","family_id":"qf-mod45-lp-standard-unrestricted-split","seed":202611146056,"semantic_template":"qf-mod45-lp-standard-unrestricted-split/{\"column\":[[6,1],[-7,1],[8,1]],\"objective_coefficient\":[9,1],\"value\":[-6,1]}"},"fields":{"title":"A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split","stem":"Use column=[6, -7, 8], objective coefficient=9, unrestricted value=-6. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).","answer":{"kind":"expression","value":"([6, -7, 8], 9, -6, 0, 6, -6, [-36, 42, -48], [-36, 42, -48], Yes, -54, -54, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)","explanation":"Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([6, -7, 8], 9, -6, 0, 6, -6, [-36, 42, -48], [-36, 42, -48], Yes, -54, -54, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)."},"skills":["A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0057","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c6146834f64f339ae8c59321d4a66eabd32e0b99a155a1ab01ee643004086cde","family_id":"qf-mod45-lp-standard-unrestricted-split","seed":202611146057,"semantic_template":"qf-mod45-lp-standard-unrestricted-split/{\"column\":[[7,1],[-8,1],[9,1]],\"objective_coefficient\":[10,1],\"value\":[7,1]}"},"fields":{"title":"A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split","stem":"Use column=[7, -8, 9], objective coefficient=10, unrestricted value=7. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).","answer":{"kind":"expression","value":"([7, -8, 9], 10, 7, 7, 0, 7, [49, -56, 63], [49, -56, 63], Yes, 70, 70, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)","explanation":"Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([7, -8, 9], 10, 7, 7, 0, 7, [49, -56, 63], [49, -56, 63], Yes, 70, 70, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)."},"skills":["A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0058","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8a2bc43203b3412cc60bb80e1132c3d6fc4c6cd3c79665ae3698da1f6c41cbe","family_id":"qf-mod45-lp-standard-unrestricted-split","seed":202611146058,"semantic_template":"qf-mod45-lp-standard-unrestricted-split/{\"column\":[[8,1],[-9,1],[10,1]],\"objective_coefficient\":[11,1],\"value\":[-8,1]}"},"fields":{"title":"A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split","stem":"Use column=[8, -9, 10], objective coefficient=11, unrestricted value=-8. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).","answer":{"kind":"expression","value":"([8, -9, 10], 11, -8, 0, 8, -8, [-64, 72, -80], [-64, 72, -80], Yes, -88, -88, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)","explanation":"Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([8, -9, 10], 11, -8, 0, 8, -8, [-64, 72, -80], [-64, 72, -80], Yes, -88, -88, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)."},"skills":["A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0059","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dcb2e259a4dbb30adf7456571e92f7fc814321ec149bd2e3bf97d82aee2db2a9","family_id":"qf-mod45-lp-standard-unrestricted-split","seed":202611146059,"semantic_template":"qf-mod45-lp-standard-unrestricted-split/{\"column\":[[9,1],[-10,1],[11,1]],\"objective_coefficient\":[12,1],\"value\":[9,1]}"},"fields":{"title":"A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split","stem":"Use column=[9, -10, 11], objective coefficient=12, unrestricted value=9. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).","answer":{"kind":"expression","value":"([9, -10, 11], 12, 9, 9, 0, 9, [81, -90, 99], [81, -90, 99], Yes, 108, 108, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)","explanation":"Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([9, -10, 11], 12, 9, 9, 0, 9, [81, -90, 99], [81, -90, 99], Yes, 108, 108, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)."},"skills":["A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0060","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b01d77ebd7a8ed4541a842876a3cfc7e7df160bb3e61d7fe86763864d9d8849","family_id":"qf-mod45-lp-standard-unrestricted-split","seed":202611146060,"semantic_template":"qf-mod45-lp-standard-unrestricted-split/{\"column\":[[10,1],[-11,1],[12,1]],\"objective_coefficient\":[13,1],\"value\":[-10,1]}"},"fields":{"title":"A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split","stem":"Use column=[10, -11, 12], objective coefficient=13, unrestricted value=-10. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).","answer":{"kind":"expression","value":"([10, -11, 12], 13, -10, 0, 10, -10, [-100, 110, -120], [-100, 110, -120], Yes, -130, -130, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)","explanation":"Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([10, -11, 12], 13, -10, 0, 10, -10, [-100, 110, -120], [-100, 110, -120], Yes, -130, -130, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)."},"skills":["A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0061","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c134fff26c4425b9f07d7dff11f9169593f9140458dfcf56b3246fddf97edf3a","family_id":"qf-mod45-lp-standard-unrestricted-split","seed":202611146061,"semantic_template":"qf-mod45-lp-standard-unrestricted-split/{\"column\":[[11,1],[-12,1],[13,1]],\"objective_coefficient\":[14,1],\"value\":[11,1]}"},"fields":{"title":"A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split","stem":"Use column=[11, -12, 13], objective coefficient=14, unrestricted value=11. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).","answer":{"kind":"expression","value":"([11, -12, 13], 14, 11, 11, 0, 11, [121, -132, 143], [121, -132, 143], Yes, 154, 154, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)","explanation":"Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([11, -12, 13], 14, 11, 11, 0, 11, [121, -132, 143], [121, -132, 143], Yes, 154, 154, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)."},"skills":["A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0062","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3854894e5935f20c2e7418fed4d7827175e2690021ceda494fa9ec5c92a74901","family_id":"qf-mod45-lp-standard-unrestricted-split","seed":202611146062,"semantic_template":"qf-mod45-lp-standard-unrestricted-split/{\"column\":[[12,1],[-13,1],[14,1]],\"objective_coefficient\":[15,1],\"value\":[-12,1]}"},"fields":{"title":"A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split","stem":"Use column=[12, -13, 14], objective coefficient=15, unrestricted value=-12. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).","answer":{"kind":"expression","value":"([12, -13, 14], 15, -12, 0, 12, -12, [-144, 156, -168], [-144, 156, -168], Yes, -180, -180, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)","explanation":"Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([12, -13, 14], 15, -12, 0, 12, -12, [-144, 156, -168], [-144, 156, -168], Yes, -180, -180, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)."},"skills":["A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0063","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30bfffe156d978e4039221e6fde0dfc85b4b8c3ee5aca84f4071e0642cf7c6dc","family_id":"qf-mod45-lp-standard-unrestricted-split","seed":202611146063,"semantic_template":"qf-mod45-lp-standard-unrestricted-split/{\"column\":[[13,1],[-14,1],[15,1]],\"objective_coefficient\":[16,1],\"value\":[13,1]}"},"fields":{"title":"A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split","stem":"Use column=[13, -14, 15], objective coefficient=16, unrestricted value=13. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).","answer":{"kind":"expression","value":"([13, -14, 15], 16, 13, 13, 0, 13, [169, -182, 195], [169, -182, 195], Yes, 208, 208, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)","explanation":"Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([13, -14, 15], 16, 13, 13, 0, 13, [169, -182, 195], [169, -182, 195], Yes, 208, 208, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)."},"skills":["A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0064","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"250fd7cc15d83d01fa8148bf6334087ba28162c1ce337605d576a722f937127f","family_id":"qf-mod45-lp-standard-unrestricted-split","seed":202611146064,"semantic_template":"qf-mod45-lp-standard-unrestricted-split/{\"column\":[[14,1],[-15,1],[16,1]],\"objective_coefficient\":[17,1],\"value\":[-14,1]}"},"fields":{"title":"A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split","stem":"Use column=[14, -15, 16], objective coefficient=17, unrestricted value=-14. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).","answer":{"kind":"expression","value":"([14, -15, 16], 17, -14, 0, 14, -14, [-196, 210, -224], [-196, 210, -224], Yes, -238, -238, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)","explanation":"Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([14, -15, 16], 17, -14, 0, 14, -14, [-196, 210, -224], [-196, 210, -224], Yes, -238, -238, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)."},"skills":["A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0065","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab45326ed73c65e6ea145e49faffc7d4cb59de32d93d53b919354a8931dddfd1","family_id":"qf-mod45-lp-standard-unrestricted-split","seed":202611146065,"semantic_template":"qf-mod45-lp-standard-unrestricted-split/{\"column\":[[15,1],[-16,1],[17,1]],\"objective_coefficient\":[18,1],\"value\":[15,1]}"},"fields":{"title":"A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split","stem":"Use column=[15, -16, 17], objective coefficient=18, unrestricted value=15. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).","answer":{"kind":"expression","value":"([15, -16, 17], 18, 15, 15, 0, 15, [225, -240, 255], [225, -240, 255], Yes, 270, 270, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)","explanation":"Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([15, -16, 17], 18, 15, 15, 0, 15, [225, -240, 255], [225, -240, 255], Yes, 270, 270, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)."},"skills":["A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0066","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"719a754f3364b41c4cdf7e8b149f3d051d7ce304fea90be1300b8b29e31ffeac","family_id":"qf-mod45-lp-standard-unrestricted-split","seed":202611146066,"semantic_template":"qf-mod45-lp-standard-unrestricted-split/{\"column\":[[16,1],[-17,1],[18,1]],\"objective_coefficient\":[19,1],\"value\":[-16,1]}"},"fields":{"title":"A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split","stem":"Use column=[16, -17, 18], objective coefficient=19, unrestricted value=-16. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).","answer":{"kind":"expression","value":"([16, -17, 18], 19, -16, 0, 16, -16, [-256, 272, -288], [-256, 272, -288], Yes, -304, -304, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)","explanation":"Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([16, -17, 18], 19, -16, 0, 16, -16, [-256, 272, -288], [-256, 272, -288], Yes, -304, -304, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)."},"skills":["A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0067","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"241c2c1657df4d3550171e8dae6d9685dee77bfe52b7b76597f09002da9f6a9b","family_id":"qf-mod45-lp-standard-unrestricted-split","seed":202611146067,"semantic_template":"qf-mod45-lp-standard-unrestricted-split/{\"column\":[[17,1],[-18,1],[19,1]],\"objective_coefficient\":[20,1],\"value\":[17,1]}"},"fields":{"title":"A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split","stem":"Use column=[17, -18, 19], objective coefficient=20, unrestricted value=17. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).","answer":{"kind":"expression","value":"([17, -18, 19], 20, 17, 17, 0, 17, [289, -306, 323], [289, -306, 323], Yes, 340, 340, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)","explanation":"Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([17, -18, 19], 20, 17, 17, 0, 17, [289, -306, 323], [289, -306, 323], Yes, 340, 340, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)."},"skills":["A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0068","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed9e46ac5ca0e4eb8aa13ca9e5b85f5bdc7cc184ce9c167b04ac290aa129426e","family_id":"qf-mod45-lp-standard-unrestricted-split","seed":202611146068,"semantic_template":"qf-mod45-lp-standard-unrestricted-split/{\"column\":[[18,1],[-19,1],[20,1]],\"objective_coefficient\":[21,1],\"value\":[-18,1]}"},"fields":{"title":"A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split","stem":"Use column=[18, -19, 20], objective coefficient=21, unrestricted value=-18. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).","answer":{"kind":"expression","value":"([18, -19, 20], 21, -18, 0, 18, -18, [-324, 342, -360], [-324, 342, -360], Yes, -378, -378, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)","explanation":"Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([18, -19, 20], 21, -18, 0, 18, -18, [-324, 342, -360], [-324, 342, -360], Yes, -378, -378, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)."},"skills":["A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0069","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9186cc18d53982f11d15bac25a4a740b9797296e05982f37a8ea7dc271a40b0a","family_id":"qf-mod45-lp-standard-unrestricted-split","seed":202611146069,"semantic_template":"qf-mod45-lp-standard-unrestricted-split/{\"column\":[[19,1],[-20,1],[21,1]],\"objective_coefficient\":[22,1],\"value\":[19,1]}"},"fields":{"title":"A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split","stem":"Use column=[19, -20, 21], objective coefficient=22, unrestricted value=19. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).","answer":{"kind":"expression","value":"([19, -20, 21], 22, 19, 19, 0, 19, [361, -380, 399], [361, -380, 399], Yes, 418, 418, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)","explanation":"Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([19, -20, 21], 22, 19, 19, 0, 19, [361, -380, 399], [361, -380, 399], Yes, 418, 418, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)."},"skills":["A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0070","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"15939e36131b5045468ec6fd9fd9c57d0aeeaadd434bbd1396c29e3a63ec47c1","family_id":"qf-mod45-lp-standard-unrestricted-split","seed":202611146070,"semantic_template":"qf-mod45-lp-standard-unrestricted-split/{\"column\":[[20,1],[-21,1],[22,1]],\"objective_coefficient\":[23,1],\"value\":[-20,1]}"},"fields":{"title":"A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split","stem":"Use column=[20, -21, 22], objective coefficient=23, unrestricted value=-20. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).","answer":{"kind":"expression","value":"([20, -21, 22], 23, -20, 0, 20, -20, [-400, 420, -440], [-400, 420, -440], Yes, -460, -460, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)","explanation":"Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([20, -21, 22], 23, -20, 0, 20, -20, [-400, 420, -440], [-400, 420, -440], Yes, -460, -460, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)."},"skills":["A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0071","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"941efaa244476f9ddb73e03aefbe747cad4ab9c8283d82ddb107fc2279a7dcd4","family_id":"qf-mod45-lp-standard-unrestricted-split","seed":202611146071,"semantic_template":"qf-mod45-lp-standard-unrestricted-split/{\"column\":[[21,1],[-22,1],[23,1]],\"objective_coefficient\":[24,1],\"value\":[21,1]}"},"fields":{"title":"A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split","stem":"Use column=[21, -22, 23], objective coefficient=24, unrestricted value=21. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).","answer":{"kind":"expression","value":"([21, -22, 23], 24, 21, 21, 0, 21, [441, -462, 483], [441, -462, 483], Yes, 504, 504, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)","explanation":"Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([21, -22, 23], 24, 21, 21, 0, 21, [441, -462, 483], [441, -462, 483], Yes, 504, 504, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)."},"skills":["A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0072","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43115532d9320caa8c36f8923ec1082a97b86a4f499bb1cb46e1e09a8c6bbe5d","family_id":"qf-mod45-lp-standard-unrestricted-split","seed":202611146072,"semantic_template":"qf-mod45-lp-standard-unrestricted-split/{\"column\":[[22,1],[-23,1],[24,1]],\"objective_coefficient\":[25,1],\"value\":[-22,1]}"},"fields":{"title":"A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split","stem":"Use column=[22, -23, 24], objective coefficient=25, unrestricted value=-22. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).","answer":{"kind":"expression","value":"([22, -23, 24], 25, -22, 0, 22, -22, [-484, 506, -528], [-484, 506, -528], Yes, -550, -550, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)","explanation":"Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([22, -23, 24], 25, -22, 0, 22, -22, [-484, 506, -528], [-484, 506, -528], Yes, -550, -550, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)."},"skills":["A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0073","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d46c891f13043d4af6e7e3052cf35984b0159fe03266d84548b986487d1dc2bb","family_id":"qf-mod45-lp-standard-unrestricted-split","seed":202611146073,"semantic_template":"qf-mod45-lp-standard-unrestricted-split/{\"column\":[[23,1],[-24,1],[25,1]],\"objective_coefficient\":[26,1],\"value\":[23,1]}"},"fields":{"title":"A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split","stem":"Use column=[23, -24, 25], objective coefficient=26, unrestricted value=23. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).","answer":{"kind":"expression","value":"([23, -24, 25], 26, 23, 23, 0, 23, [529, -552, 575], [529, -552, 575], Yes, 598, 598, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)","explanation":"Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([23, -24, 25], 26, 23, 23, 0, 23, [529, -552, 575], [529, -552, 575], Yes, 598, 598, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)."},"skills":["A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0074","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d824061003da531c3eb33fd29a2963d8ab743588d5f07a0c3ce35a8c4f83ac0e","family_id":"qf-mod45-lp-standard-unrestricted-split","seed":202611146074,"semantic_template":"qf-mod45-lp-standard-unrestricted-split/{\"column\":[[24,1],[-25,1],[26,1]],\"objective_coefficient\":[27,1],\"value\":[-24,1]}"},"fields":{"title":"A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split","stem":"Use column=[24, -25, 26], objective coefficient=27, unrestricted value=-24. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).","answer":{"kind":"expression","value":"([24, -25, 26], 27, -24, 0, 24, -24, [-576, 600, -624], [-576, 600, -624], Yes, -648, -648, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)","explanation":"Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([24, -25, 26], 27, -24, 0, 24, -24, [-576, 600, -624], [-576, 600, -624], Yes, -648, -648, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)."},"skills":["A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0075","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3261f519464eda2beb925d992617058f1227da3f972c6fe0cf87b5d685c9a77","family_id":"qf-mod45-lp-standard-unrestricted-split","seed":202611146075,"semantic_template":"qf-mod45-lp-standard-unrestricted-split/{\"column\":[[25,1],[-26,1],[27,1]],\"objective_coefficient\":[28,1],\"value\":[25,1]}"},"fields":{"title":"A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split","stem":"Use column=[25, -26, 27], objective coefficient=28, unrestricted value=25. Give (column, objective_coefficient, value, x_plus, x_minus, reconstructed_value, original_column_contribution, split_column_contribution, columns_equal, original_objective_contribution, split_objective_contribution, objectives_equal, certificate).","answer":{"kind":"expression","value":"([25, -26, 27], 28, 25, 25, 0, 25, [625, -650, 675], [625, -650, 675], Yes, 700, 700, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)","explanation":"Choose canonical x_plus=max(value,0), x_minus=max(−value,0), reconstruct value=x_plus−x_minus and compare original/split column and objective contributions. Reverse projection is valid, but adding the same nonnegative amount to both parts gives other representations, so the unrestricted split is not a bijection of all pairs. The exact result, retaining the requested order, is ([25, -26, 27], 28, 25, 25, 0, 25, [625, -650, 675], [625, -650, 675], Yes, 700, 700, Yes, x=x_plus-x_minus_with_both_nonnegative_preserves_column_and_objective_contribution)."},"skills":["A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated 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-20261114-r06-linear-program-standard-form-0076","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b107c56b2ea8071788f25f1c82dbb9e2eb2f1b1b7528cce350521fec8014a829","family_id":"qf-mod45-lp-standard-convention-boundary","seed":202611146076,"semantic_template":"qf-mod45-lp-standard-convention-boundary/{\"coefficients\":[[1,1],[-2,1],[1,1]],\"mode\":\"greater_equal\",\"rhs\":[3,1]}"},"fields":{"title":"Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles","stem":"Use mode=greater_equal, coefficients=[1, -2, 1], rhs=3. The fixed target convention is maximize/≤/nonnegative. Give (mode, coefficients, rhs, negated_coefficients, negated_rhs, certificate).","answer":{"kind":"expression","value":"(greater_equal, [1, -2, 1], 3, [-1, 2, -1], -3, multiply_constraint_by_minus_one_to_obtain_less_equal)","explanation":"For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (greater_equal, [1, -2, 1], 3, [-1, 2, -1], -3, multiply_constraint_by_minus_one_to_obtain_less_equal)."},"skills":["Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r06-linear-program-standard-form-0077","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13646903fc8f4958fe8756e872468bdbefa72289870e584487abbd03ed69d775","family_id":"qf-mod45-lp-standard-convention-boundary","seed":202611146077,"semantic_template":"qf-mod45-lp-standard-convention-boundary/{\"coefficients\":[[2,1],[-3,1],[1,1]],\"mode\":\"equality\",\"rhs\":[4,1]}"},"fields":{"title":"Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles","stem":"Use mode=equality, coefficients=[2, -3, 1], rhs=4. The fixed target convention is maximize/≤/nonnegative. Give (mode, coefficients, rhs, less_equal_row, opposite_less_equal_row, certificate).","answer":{"kind":"expression","value":"(equality, [2, -3, 1], 4, [[2, -3, 1], 4], [[-2, 3, -1], -4], replace_equality_by_two_opposite_less_equal_constraints)","explanation":"For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (equality, [2, -3, 1], 4, [[2, -3, 1], 4], [[-2, 3, -1], -4], replace_equality_by_two_opposite_less_equal_constraints)."},"skills":["Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r06-linear-program-standard-form-0078","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16994b269f6a967d772465a52860028d71f0014e82d15e7df0b78b41a9c2dbc6","family_id":"qf-mod45-lp-standard-convention-boundary","seed":202611146078,"semantic_template":"qf-mod45-lp-standard-convention-boundary/{\"coefficients\":[[3,1],[-4,1],[1,1]],\"mode\":\"minimize\",\"rhs\":[0,1]}"},"fields":{"title":"Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles","stem":"Use mode=minimize, coefficients=[3, -4, 1], rhs=0. The fixed target convention is maximize/≤/nonnegative. Give (mode, objective_coefficients, maximization_coefficients, conversion_rule, optimality_certificate, boundary).","answer":{"kind":"expression","value":"(minimize, [3, -4, 1], [-3, 4, -1], negate_objective_to_convert_minimize_to_maximize, No, conversion_does_not_prove_feasible_bounded_or_optimal)","explanation":"For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (minimize, [3, -4, 1], [-3, 4, -1], negate_objective_to_convert_minimize_to_maximize, No, conversion_does_not_prove_feasible_bounded_or_optimal)."},"skills":["Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r06-linear-program-standard-form-0079","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cee532633a14c2435fca09c1d9eaecf72c3dfc4b41a3170eac9e52ca5a2143a4","family_id":"qf-mod45-lp-standard-convention-boundary","seed":202611146079,"semantic_template":"qf-mod45-lp-standard-convention-boundary/{\"coefficients\":[[4,1],[-5,1],[1,1]],\"mode\":\"greater_equal\",\"rhs\":[6,1]}"},"fields":{"title":"Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles","stem":"Use mode=greater_equal, coefficients=[4, -5, 1], rhs=6. The fixed target convention is maximize/≤/nonnegative. Give (mode, coefficients, rhs, negated_coefficients, negated_rhs, certificate).","answer":{"kind":"expression","value":"(greater_equal, [4, -5, 1], 6, [-4, 5, -1], -6, multiply_constraint_by_minus_one_to_obtain_less_equal)","explanation":"For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (greater_equal, [4, -5, 1], 6, [-4, 5, -1], -6, multiply_constraint_by_minus_one_to_obtain_less_equal)."},"skills":["Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r06-linear-program-standard-form-0080","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c85d9a11de679637835afd8fce7f89fdd324142ac071453224fd3a64a572fab","family_id":"qf-mod45-lp-standard-convention-boundary","seed":202611146080,"semantic_template":"qf-mod45-lp-standard-convention-boundary/{\"coefficients\":[[5,1],[-6,1],[1,1]],\"mode\":\"equality\",\"rhs\":[7,1]}"},"fields":{"title":"Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles","stem":"Use mode=equality, coefficients=[5, -6, 1], rhs=7. The fixed target convention is maximize/≤/nonnegative. Give (mode, coefficients, rhs, less_equal_row, opposite_less_equal_row, certificate).","answer":{"kind":"expression","value":"(equality, [5, -6, 1], 7, [[5, -6, 1], 7], [[-5, 6, -1], -7], replace_equality_by_two_opposite_less_equal_constraints)","explanation":"For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (equality, [5, -6, 1], 7, [[5, -6, 1], 7], [[-5, 6, -1], -7], replace_equality_by_two_opposite_less_equal_constraints)."},"skills":["Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r06-linear-program-standard-form-0081","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0aa54deebee15f5c7a5210a368d9d27584dae58af7ed253fc384d8260fdc89e","family_id":"qf-mod45-lp-standard-convention-boundary","seed":202611146081,"semantic_template":"qf-mod45-lp-standard-convention-boundary/{\"coefficients\":[[6,1],[-7,1],[1,1]],\"mode\":\"minimize\",\"rhs\":[0,1]}"},"fields":{"title":"Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles","stem":"Use mode=minimize, coefficients=[6, -7, 1], rhs=0. The fixed target convention is maximize/≤/nonnegative. Give (mode, objective_coefficients, maximization_coefficients, conversion_rule, optimality_certificate, boundary).","answer":{"kind":"expression","value":"(minimize, [6, -7, 1], [-6, 7, -1], negate_objective_to_convert_minimize_to_maximize, No, conversion_does_not_prove_feasible_bounded_or_optimal)","explanation":"For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (minimize, [6, -7, 1], [-6, 7, -1], negate_objective_to_convert_minimize_to_maximize, No, conversion_does_not_prove_feasible_bounded_or_optimal)."},"skills":["Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r06-linear-program-standard-form-0082","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d97f14a78a6eca8103a07391775cee3931124ca9d1c6803d28bf5ea27887fa6","family_id":"qf-mod45-lp-standard-convention-boundary","seed":202611146082,"semantic_template":"qf-mod45-lp-standard-convention-boundary/{\"coefficients\":[[7,1],[-8,1],[1,1]],\"mode\":\"greater_equal\",\"rhs\":[9,1]}"},"fields":{"title":"Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles","stem":"Use mode=greater_equal, coefficients=[7, -8, 1], rhs=9. The fixed target convention is maximize/≤/nonnegative. Give (mode, coefficients, rhs, negated_coefficients, negated_rhs, certificate).","answer":{"kind":"expression","value":"(greater_equal, [7, -8, 1], 9, [-7, 8, -1], -9, multiply_constraint_by_minus_one_to_obtain_less_equal)","explanation":"For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (greater_equal, [7, -8, 1], 9, [-7, 8, -1], -9, multiply_constraint_by_minus_one_to_obtain_less_equal)."},"skills":["Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r06-linear-program-standard-form-0083","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"653621fa3757f53ae77f67936cd7d9c0beda4150e53e026ebc782e79ec1169e1","family_id":"qf-mod45-lp-standard-convention-boundary","seed":202611146083,"semantic_template":"qf-mod45-lp-standard-convention-boundary/{\"coefficients\":[[8,1],[-9,1],[1,1]],\"mode\":\"equality\",\"rhs\":[10,1]}"},"fields":{"title":"Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles","stem":"Use mode=equality, coefficients=[8, -9, 1], rhs=10. The fixed target convention is maximize/≤/nonnegative. Give (mode, coefficients, rhs, less_equal_row, opposite_less_equal_row, certificate).","answer":{"kind":"expression","value":"(equality, [8, -9, 1], 10, [[8, -9, 1], 10], [[-8, 9, -1], -10], replace_equality_by_two_opposite_less_equal_constraints)","explanation":"For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (equality, [8, -9, 1], 10, [[8, -9, 1], 10], [[-8, 9, -1], -10], replace_equality_by_two_opposite_less_equal_constraints)."},"skills":["Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r06-linear-program-standard-form-0084","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7773934b19451c5fb18f4f47c42d40e4bab57af38e05afb1eccb43a084aa6d6a","family_id":"qf-mod45-lp-standard-convention-boundary","seed":202611146084,"semantic_template":"qf-mod45-lp-standard-convention-boundary/{\"coefficients\":[[9,1],[-10,1],[1,1]],\"mode\":\"minimize\",\"rhs\":[0,1]}"},"fields":{"title":"Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles","stem":"Use mode=minimize, coefficients=[9, -10, 1], rhs=0. The fixed target convention is maximize/≤/nonnegative. Give (mode, objective_coefficients, maximization_coefficients, conversion_rule, optimality_certificate, boundary).","answer":{"kind":"expression","value":"(minimize, [9, -10, 1], [-9, 10, -1], negate_objective_to_convert_minimize_to_maximize, No, conversion_does_not_prove_feasible_bounded_or_optimal)","explanation":"For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (minimize, [9, -10, 1], [-9, 10, -1], negate_objective_to_convert_minimize_to_maximize, No, conversion_does_not_prove_feasible_bounded_or_optimal)."},"skills":["Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r06-linear-program-standard-form-0085","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9fd1936e16118a075dbf3ad41a3d0a07e8edf54e17304bcca02801f13ab3543f","family_id":"qf-mod45-lp-standard-convention-boundary","seed":202611146085,"semantic_template":"qf-mod45-lp-standard-convention-boundary/{\"coefficients\":[[10,1],[-11,1],[1,1]],\"mode\":\"greater_equal\",\"rhs\":[12,1]}"},"fields":{"title":"Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles","stem":"Use mode=greater_equal, coefficients=[10, -11, 1], rhs=12. The fixed target convention is maximize/≤/nonnegative. Give (mode, coefficients, rhs, negated_coefficients, negated_rhs, certificate).","answer":{"kind":"expression","value":"(greater_equal, [10, -11, 1], 12, [-10, 11, -1], -12, multiply_constraint_by_minus_one_to_obtain_less_equal)","explanation":"For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (greater_equal, [10, -11, 1], 12, [-10, 11, -1], -12, multiply_constraint_by_minus_one_to_obtain_less_equal)."},"skills":["Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r06-linear-program-standard-form-0086","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a32444908fbd0a7c642c03d80a1a1a91725822c683474d34abef556ca7bea11a","family_id":"qf-mod45-lp-standard-convention-boundary","seed":202611146086,"semantic_template":"qf-mod45-lp-standard-convention-boundary/{\"coefficients\":[[11,1],[-12,1],[1,1]],\"mode\":\"equality\",\"rhs\":[13,1]}"},"fields":{"title":"Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles","stem":"Use mode=equality, coefficients=[11, -12, 1], rhs=13. The fixed target convention is maximize/≤/nonnegative. Give (mode, coefficients, rhs, less_equal_row, opposite_less_equal_row, certificate).","answer":{"kind":"expression","value":"(equality, [11, -12, 1], 13, [[11, -12, 1], 13], [[-11, 12, -1], -13], replace_equality_by_two_opposite_less_equal_constraints)","explanation":"For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (equality, [11, -12, 1], 13, [[11, -12, 1], 13], [[-11, 12, -1], -13], replace_equality_by_two_opposite_less_equal_constraints)."},"skills":["Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r06-linear-program-standard-form-0087","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"15a1d1fa65c96ba659955d3c1b3ad627fd8f3ca67a29abe9116e2ac3671d1952","family_id":"qf-mod45-lp-standard-convention-boundary","seed":202611146087,"semantic_template":"qf-mod45-lp-standard-convention-boundary/{\"coefficients\":[[12,1],[-13,1],[1,1]],\"mode\":\"minimize\",\"rhs\":[0,1]}"},"fields":{"title":"Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles","stem":"Use mode=minimize, coefficients=[12, -13, 1], rhs=0. The fixed target convention is maximize/≤/nonnegative. Give (mode, objective_coefficients, maximization_coefficients, conversion_rule, optimality_certificate, boundary).","answer":{"kind":"expression","value":"(minimize, [12, -13, 1], [-12, 13, -1], negate_objective_to_convert_minimize_to_maximize, No, conversion_does_not_prove_feasible_bounded_or_optimal)","explanation":"For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (minimize, [12, -13, 1], [-12, 13, -1], negate_objective_to_convert_minimize_to_maximize, No, conversion_does_not_prove_feasible_bounded_or_optimal)."},"skills":["Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r06-linear-program-standard-form-0088","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d0c8802d3b9badbfbf4909a91433ec5dc7dd9f3fdf6e0eda3520873336d9373","family_id":"qf-mod45-lp-standard-convention-boundary","seed":202611146088,"semantic_template":"qf-mod45-lp-standard-convention-boundary/{\"coefficients\":[[13,1],[-14,1],[1,1]],\"mode\":\"greater_equal\",\"rhs\":[15,1]}"},"fields":{"title":"Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles","stem":"Use mode=greater_equal, coefficients=[13, -14, 1], rhs=15. The fixed target convention is maximize/≤/nonnegative. Give (mode, coefficients, rhs, negated_coefficients, negated_rhs, certificate).","answer":{"kind":"expression","value":"(greater_equal, [13, -14, 1], 15, [-13, 14, -1], -15, multiply_constraint_by_minus_one_to_obtain_less_equal)","explanation":"For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (greater_equal, [13, -14, 1], 15, [-13, 14, -1], -15, multiply_constraint_by_minus_one_to_obtain_less_equal)."},"skills":["Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r06-linear-program-standard-form-0089","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb235828e92a3aaa36843d609b1cb0b8af6c63396399a489685f558bb3a293f9","family_id":"qf-mod45-lp-standard-convention-boundary","seed":202611146089,"semantic_template":"qf-mod45-lp-standard-convention-boundary/{\"coefficients\":[[14,1],[-15,1],[1,1]],\"mode\":\"equality\",\"rhs\":[16,1]}"},"fields":{"title":"Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles","stem":"Use mode=equality, coefficients=[14, -15, 1], rhs=16. The fixed target convention is maximize/≤/nonnegative. Give (mode, coefficients, rhs, less_equal_row, opposite_less_equal_row, certificate).","answer":{"kind":"expression","value":"(equality, [14, -15, 1], 16, [[14, -15, 1], 16], [[-14, 15, -1], -16], replace_equality_by_two_opposite_less_equal_constraints)","explanation":"For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (equality, [14, -15, 1], 16, [[14, -15, 1], 16], [[-14, 15, -1], -16], replace_equality_by_two_opposite_less_equal_constraints)."},"skills":["Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r06-linear-program-standard-form-0090","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9690e16ed474da5ca2fb6b5a7cd14b0315f14994098c3479a76baa5026bf6e76","family_id":"qf-mod45-lp-standard-convention-boundary","seed":202611146090,"semantic_template":"qf-mod45-lp-standard-convention-boundary/{\"coefficients\":[[15,1],[-16,1],[1,1]],\"mode\":\"minimize\",\"rhs\":[0,1]}"},"fields":{"title":"Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles","stem":"Use mode=minimize, coefficients=[15, -16, 1], rhs=0. The fixed target convention is maximize/≤/nonnegative. Give (mode, objective_coefficients, maximization_coefficients, conversion_rule, optimality_certificate, boundary).","answer":{"kind":"expression","value":"(minimize, [15, -16, 1], [-15, 16, -1], negate_objective_to_convert_minimize_to_maximize, No, conversion_does_not_prove_feasible_bounded_or_optimal)","explanation":"For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (minimize, [15, -16, 1], [-15, 16, -1], negate_objective_to_convert_minimize_to_maximize, No, conversion_does_not_prove_feasible_bounded_or_optimal)."},"skills":["Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r06-linear-program-standard-form-0091","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc410b9a74640911d108e8dcf0ee28b0900ba1a9692023a27506fbdccb1e1634","family_id":"qf-mod45-lp-standard-convention-boundary","seed":202611146091,"semantic_template":"qf-mod45-lp-standard-convention-boundary/{\"coefficients\":[[16,1],[-17,1],[1,1]],\"mode\":\"greater_equal\",\"rhs\":[18,1]}"},"fields":{"title":"Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles","stem":"Use mode=greater_equal, coefficients=[16, -17, 1], rhs=18. The fixed target convention is maximize/≤/nonnegative. Give (mode, coefficients, rhs, negated_coefficients, negated_rhs, certificate).","answer":{"kind":"expression","value":"(greater_equal, [16, -17, 1], 18, [-16, 17, -1], -18, multiply_constraint_by_minus_one_to_obtain_less_equal)","explanation":"For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (greater_equal, [16, -17, 1], 18, [-16, 17, -1], -18, multiply_constraint_by_minus_one_to_obtain_less_equal)."},"skills":["Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r06-linear-program-standard-form-0092","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d86f63b13962a0a3ec9498aa7f2e672a54a1dc599c6c356c3ef0d05223414fe","family_id":"qf-mod45-lp-standard-convention-boundary","seed":202611146092,"semantic_template":"qf-mod45-lp-standard-convention-boundary/{\"coefficients\":[[17,1],[-18,1],[1,1]],\"mode\":\"equality\",\"rhs\":[19,1]}"},"fields":{"title":"Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles","stem":"Use mode=equality, coefficients=[17, -18, 1], rhs=19. The fixed target convention is maximize/≤/nonnegative. Give (mode, coefficients, rhs, less_equal_row, opposite_less_equal_row, certificate).","answer":{"kind":"expression","value":"(equality, [17, -18, 1], 19, [[17, -18, 1], 19], [[-17, 18, -1], -19], replace_equality_by_two_opposite_less_equal_constraints)","explanation":"For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (equality, [17, -18, 1], 19, [[17, -18, 1], 19], [[-17, 18, -1], -19], replace_equality_by_two_opposite_less_equal_constraints)."},"skills":["Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r06-linear-program-standard-form-0093","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa1dc0d9e409fe098c1da287b3dd6460aac93a43a24dcb5618d1b0a0557a7b8b","family_id":"qf-mod45-lp-standard-convention-boundary","seed":202611146093,"semantic_template":"qf-mod45-lp-standard-convention-boundary/{\"coefficients\":[[18,1],[-19,1],[1,1]],\"mode\":\"minimize\",\"rhs\":[0,1]}"},"fields":{"title":"Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles","stem":"Use mode=minimize, coefficients=[18, -19, 1], rhs=0. The fixed target convention is maximize/≤/nonnegative. Give (mode, objective_coefficients, maximization_coefficients, conversion_rule, optimality_certificate, boundary).","answer":{"kind":"expression","value":"(minimize, [18, -19, 1], [-18, 19, -1], negate_objective_to_convert_minimize_to_maximize, No, conversion_does_not_prove_feasible_bounded_or_optimal)","explanation":"For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (minimize, [18, -19, 1], [-18, 19, -1], negate_objective_to_convert_minimize_to_maximize, No, conversion_does_not_prove_feasible_bounded_or_optimal)."},"skills":["Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r06-linear-program-standard-form-0094","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f72ae047874132dbe80b3d5d4a4a4016fdc08127d6c7cee6ea4cb9edbaffced8","family_id":"qf-mod45-lp-standard-convention-boundary","seed":202611146094,"semantic_template":"qf-mod45-lp-standard-convention-boundary/{\"coefficients\":[[19,1],[-20,1],[1,1]],\"mode\":\"greater_equal\",\"rhs\":[21,1]}"},"fields":{"title":"Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles","stem":"Use mode=greater_equal, coefficients=[19, -20, 1], rhs=21. The fixed target convention is maximize/≤/nonnegative. Give (mode, coefficients, rhs, negated_coefficients, negated_rhs, certificate).","answer":{"kind":"expression","value":"(greater_equal, [19, -20, 1], 21, [-19, 20, -1], -21, multiply_constraint_by_minus_one_to_obtain_less_equal)","explanation":"For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (greater_equal, [19, -20, 1], 21, [-19, 20, -1], -21, multiply_constraint_by_minus_one_to_obtain_less_equal)."},"skills":["Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r06-linear-program-standard-form-0095","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7470addeecafb0d6bf8d0f1e1ea594a63a0dde174658270d29fa0d48b733ece","family_id":"qf-mod45-lp-standard-convention-boundary","seed":202611146095,"semantic_template":"qf-mod45-lp-standard-convention-boundary/{\"coefficients\":[[20,1],[-21,1],[1,1]],\"mode\":\"equality\",\"rhs\":[22,1]}"},"fields":{"title":"Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles","stem":"Use mode=equality, coefficients=[20, -21, 1], rhs=22. The fixed target convention is maximize/≤/nonnegative. Give (mode, coefficients, rhs, less_equal_row, opposite_less_equal_row, certificate).","answer":{"kind":"expression","value":"(equality, [20, -21, 1], 22, [[20, -21, 1], 22], [[-20, 21, -1], -22], replace_equality_by_two_opposite_less_equal_constraints)","explanation":"For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (equality, [20, -21, 1], 22, [[20, -21, 1], 22], [[-20, 21, -1], -22], replace_equality_by_two_opposite_less_equal_constraints)."},"skills":["Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r06-linear-program-standard-form-0096","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b85e430ee43d6f4b59c4fb5370cc151482c82c299c8a1019d15a0fef05aafbd7","family_id":"qf-mod45-lp-standard-convention-boundary","seed":202611146096,"semantic_template":"qf-mod45-lp-standard-convention-boundary/{\"coefficients\":[[21,1],[-22,1],[1,1]],\"mode\":\"minimize\",\"rhs\":[0,1]}"},"fields":{"title":"Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles","stem":"Use mode=minimize, coefficients=[21, -22, 1], rhs=0. The fixed target convention is maximize/≤/nonnegative. Give (mode, objective_coefficients, maximization_coefficients, conversion_rule, optimality_certificate, boundary).","answer":{"kind":"expression","value":"(minimize, [21, -22, 1], [-21, 22, -1], negate_objective_to_convert_minimize_to_maximize, No, conversion_does_not_prove_feasible_bounded_or_optimal)","explanation":"For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (minimize, [21, -22, 1], [-21, 22, -1], negate_objective_to_convert_minimize_to_maximize, No, conversion_does_not_prove_feasible_bounded_or_optimal)."},"skills":["Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r06-linear-program-standard-form-0097","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4971a5bc3881019322d3ef052f9143190e9c6403326f68984670bbb58ddaf76","family_id":"qf-mod45-lp-standard-convention-boundary","seed":202611146097,"semantic_template":"qf-mod45-lp-standard-convention-boundary/{\"coefficients\":[[22,1],[-23,1],[1,1]],\"mode\":\"greater_equal\",\"rhs\":[24,1]}"},"fields":{"title":"Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles","stem":"Use mode=greater_equal, coefficients=[22, -23, 1], rhs=24. The fixed target convention is maximize/≤/nonnegative. Give (mode, coefficients, rhs, negated_coefficients, negated_rhs, certificate).","answer":{"kind":"expression","value":"(greater_equal, [22, -23, 1], 24, [-22, 23, -1], -24, multiply_constraint_by_minus_one_to_obtain_less_equal)","explanation":"For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (greater_equal, [22, -23, 1], 24, [-22, 23, -1], -24, multiply_constraint_by_minus_one_to_obtain_less_equal)."},"skills":["Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r06-linear-program-standard-form-0098","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f9a1d7231a64fb92ee57210413486a2bfaf682f897d86ed9a4cfd787049fbfe7","family_id":"qf-mod45-lp-standard-convention-boundary","seed":202611146098,"semantic_template":"qf-mod45-lp-standard-convention-boundary/{\"coefficients\":[[23,1],[-24,1],[1,1]],\"mode\":\"equality\",\"rhs\":[25,1]}"},"fields":{"title":"Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles","stem":"Use mode=equality, coefficients=[23, -24, 1], rhs=25. The fixed target convention is maximize/≤/nonnegative. Give (mode, coefficients, rhs, less_equal_row, opposite_less_equal_row, certificate).","answer":{"kind":"expression","value":"(equality, [23, -24, 1], 25, [[23, -24, 1], 25], [[-23, 24, -1], -25], replace_equality_by_two_opposite_less_equal_constraints)","explanation":"For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (equality, [23, -24, 1], 25, [[23, -24, 1], 25], [[-23, 24, -1], -25], replace_equality_by_two_opposite_less_equal_constraints)."},"skills":["Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r06-linear-program-standard-form-0099","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93ce63d77848cb5f86d9051dc177831ae56e62ffae24f1fc447b71cf4cfad892","family_id":"qf-mod45-lp-standard-convention-boundary","seed":202611146099,"semantic_template":"qf-mod45-lp-standard-convention-boundary/{\"coefficients\":[[24,1],[-25,1],[1,1]],\"mode\":\"minimize\",\"rhs\":[0,1]}"},"fields":{"title":"Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles","stem":"Use mode=minimize, coefficients=[24, -25, 1], rhs=0. The fixed target convention is maximize/≤/nonnegative. Give (mode, objective_coefficients, maximization_coefficients, conversion_rule, optimality_certificate, boundary).","answer":{"kind":"expression","value":"(minimize, [24, -25, 1], [-24, 25, -1], negate_objective_to_convert_minimize_to_maximize, No, conversion_does_not_prove_feasible_bounded_or_optimal)","explanation":"For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (minimize, [24, -25, 1], [-24, 25, -1], negate_objective_to_convert_minimize_to_maximize, No, conversion_does_not_prove_feasible_bounded_or_optimal)."},"skills":["Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r06-linear-program-standard-form-0100","batch_id":"batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"382225eb56cf5ca49d746a6d679d2a76b20a9fef70ce422dd5afbde278288c53","family_id":"qf-mod45-lp-standard-convention-boundary","seed":202611146100,"semantic_template":"qf-mod45-lp-standard-convention-boundary/{\"coefficients\":[[25,1],[-26,1],[1,1]],\"mode\":\"greater_equal\",\"rhs\":[27,1]}"},"fields":{"title":"Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles","stem":"Use mode=greater_equal, coefficients=[25, -26, 1], rhs=27. The fixed target convention is maximize/≤/nonnegative. Give (mode, coefficients, rhs, negated_coefficients, negated_rhs, certificate).","answer":{"kind":"expression","value":"(greater_equal, [25, -26, 1], 27, [-25, 26, -1], -27, multiply_constraint_by_minus_one_to_obtain_less_equal)","explanation":"For greater_equal negate the constraint and rhs; for equality retain both opposite ≤ rows; for minimize negate objective coefficients. All have6 fields but different roles, so select only from source mode, never answer shape. The minimize rhs0 is a required unused-field sentinel, not a constraint. No conversion establishes feasibility, boundedness, an optimum or duality. The exact result, retaining the requested order, is (greater_equal, [25, -26, 1], 27, [-25, 26, -1], -27, multiply_constraint_by_minus_one_to_obtain_less_equal)."},"skills":["Same-Length LP Conversion Tuples Still Need Mode-Specific Field Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}