# Question 25846: Equal Objective Values and Finite Grids Do Not Replace Theorem Premises [Return to catalogue page 130](https://ocme.evemisslab.com/en/questions/page/130/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261115-r04-linear-program-duality-foundations/#q-20261015-advance-20261115-r04-linear-program-duality-0096) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0050-25846 · q-20261015-advance-20261115-r04-linear-program-duality-0096 ## Original question and givens Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[22], c=[1]. Candidate x=[1], y=[2]. Boundary mode=finite_grid_not_general. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary). - G001 — Fix only primal maximize c^T*x with A*x≤b,x≥0 and dual minimize b^T*y with A^T*y≥c,y≥0. Exact A=[[1]], b=[22], c=[1]. Candidate x=[1], y=[2]. Boundary mode=finite_grid_not_general. Give (mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-linear-program-duality](https://ocme.evemisslab.com/en/mko/mko-linear-program-duality/) ## Step-by-step derivation S001 · impliesBefore source roles = [lpdual-strong-boundary] After derived = [[mode, x, y, Ax, transpose_A_y, primal_feasible, dual_feasible, primal_objective, middle_value, dual_objective, first_inequality, second_inequality, zero_gap, unsupported_optimality_or_general_strong_duality_claim, boundary]] Conditions: none List the full source field order without JavaScript.LP weak uses13 fields,zero/strong15;premises and roles follow the original operation/mode without padding. Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · impliesBefore source roles = [[[1]], [22], [1]] After derived = [1, 1, [[1]], primal:max_cTx_Ax_le_b_x_ge_0, dual:min_bTy_ATy_ge_c_y_ge_0, Yes] Conditions: none The fixed max/<=/x>=0 convention pairs with min/>=/y>=0.The dual has m variables and n constraints.Transpose every position;either coefficient sign is legal and dimensions do not imply feasibility. Step identifiers and dependenciesRule: LPdual_fixed_conventions_complete_transpose_and_dimensions · Uses: G001, S001 S003 · calculationBefore LPdual ORIGINAL Ax row0 product0 := mul(1, 1) After LPdual ORIGINAL Ax row0 product0 = 1 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore LPdual ORIGINAL Ax row0 sum := sum(1) After LPdual ORIGINAL Ax row0 sum = 1 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore LPdual ORIGINAL ATy row0 product0 := mul(1, 2) After LPdual ORIGINAL ATy row0 product0 = 2 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore LPdual ORIGINAL ATy row0 sum := sum(2) After LPdual ORIGINAL ATy row0 sum = 2 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore LPdual ORIGINAL primal objective product0 := mul(1, 1) After LPdual ORIGINAL primal objective product0 = 1 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore LPdual ORIGINAL primal objective sum := sum(1) After LPdual ORIGINAL primal objective sum = 1 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore LPdual ORIGINAL middle yTAx product0 := mul(2, 1) After LPdual ORIGINAL middle yTAx product0 = 2 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore LPdual ORIGINAL middle yTAx sum := sum(2) After LPdual ORIGINAL middle yTAx sum = 2 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore LPdual ORIGINAL dual objective product0 := mul(22, 2) After LPdual ORIGINAL dual objective product0 = 44 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore LPdual ORIGINAL dual objective sum := sum(44) After LPdual ORIGINAL dual objective sum = 44 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · impliesBefore source roles = [[1], [2], [1], [2], [22], [1]] After derived = [Yes, Yes, 1, 2, 44, [[0, 0, 2, 1, 1, 2]], 2, [[0, 1, 1, 1]], 1, [[0, 2, 21, 42]], 42, Yes, Yes, No] Conditions: none Check both feasibilities separately.Expanding the double sum proves(ATy)Tx=yTAx.The first gap is sum x_j((ATy)_j−c_j),the second sum y_i(b_i−(Ax)_i).Only with both feasibility premises do nonnegative factors establish the weak chain.With missing premises,numerical inequalities may be accidental.Teaching residuals impose no new source numeric bounds. Step identifiers and dependenciesRule: LPdual_ALL_feasibilities_exact_pairings_residuals_and_conditional_weak_chain · Uses: G001, S012 S014 · impliesBefore source roles = [finite_grid_not_general, Yes, Yes, 1, 44, No] After derived = [No] Conditions: none Primal-infeasible mode does not additionally require dual feasibility;dual-infeasible mode requires primal feasibility;finite-grid mode requires both but may have nonzero gap.No rejects unsupported optimality/general strong-duality inference,not actual optimality of a particular feasible zero-gap pair. Step identifiers and dependenciesRule: LPdual_original_boundary_mode_premises_NOT_bare_equality_general_proof · Uses: G001, S013 ## Reverse checks - S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S012 · premise_and_result_audit — Recompute LPdual ORIGINAL dual objective sum with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute LPdual ORIGINAL dual objective product0 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute LPdual ORIGINAL middle yTAx sum with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute LPdual ORIGINAL middle yTAx product0 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute LPdual ORIGINAL primal objective sum with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute LPdual ORIGINAL primal objective product0 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute LPdual ORIGINAL ATy row0 sum with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute LPdual ORIGINAL ATy row0 product0 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute LPdual ORIGINAL Ax row0 sum with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute LPdual ORIGINAL Ax row0 product0 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result (finite_grid_not_general, [1], [2], [1], [2], Yes, Yes, 1, 2, 44, Yes, Yes, No, No, objective_equality_without_both_feasibilities_is_not_optimality_and_finite_grids_do_not_prove_general_strong_duality_or_handle_infeasible_unbounded_cases) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 841a1d3d026622b3e676c7c43e31c527b746a8961bf6600b4a525714ec3b1405 - candidate record SHA-256: 460af8e61c0c9240f3dff33a058afaee75c1ca50917741ca2bd9ccaa6361a768 - semantic review receipt SHA-256: 74c63a2c12285a3b2073f8d9f102bc2ba5fb7ed890c5b7c3b8afa94f032eda82 - question-first witness SHA-256: 6d56428610f8e86c45fd00a2ad3b27ad9c052956946f0b91d71a68536f79d95f - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261115-r04-linear-program-duality-0096.json)