# Question 2475: All Tied Optimizers among the Explicit Feasible Candidates [Return to catalogue page 13](https://ocme.evemisslab.com/en/questions/page/13/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261021-r06-linear-programming-foundations/#q-20261013-advance-20261021-r06-linear-programming-0100) 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. R0003-02475 · q-20261013-advance-20261021-r06-linear-programming-0100 ## Original question and givens Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 0], [2, 4], [0, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 6]], objective=[2, 2], direction=max. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. At least one canonical candidate must be feasible. Give (canonical constraints,objective,direction,every [index,x,y,constraint truth list,feasible?,objective value] row,best feasible value,all tied optimal canonical indices). - G001 — Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. points=[[0, 0], [1, 0], [2, 4], [0, 2]], constraints=[[-1, 0, 0], [0, -1, 0], [1, 1, 6]], objective=[2, 2], direction=max. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. At least one canonical candidate must be feasible. Give (canonical constraints,objective,direction,every [index,x,y,constraint truth list,feasible?,objective value] row,best feasible value,all tied optimal canonical indices). ## 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-programming](https://ocme.evemisslab.com/en/mko/mko-linear-programming/) ## Step-by-step derivation S001 · impliesBefore explicit finite candidate grid and linear rows a·x+b·y≤c After keep candidate-grid optimization separate from continuous LP Conditions: none Audit only the listed finite candidates;without a complete-vertex or full-feasible-domain certificate,this is not a general continuous-LP optimum claim. Step identifiers and dependenciesRule: finite_LP_scope · Uses: G001 S002 · impliesBefore [[0, 0], [1, 0], [2, 4], [0, 2]] After deduplicated canonical indexed points=[[0, 0, 0], [1, 0, 2], [2, 1, 0], [3, 2, 4]] Conditions: none Keep the original data;output indices follow the source contract's sorted distinct points,not original input positions. Step identifiers and dependenciesRule: candidate_index_normalization · Uses: G001, S001 S003 · calculationBefore point0 objective first term := mul(2, 0) After point0 objective first term = 0 Conditions: none Use the same coordinate order in the objective. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore point0 objective second term := mul(2, 0) After point0 objective second term = 0 Conditions: none Multiply the second objective coefficient by the second coordinate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore point0 objective := sum(0, 0) After point0 objective = 0 Conditions: none Sum the complete two products. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore point0 row0 a·x := mul(-1, 0) After point0 row0 a·x = 0 Conditions: none Multiply the first coordinate by its row coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore point0 row0 b·y := mul(0, 0) After point0 row0 b·y = 0 Conditions: none Compute the second coordinate in the same declared order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore point0 row0 LHS := sum(0, 0) After point0 row0 LHS = 0 Conditions: none Add before comparing the complete row inequality. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · impliesBefore row0: 0≤0 After satisfied=Yes Conditions: none Equality is allowed and negative coefficients remain in the products;do not arbitrarily reverse the inequality. Step identifiers and dependenciesRule: inclusive_constraint · Uses: G001, S008 S010 · calculationBefore point0 row1 a·x := mul(0, 0) After point0 row1 a·x = 0 Conditions: none Multiply the first coordinate by its row coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore point0 row1 b·y := mul(-1, 0) After point0 row1 b·y = 0 Conditions: none Compute the second coordinate in the same declared order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore point0 row1 LHS := sum(0, 0) After point0 row1 LHS = 0 Conditions: none Add before comparing the complete row inequality. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · impliesBefore row1: 0≤0 After satisfied=Yes Conditions: none Equality is allowed and negative coefficients remain in the products;do not arbitrarily reverse the inequality. Step identifiers and dependenciesRule: inclusive_constraint · Uses: G001, S012 S014 · calculationBefore point0 row2 a·x := mul(1, 0) After point0 row2 a·x = 0 Conditions: none Multiply the first coordinate by its row coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore point0 row2 b·y := mul(1, 0) After point0 row2 b·y = 0 Conditions: none Compute the second coordinate in the same declared order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore point0 row2 LHS := sum(0, 0) After point0 row2 LHS = 0 Conditions: none Add before comparing the complete row inequality. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · impliesBefore row2: 0≤6 After satisfied=Yes Conditions: none Equality is allowed and negative coefficients remain in the products;do not arbitrarily reverse the inequality. Step identifiers and dependenciesRule: inclusive_constraint · Uses: G001, S016 S018 · calculationBefore point1 objective first term := mul(2, 0) After point1 objective first term = 0 Conditions: none Use the same coordinate order in the objective. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore point1 objective second term := mul(2, 2) After point1 objective second term = 4 Conditions: none Multiply the second objective coefficient by the second coordinate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore point1 objective := sum(0, 4) After point1 objective = 4 Conditions: none Sum the complete two products. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore point1 row0 a·x := mul(-1, 0) After point1 row0 a·x = 0 Conditions: none Multiply the first coordinate by its row coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore point1 row0 b·y := mul(0, 2) After point1 row0 b·y = 0 Conditions: none Compute the second coordinate in the same declared order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore point1 row0 LHS := sum(0, 0) After point1 row0 LHS = 0 Conditions: none Add before comparing the complete row inequality. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · impliesBefore row0: 0≤0 After satisfied=Yes Conditions: none Equality is allowed and negative coefficients remain in the products;do not arbitrarily reverse the inequality. Step identifiers and dependenciesRule: inclusive_constraint · Uses: G001, S023 S025 · calculationBefore point1 row1 a·x := mul(0, 0) After point1 row1 a·x = 0 Conditions: none Multiply the first coordinate by its row coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore point1 row1 b·y := mul(-1, 2) After point1 row1 b·y = -2 Conditions: none Compute the second coordinate in the same declared order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore point1 row1 LHS := sum(0, -2) After point1 row1 LHS = -2 Conditions: none Add before comparing the complete row inequality. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · impliesBefore row1: -2≤0 After satisfied=Yes Conditions: none Equality is allowed and negative coefficients remain in the products;do not arbitrarily reverse the inequality. Step identifiers and dependenciesRule: inclusive_constraint · Uses: G001, S027 S029 · calculationBefore point1 row2 a·x := mul(1, 0) After point1 row2 a·x = 0 Conditions: none Multiply the first coordinate by its row coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · calculationBefore point1 row2 b·y := mul(1, 2) After point1 row2 b·y = 2 Conditions: none Compute the second coordinate in the same declared order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · calculationBefore point1 row2 LHS := sum(0, 2) After point1 row2 LHS = 2 Conditions: none Add before comparing the complete row inequality. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · impliesBefore row2: 2≤6 After satisfied=Yes Conditions: none Equality is allowed and negative coefficients remain in the products;do not arbitrarily reverse the inequality. Step identifiers and dependenciesRule: inclusive_constraint · Uses: G001, S031 S033 · calculationBefore point2 objective first term := mul(2, 1) After point2 objective first term = 2 Conditions: none Use the same coordinate order in the objective. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · calculationBefore point2 objective second term := mul(2, 0) After point2 objective second term = 0 Conditions: none Multiply the second objective coefficient by the second coordinate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · calculationBefore point2 objective := sum(2, 0) After point2 objective = 2 Conditions: none Sum the complete two products. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034 S036 · calculationBefore point2 row0 a·x := mul(-1, 1) After point2 row0 a·x = -1 Conditions: none Multiply the first coordinate by its row coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · calculationBefore point2 row0 b·y := mul(0, 0) After point2 row0 b·y = 0 Conditions: none Compute the second coordinate in the same declared order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036 S038 · calculationBefore point2 row0 LHS := sum(-1, 0) After point2 row0 LHS = -1 Conditions: none Add before comparing the complete row inequality. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S037 S039 · impliesBefore row0: -1≤0 After satisfied=Yes Conditions: none Equality is allowed and negative coefficients remain in the products;do not arbitrarily reverse the inequality. Step identifiers and dependenciesRule: inclusive_constraint · Uses: G001, S038 S040 · calculationBefore point2 row1 a·x := mul(0, 1) After point2 row1 a·x = 0 Conditions: none Multiply the first coordinate by its row coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S039 S041 · calculationBefore point2 row1 b·y := mul(-1, 0) After point2 row1 b·y = 0 Conditions: none Compute the second coordinate in the same declared order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S040 S042 · calculationBefore point2 row1 LHS := sum(0, 0) After point2 row1 LHS = 0 Conditions: none Add before comparing the complete row inequality. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S041 S043 · impliesBefore row1: 0≤0 After satisfied=Yes Conditions: none Equality is allowed and negative coefficients remain in the products;do not arbitrarily reverse the inequality. Step identifiers and dependenciesRule: inclusive_constraint · Uses: G001, S042 S044 · calculationBefore point2 row2 a·x := mul(1, 1) After point2 row2 a·x = 1 Conditions: none Multiply the first coordinate by its row coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S043 S045 · calculationBefore point2 row2 b·y := mul(1, 0) After point2 row2 b·y = 0 Conditions: none Compute the second coordinate in the same declared order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S044 S046 · calculationBefore point2 row2 LHS := sum(1, 0) After point2 row2 LHS = 1 Conditions: none Add before comparing the complete row inequality. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S045 S047 · impliesBefore row2: 1≤6 After satisfied=Yes Conditions: none Equality is allowed and negative coefficients remain in the products;do not arbitrarily reverse the inequality. Step identifiers and dependenciesRule: inclusive_constraint · Uses: G001, S046 S048 · calculationBefore point3 objective first term := mul(2, 2) After point3 objective first term = 4 Conditions: none Use the same coordinate order in the objective. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S047 S049 · calculationBefore point3 objective second term := mul(2, 4) After point3 objective second term = 8 Conditions: none Multiply the second objective coefficient by the second coordinate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S048 S050 · calculationBefore point3 objective := sum(4, 8) After point3 objective = 12 Conditions: none Sum the complete two products. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S049 S051 · calculationBefore point3 row0 a·x := mul(-1, 2) After point3 row0 a·x = -2 Conditions: none Multiply the first coordinate by its row coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S050 S052 · calculationBefore point3 row0 b·y := mul(0, 4) After point3 row0 b·y = 0 Conditions: none Compute the second coordinate in the same declared order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S051 S053 · calculationBefore point3 row0 LHS := sum(-2, 0) After point3 row0 LHS = -2 Conditions: none Add before comparing the complete row inequality. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S052 S054 · impliesBefore row0: -2≤0 After satisfied=Yes Conditions: none Equality is allowed and negative coefficients remain in the products;do not arbitrarily reverse the inequality. Step identifiers and dependenciesRule: inclusive_constraint · Uses: G001, S053 S055 · calculationBefore point3 row1 a·x := mul(0, 2) After point3 row1 a·x = 0 Conditions: none Multiply the first coordinate by its row coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S054 S056 · calculationBefore point3 row1 b·y := mul(-1, 4) After point3 row1 b·y = -4 Conditions: none Compute the second coordinate in the same declared order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S055 S057 · calculationBefore point3 row1 LHS := sum(0, -4) After point3 row1 LHS = -4 Conditions: none Add before comparing the complete row inequality. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S056 S058 · impliesBefore row1: -4≤0 After satisfied=Yes Conditions: none Equality is allowed and negative coefficients remain in the products;do not arbitrarily reverse the inequality. Step identifiers and dependenciesRule: inclusive_constraint · Uses: G001, S057 S059 · calculationBefore point3 row2 a·x := mul(1, 2) After point3 row2 a·x = 2 Conditions: none Multiply the first coordinate by its row coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S058 S060 · calculationBefore point3 row2 b·y := mul(1, 4) After point3 row2 b·y = 4 Conditions: none Compute the second coordinate in the same declared order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S059 S061 · calculationBefore point3 row2 LHS := sum(2, 4) After point3 row2 LHS = 6 Conditions: none Add before comparing the complete row inequality. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S060 S062 · impliesBefore row2: 6≤6 After satisfied=Yes Conditions: none Equality is allowed and negative coefficients remain in the products;do not arbitrarily reverse the inequality. Step identifiers and dependenciesRule: inclusive_constraint · Uses: G001, S061 S063 · impliesBefore direction=max; feasible objective rows=[[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 0, 2, [Yes, Yes, Yes], Yes, 4], [2, 1, 0, [Yes, Yes, Yes], Yes, 2], [3, 2, 4, [Yes, Yes, Yes], Yes, 12]] After best among listed feasible points=12; all attaining indices=[3] Conditions: none Filter infeasible points before optimizing in the specified direction;retain every tied attaining index,not just the first. Step identifiers and dependenciesRule: finite_optimum_all_ties · Uses: G001, S062 S064 · impliesBefore finite candidates have been completely audited After no unlisted-point or general-LP optimality claim Conditions: none A best value in the finite list does not exclude a better unlisted feasible point;script PASS does not lift this boundary. Step identifiers and dependenciesRule: finite_grid_boundary · Uses: G001, S063 ## Reverse checks - S064 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S063 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S061 · premise_and_result_audit — Recompute point3 row2 LHS with exact rational arithmetic, not rounded decimals. - S060 · premise_and_result_audit — Recompute point3 row2 b·y with exact rational arithmetic, not rounded decimals. - S059 · premise_and_result_audit — Recompute point3 row2 a·x with exact rational arithmetic, not rounded decimals. - S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S057 · premise_and_result_audit — Recompute point3 row1 LHS with exact rational arithmetic, not rounded decimals. - S056 · premise_and_result_audit — Recompute point3 row1 b·y with exact rational arithmetic, not rounded decimals. - S055 · premise_and_result_audit — Recompute point3 row1 a·x with exact rational arithmetic, not rounded decimals. - S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S053 · premise_and_result_audit — Recompute point3 row0 LHS with exact rational arithmetic, not rounded decimals. - S052 · premise_and_result_audit — Recompute point3 row0 b·y with exact rational arithmetic, not rounded decimals. - S051 · premise_and_result_audit — Recompute point3 row0 a·x with exact rational arithmetic, not rounded decimals. - S050 · premise_and_result_audit — Recompute point3 objective with exact rational arithmetic, not rounded decimals. - S049 · premise_and_result_audit — Recompute point3 objective second term with exact rational arithmetic, not rounded decimals. - S048 · premise_and_result_audit — Recompute point3 objective first term with exact rational arithmetic, not rounded decimals. - S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S046 · premise_and_result_audit — Recompute point2 row2 LHS with exact rational arithmetic, not rounded decimals. - S045 · premise_and_result_audit — Recompute point2 row2 b·y with exact rational arithmetic, not rounded decimals. - S044 · premise_and_result_audit — Recompute point2 row2 a·x with exact rational arithmetic, not rounded decimals. - S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S042 · premise_and_result_audit — Recompute point2 row1 LHS with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Recompute point2 row1 b·y with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Recompute point2 row1 a·x with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S038 · premise_and_result_audit — Recompute point2 row0 LHS with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Recompute point2 row0 b·y with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Recompute point2 row0 a·x with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute point2 objective with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute point2 objective second term with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute point2 objective first term with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S031 · premise_and_result_audit — Recompute point1 row2 LHS with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute point1 row2 b·y with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute point1 row2 a·x with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S027 · premise_and_result_audit — Recompute point1 row1 LHS with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute point1 row1 b·y with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute point1 row1 a·x with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S023 · premise_and_result_audit — Recompute point1 row0 LHS with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute point1 row0 b·y with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute point1 row0 a·x with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute point1 objective with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute point1 objective second term with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute point1 objective first term with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S016 · premise_and_result_audit — Recompute point0 row2 LHS with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute point0 row2 b·y with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute point0 row2 a·x with exact rational arithmetic, not rounded decimals. - 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 point0 row1 LHS with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute point0 row1 b·y with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute point0 row1 a·x with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Recompute point0 row0 LHS with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute point0 row0 b·y with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute point0 row0 a·x with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute point0 objective with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute point0 objective second term with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute point0 objective first term 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 ([[-1, 0, 0], [0, -1, 0], [1, 1, 6]], [2, 2], max, [[0, 0, 0, [Yes, Yes, Yes], Yes, 0], [1, 0, 2, [Yes, Yes, Yes], Yes, 4], [2, 1, 0, [Yes, Yes, Yes], Yes, 2], [3, 2, 4, [Yes, Yes, Yes], Yes, 12]], 12, [3]) 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: 701afb1109c3480dfcc25caebcd98f1470aed7573d949083b3279687bb522743 - candidate record SHA-256: 60dc9cd4065a02810473a36d347ba141c760f6cdf59f4426e0f7b790f46b13e5 - semantic review receipt SHA-256: 5dbc70ebd5f74b23384c93b63a2b1d6519e206c79f22ae16a6c2d7483cea783f - question-first witness SHA-256: cf121c8dd2fec13aa118b04b049ee39494f1d6d9ecdc13514cafb78c0fc021fa - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261021-r06-linear-programming-0100.json)