# Question 2476: A Linear Objective Does Not by Itself Establish Feasibility [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-0051) 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. R0003-02476 · q-20261013-advance-20261021-r06-linear-programming-0051 ## 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, 0], [0, 2]], objective [u,v]=[-2, 2]. 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. Give (objective,every [canonical index,x,y,u*x+v*y] row). No constraints are checked in this family. - 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, 0], [0, 2]], objective [u,v]=[-2, 2]. 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. Give (objective,every [canonical index,x,y,u*x+v*y] row). No constraints are checked in this family. ## 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, 0], [0, 2]] After deduplicated canonical indexed points=[[0, 0, 0], [1, 0, 2], [2, 1, 0], [3, 2, 0]] 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 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, S005 S007 · 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, S006 S008 · 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, S007 S009 · 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, S008 S010 · 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, S009 S011 · 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, S010 S012 · 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, S011 S013 · calculationBefore point3 objective second term := mul(2, 0) After point3 objective second term = 0 Conditions: none Multiply the second objective coefficient by the second coordinate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore point3 objective := sum(-4, 0) After point3 objective = -4 Conditions: none Sum the complete two products. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · impliesBefore [[0, 0, 0, 0], [1, 0, 2, 4], [2, 1, 0, -2], [3, 2, 0, -4]] After objective values alone do not certify feasibility Conditions: none This operation supplies no constraints;objective values alone establish neither feasibility nor optimality. Step identifiers and dependenciesRule: objective_not_constraint · Uses: G001, S014 S016 · 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, S015 ## Reverse checks - S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S014 · premise_and_result_audit — Recompute point3 objective with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute point3 objective second term with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute point3 objective first term with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute point2 objective with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute point2 objective second term with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute point2 objective first term with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute point1 objective with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute point1 objective second term with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute point1 objective first term 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 ([-2, 2], [[0, 0, 0, 0], [1, 0, 2, 4], [2, 1, 0, -2], [3, 2, 0, -4]]) 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: 097b50d233922d21117f8a2c289535ce5bb46079f972058f3a47879147a5477c - candidate record SHA-256: 8ffbf845498c998db3f0d53c8cadc9fc4745ddc171fc15add038b09102916bf0 - 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-0051.json)