# Question 25095: A Free Variable Has a Nonnegative Difference Representation, Not a Unique Split [Return to catalogue page 126](https://ocme.evemisslab.com/en/questions/page/126/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261114-r06-linear-program-standard-form-foundations/#q-20261015-advance-20261114-r06-linear-program-standard-form-0070) 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. R0049-25095 · q-20261015-advance-20261114-r06-linear-program-standard-form-0070 ## Original question and givens 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). - G001 — 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). ## 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-standard-form](https://ocme.evemisslab.com/en/mko/mko-linear-program-standard-form/) ## Step-by-step derivation S001 · impliesBefore source roles = [lp-standard-unrestricted-split] After derived = [[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]] Conditions: none List the full source field order without JavaScript.LP boundary modes all have6 fields but select different roles from the original mode,not answer shape. Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · calculationBefore LP unrestricted reconstruct value := sub(0, 20) After LP unrestricted reconstruct value = -20 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore LP unrestricted ORIGINAL column0 := mul(20, -20) After LP unrestricted ORIGINAL column0 = -400 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore LP unrestricted positive column0 := mul(20, 0) After LP unrestricted positive column0 = 0 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore LP unrestricted negative column0 := mul(-20, 20) After LP unrestricted negative column0 = -400 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore LP unrestricted split column0 := sum(0, -400) After LP unrestricted split column0 = -400 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore LP unrestricted ORIGINAL column1 := mul(-21, -20) After LP unrestricted ORIGINAL column1 = 420 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore LP unrestricted positive column1 := mul(-21, 0) After LP unrestricted positive column1 = 0 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore LP unrestricted negative column1 := mul(21, 20) After LP unrestricted negative column1 = 420 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore LP unrestricted split column1 := sum(0, 420) After LP unrestricted split column1 = 420 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore LP unrestricted ORIGINAL column2 := mul(22, -20) After LP unrestricted ORIGINAL column2 = -440 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore LP unrestricted positive column2 := mul(22, 0) After LP unrestricted positive column2 = 0 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore LP unrestricted negative column2 := mul(-22, 20) After LP unrestricted negative column2 = -440 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore LP unrestricted split column2 := sum(0, -440) After LP unrestricted split column2 = -440 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore LP unrestricted ORIGINAL objective contribution := mul(23, -20) After LP unrestricted ORIGINAL objective contribution = -460 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore LP unrestricted positive objective := mul(23, 0) After LP unrestricted positive objective = 0 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore LP unrestricted negative objective := mul(-23, 20) After LP unrestricted negative objective = -460 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore LP unrestricted split objective sum := sum(0, -460) After LP unrestricted split objective sum = -460 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · impliesBefore source roles = [[20, -21, 22], 23, -20, 0, 20] After derived = [-20, [-400, 420, -440], [-400, 420, -440], Yes, -460, -460, Yes] Conditions: none Both parts are nonnegative and v=plus−minus.Each row satisfies a*v=a*plus+(−a)*minus,and so does the objective.Projection back is valid,but adding any s>=0 to both parts gives the same v,so this is not a bijection of all split pairs. Step identifiers and dependenciesRule: LP_nonnegative_split_preserves_contributions_NOT_all_pair_bijection · Uses: G001, S018 ## Reverse checks - S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S018 · premise_and_result_audit — Recompute LP unrestricted split objective sum with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute LP unrestricted negative objective with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute LP unrestricted positive objective with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute LP unrestricted ORIGINAL objective contribution with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute LP unrestricted split column2 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute LP unrestricted negative column2 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute LP unrestricted positive column2 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute LP unrestricted ORIGINAL column2 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute LP unrestricted split column1 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute LP unrestricted negative column1 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute LP unrestricted positive column1 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute LP unrestricted ORIGINAL column1 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute LP unrestricted split column0 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute LP unrestricted negative column0 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute LP unrestricted positive column0 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute LP unrestricted ORIGINAL column0 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute LP unrestricted reconstruct value with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([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) 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: 15939e36131b5045468ec6fd9fd9c57d0aeeaadd434bbd1396c29e3a63ec47c1 - candidate record SHA-256: 301a45beb0f496c558db93a2685fd2a04a3ed393ba946d8ec5b1db9c1527eac9 - semantic review receipt SHA-256: 6212b8b0b3b273f3a514e26ebd17d7e81219c56a8f47e40047ee2b6bf01d1ba3 - question-first witness SHA-256: f32a23d5e1b66c7794c60def58741bcdfacff24f3cdfc9bc48109c8f71543cf3 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261114-r06-linear-program-standard-form-0070.json)